Central European Journal of Mathematics

CEJM 4 (2003) 418{434 Schur and Schubert polynomials as Thom polynomials|cohomology of moduli spaces L¶aszl¶o M. Feh¶er...

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CEJM 4 (2003) 418{434

Schur and Schubert polynomials as Thom polynomials|cohomology of moduli spaces L¶aszl¶o M. Feh¶er¤ , Rich¶ard Rim¶anyiy 1

R¶enyi Institute, Re¶altanoda u. 13-15, 1053, Hungary 2 Department of Mathematics, Ohio State University

Received 9 April 2003; accepted 25 June 2003 Abstract: The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new de nition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems. c Central European Science Journals. All rights reserved. ® Keywords: Thom polynomials, global singularity theory, Schur and Schubert polynomials, double Schubert polynomials, cohomology of quotients AMS: 14N10, 57R45

1

Introduction

y

¤

In this paper we approach the theory of Schur and Schubert polynomials from the the point of view of generalized Thom polynomials. The starting point is that we realize Schur and Schubert polynomials|of Chern classes|are ¯rst obstructions of certain ¯ber bundles (in the spirit of Stiefel, Whitney and Steenrod who de¯ned Chern classes as ¯rst obstructions). This realization makes Schur and Schubert polynomials a special case of the generalized Thom polynomials. Applying a general method to compute them as Thom polynomials we obtain new de¯nitions for these polynomials. Along the way we also rede¯ne the double Schubert polynomials and the Kempf-Laksov-Schur (or °agged Schur) polynomials which are also ¯rst obstructions. We show how these results are related to [email protected] [email protected]

L.M. Feh´ er, R. Rimányi / Central European Journal of Mathematics 4 (2003) 418{434

419

the structure of the cohomology ring of the Grassmannian and the °ag manifold. The authors believe that this Thom polynomial technique may turn out to be useful in the study of the cohomology ring structure of various moduli spaces. See section 6 for a discussion and refer to [FRN03] where we obtained new results for the cohomology ring of the moduli space of binary forms. We will de¯ne Schur and Schubert polynomials as unique polynomials vanishing at certain substitutions, ie. results of interpolation. This might seem somewhat implicit, but it allows us to give closed formulas for the Kempf-Laksov-Schur polynomials (Theorem 5.1) and to give a natural deduction of the recursion formula of Lascoux and SchÄutzenberger [LS82] for double Schubert polynomials (Theorem 4.2). Some of these results follow from each other via sophisticated algebraic combinatorics (see e.g. [Mac91]), but we avoided such reasonings to put the emphasis on the strength of Theorem 3.3 in building up the theory of Schur and Schubert polynomials. Also, our interpolation approach to double Schubert polynomials turned out to be very fruitful in ¯nding closed formulas for quiver Thom polynomials (the problem studied in [BF99]) in terms of double Schubert polynomials, a result we are presenting in a separate paper with A. Buch. For geometrically de¯ned ¯ber bundles|by which we mean ¯ber bundles associated with a principal G-bundle where G is a Lie group|¯rst obstructions are usually called Thom polynomials (see [FR03] on this subject). The word polynomial is justi¯ed since the ¯rst obstruction of the universal bundle is an element in H ¤ (BG) which is a polynomial ring, at least rationally. Calculating Thom polynomials has a long history. It was Ren¶e Thom who initiated their study in the case of singularities of smooth maps. The major tool for calculating them was the method of resolution (see [AVGL91] for an account of the method and results). Works of V. Vassiliev [Vas88] and M. Kazarian [Kaz95], [Kaz97] clari¯ed the connection of Thom polynomials to the underlying symmetry groups. Their works also show that the so-called degeneracy loci formulas in algebraic geometry are also Thom polynomials for group actions (see [Ful98] for many examples). In particular in [BF99] double Schubert polynomials are described as degeneracy loci formulas for certain quiver representations. In [FR02] we showed how to describe and calculate quiver type formulas as Thom polynomials for group actions. In this paper however we use a more \economic" group action. In Remark 6.12 we compare the two methods. Based on works of A. Sz}ucs ([Sz} u79]), the second author introduced a di®erent method (called the method of restriction equations) to calculate Thom polynomials for singularities of smooth maps ([Rim01]). In [FR03] we generalized the method to the case of Thom polynomials for group actions. It turned out that though the idea is quite simple the method of restriction equations is very powerful. We reproduced and improved earlier results in several directions ([FR03]). In all these applications we start from a representation of G with the property that G has only ¯nitely many orbits (at least up to a given codimension). Recently we realized that our work is closely related to the approach of M. F. Atiyah and R. Bott [AB82] continued by F. Kirwan [Kir84]. Their goal is to calculate

420

L.M. Fehér, R. Rimányi / Central European Journal of Mathematics 4 (2003) 418{434

the cohomology rings of certain moduli spaces. Motivated by these results, in the ¯nal section of this paper we make an attempt to generalize the de¯nitions and methods to the case where continuous families of orbits occur|the case of \moduli". We also propose that via Thom polynomials we can calculate the cohomology rings of some moduli spaces. This method is related to but di®erent from the method used by Kirwan in [Kir92]. The idea of the paper originated from a question of Tam¶as Hausel who drew our attention to the problem of calculating cohomology rings of moduli spaces.

2

Thom polynomials for group actions

In this section we give a short introduction to the theory of Thom polynomials for group actions. They were de¯ned by M. Kazarian in [Kaz97]. The approach given here is somewhat di®erent since we concentrate on complex representations, see [FR03] for a detailed introduction along these lines. Given a complex representation ½ : G ! GL(V ) and a principal G-bundle P ! M we can look for an obstruction for having a section of the V -bundle E = P £G V associated to this representation avoiding a certain orbit ´ (or more generally a G-invariant subset of V ). Of course the zero section avoids any orbit di®erent from the zero orbit but this is pathological: we want obstructions for a generic section. In e®ect we want to avoid the closure of ´. The obstruction we will deal with is the cohomology class represented by ´¹(s) » M for a generic section s. Thus this class is an obstruction for having a section in the complement V n ´¹. Let us call this class the Thom polynomial Tp(´)(E). In [FR03] we explain that Tp(´)(E) is equal to the ¯rst obstruction class (in the homotopy theoretic sense) of the ¯ber bundle P £G (V n ´¹). By naturality it is enough to look at the universal V -bundle BV = EG£G V (where B refers to the Borel construction), and the value of the Thom polynomial here: let Tp(´) denote Tp(´)(BV ) 2 H ¤ (BG) = HG¤ (pt). This characteristic class Tp(´) can be thought of as the G-equivariant Poincar¶e dual of ´¹ (see [FR03]). One of the early examples is the case that Stiefel, Whitney and Steenrod studied (see e.g. [Sti36]) to de¯ne Chern classes: Example 2.1. Let V = Hom(Cn ; Cn+k ) and let G = GL(n+ k) act on V by composition. We de¯ne a G-invariant subset of V : §1 (k) := fv 2 V : dimC (ker v) = 1g: ¹ 1 (k). This set can be Then Tp(§ 1 (k)) is the ¯rst obstruction of the bundle with ¯ber V n § identi¯ed with the set of n-frames in Cn+k i.e. the complex Stiefel manifold St(n; n + k). So, given a vector bundle E = P £GL(n+k) Cn+k , the cohomology class Tp(§ 1 (k)) is the ¯rst obstruction for having n linearly independent sections of E. In other words if we have n sections s1 ; : : : ; sn in generic position then Tp(§ 1 (k)) is the cohomology class represented by the subset of the base where s1 ; : : : ; sn are not linearly independent. Notice that s1 ; : : : ; sn de¯nes a section of Hom(Cn ; E) = P £GL(n+k) V ).


Theorem 2.2 (Stiefel, Whitney and Steenrod).

3

421

Tp(§ 1 (k)) = ck+1 :

Calculation of Thom polynomials via the method of restriction equations

The G-equivariant Poincar¶e dual has similar properties as the ordinary Poincar¶e dual. Since Tp(´) is the G-equivariant Poincar¶e dual of ´¹ we get: Proposition 3.1 ([FR03]). Suppose that µ and ´ are orbits of ½ and jµ¤ : HG¤ (V ) ! HG¤ (µ) is induced by the inclusion µ » V . Then ( e(º´ ) if µ = ´ jµ¤ Tp(´) = ¹ 0 if ´ 6» µ; where º´ is the normal bundle of ´ in V and e denotes the equivariant Euler class. Remark 3.2. º´ is a G-equivariant bundle so it has an equivariant Euler class in HG¤ (´). A simple calculation shows (see e.g. [AB82, x1.] or [FR03]) that HG¤ (´) ¹= HG¤ ² (pt) and e(º´ ) is equal to the Euler class of the representation of G´ on a normal space of ´. The Euler class of a representation|using the description of HG¤ ² (pt) as symmetric polynomials|is simply the product of all weights. In certain cases Tp(´) is the unique solution of these equations. In fact even fewer equations are enough to determine Tp(´): Theorem 3.3 ([FR03]). Let ½ : G ! GL(V ) be a linear representation on a complex vector space V with ¯nitely many orbits. Suppose that for every orbit ´ we have e(º´ ) 6= 0. Then the restriction equations ( e(º´ ) if µ = ´ `principal equation’ jµ¤ Tp(´) = 0 if µ 6= ´; codim µ µ codim ´ `homogeneous equations’ have a unique solution. 6 0 ( more exactly that e(º´ ) is not a zero divisor) Remark 3.4. The condition e(º´ ) = ¯rst appeared in [AB82, Prop.1.9] as a su±cient condition for equivariant perfectness. A slightly weaker version of this theorem|jµ¤ Tp(´) = 0 is required for all µ not in ´|can be found in works of Kirwan (see e.g. [Kir92, p.867]).

4

Schubert polynomials

Let the group G := (n)£ (n)|where (n) = fupper triangular n by n matricesg and (n) = flower triangular n by n matricesg|act on the vector space V := Hom(Cn ; Cn ) by (Y; X) ¢ M := X M Y ¡1 . The orbits of this action are in a one-to-one correspondence

422


with the rook-arrangements on an n by n chessboard (c.f. Bruhat-form of matrices). A rook-arrangement is a number of rooks placed on a chessboard with the property that no two are in one row or column. Associated to such a rook-arrangement we consider the n £ n matrix of 0’s and 1’s encoding the positions of rooks. We will also encode such a rook-arrangement by a permutation ¼ 2 Sm for some m ¶ n as follows (and as is in [FP89, p.9.]). Expand the matrix (to the right and down) to an m £ m matrix and add 1’s to rows which do not have any, starting from the top down, putting a 1 in the left-most column outside the n £ n matrix that does not yet have a 1 in it. Then set ¼(i) = j if in the ith row the 1 is in the j th column. For example for the rook-arrangement ¡ ¢ 0 1 the permutation ¼ 2 S is 2; 3; 1 or ¼ = 2; 3; 1; 4; 5; 6 in S . We don’t have to pick 3 6 0 0 the minimal m for which we can encode the arrangement. The choice will not a®ect our reasoning or formulas. Let M¼ denote the n £ n rook-arrangement matrix encoded by the permutation ¼ 2 Sm¸n and let §¼ be its orbit. Easy computation (matrix multiplication) shows that the tangent space to §¼ at M¼ can be obtained as follows. Let T¼ be the set of boxes in the matrix which are either directly below or right of any 1 in M¼ (including where the 1’s are). The set of the remaining boxes will be denoted by N¼ . (Observe that N¼ does not depend on n only on ¼ 2 Sm .) Then the mentioned tangent space is CTº , so for a normal slice to its orbit at M¼ we can take CNº . This also gives us the codimension of the orbit §¼ : codim §¼ = jN¼ j = jf(i; j)jthere are 0’s directly above and to the left of (i; j) in M¼ gj = = jf(i; j)j¼(i) > j; ¼(l) 6= j for all l = 1; 2; : : : ; i ¡

1gj =

= jf(i; j)j¼(i) > j; ¼ ¡1 (j) > igj = l(¼); where the length l(¼) of ¼ is the number of inversions in ¼, i.e. the number of i < j such that ¼(i) > ¼(j). According to the general theory now we need to determine the maximal compact symmetry group of M¼ together with its representation on an invariant normal slice at M¼ to §¼ . This is also an easy computation. For the sake of simplicity in our formulas, instead of considering the actual maximal compact symmetry group we will m ap a group onto the maximal compact symmetry group. Here diag(a; b; : : : ) means the diagonal matrix with a; b; : : : in the diagonal. Proposition 4.1. The homomorphism G¼ := fdiag(®1 ; ®2 ; : : : ; ®m ) : j®i j = 1g ! ©¡

diag(®1 ; ®2 ; : : : ; ®n ); diag(®¼(1) ; ®¼(2) ; : : : ; ®¼(n) )

¢ª

maps G¼ ¹= U (1)m onto the maximal compact symmetry group of M¼ . Fortunately, the normal slice CNº is invariant under the action of G¼ , so now we have the two ingredients for the restriction equations:


° the map

H ¤ (BG)

jº ¤

423

H ¤ (BG¼ )

» =

» =

Z[y1 ; : : : ; yn ; x1 ; : : : ; xn ] Z[a1 ; : : : ; am ] is given by yi 7! ai ; xi 7! a¼(i) . ° the Euler class of the representation of G¼ on CNº is Y ¡ Y ¢ e¼ = j¼¤ (xi ) ¡ j¼¤ (yj ) = (a¼(i) ¡ (i;j)2Nº

aj ):

(1)

(i;j)2Nº

Observe that this latter Euler class is never zero (since (i; j) 2 N¼ implies ¼(i) > j). So, according to Theorem 3.3, the restriction equations are uniquely solvable, giving the Thom polynomial Tp(¼) := Tp(§ ¼ ) associated to the orbit §¼ . If we consider a permutation ¼ 2 Sn in a larger permutation group Sm (where the permutation acts by identity on the extra m ¡ n elements) then the equations do not change, so Tp does not change either. That is, the map Tp is really de¯ned on S1 := [Si (here Si » Sj is meant if i < j). These polynomials were studied in geometry (see the paper of Fulton [Ful92]) under the name double Schubert polynom ials. So our approach can be regarded as a new way to compute double Schubert polynomials as solutions of (large) linear equation systems. In enumerative geometry the standard procedure is to compute double Schubert polynomials by recursion. In the rest of this section we will show that these recursion formulas occur naturally in our approach. Theorem 4.2. The recursion formulas of Lascoux and SchÄutzenberger ([LS82]) hold for Tp(¼)’s, i.e. Q (1) Tp(n; n ¡ 1; : : : ; 2; 1) = i+j ¼(i + 1) holds and ½ = ¼ ¢ si , where si is the transposition (i; i + 1), then Tp(½) = @i Tp(¼) :=

Tp(¼) ¡ si (Tp(¼)) ; xi ¡ xi+1

where si (Tp(¼)) is obtained from Tp(¼) by interchanging xi and xi+1 . Proof 4.3. Let ¼n be the permutation n; n ¡ 1; : : : ; 2; 1, and let Ei;j be the matrix which is 1 at the (i; j) position and 0 otherwise. Matrix multiplication shows that any matrix in the orbit of M¼n has zeros in the (i; j) entries if i + j < n. It means that for such (i; j)’s Ei;j does not belong to the closure of §¼n , so the homogeneous equation j¤ (Tp(¼n )) = 0 holds. This j ¤ assigns di®erent indeterminates to xk ’s and yl ’s orbit of Ei;j (k; l = 1; : : : ; n) except xi and yj are both mapped to aj . This means that Tp(¼n ) must Q be divisible by (xi ¡ yj ), and so in e®ect it must be a constant times i+j r correspond to the maximal torus of the U (d ¡ r) part. From these we can compute j¶ ¤ ° the map H ¤ (BG) H ¤ (BG¸ ) » =

Z[x1 ; : : : ; xn ; y1 ; : : : ; yd ]Sd is given by xi 7! ai ; yi 7! a¹i .

» =

Z[a1 ; : : : ; an ; an+1 ; : : : ; an+d¡r ]Sd¡ r


427

° the Euler class of the representation of G¸ on CN¶ is Y e¸ = (ai ¡ a¹j ): (i;j)2N¶

Observe that the latter is never zero, so according to Theorem 3.3 the restriction equations are uniquely solvable, giving the Thom polynomial Tp(¸) := Tp(§ ¸ ) associated to the orbit §¸ . As we mentioned earlier these polynomials were ¯rst calculated by Kempf and Laksov in [KL74] and Lascoux in [Las74]. See also [CLL02]. In the rest of this section we prove their formula for Tp(¸). Theorem 5.1. Tp(¸) = ¢¸ (c(d) ; c(d¡1) ; : : : ; c(1) ); (j)

(j)

(j)

where c(j) = c0 + c1 t + c2 t2 + : : : is the Taylor expansion of Q i 0 and jTd j + jTd¡1 j > 1, . . . . But if these all hold then the de¯nitions of the index sets Ti gives that either ¸ = ¸0 or j¸0 j > j¸j, which we wanted to prove. Remark 5.3. The ordinary Schur polynomials can be obtained from our double Schur polynomials by substituting 0 for all xi . This can be seen either from the de¯nition (only unipotent triangular matrices act in the target space) or from the concrete form of these polynomials. Interestingly enough the ordinary Schur polynomials can not be obtained directly by our method (c.f. Remark 4.5), only via the double ones.

6

Cohomology of moduli spaces

The computation of the cohomology ring of various quotients spaces is a hot area in numerous branches of geometry and topology, see e.g. [Kir84]. In this section we show how the theory of Thom polynomials for group actions can be applied to this problem; and we also show how this approach works in the case of the Grassmannians.

6.1 Obstructions as relations ¤ Suppose that X is a G-space. Then we have a map jX : HG¤ (pt) ! HG¤ (X ) induced by the G-equivariant map X ! pt. In [FR03] we called the kernel OX of this map the obstruction ideal of X since the elements of OX are the G-characteristic classes which are obstructions for having a section of an X-bundle associated to a principal G-bundle. Let us mention that obstruction ideals were ¯rst de¯ned and computed (for Hom(V; W ), S 2 V , ¤ ¤2 V representations) by P. Pragacz, see [Pra88]. If in addition the map jX is surjective, then the generators of HG¤ (pt) are also generators for the ring HG¤ (X ), and the elements of OX are the relations for HG¤ (X ). If the G-action is free, then HG¤ (X) ¹= H ¤ (X=G) i.e. we calculate the ordinary cohomology ring of the space X=G. (In this case X itself is a principal G-bundle and elements of OX are the vanishing G-characteristic classes of X.)

6.2 Using Thom polynomials If there is a geometric realization of the map X ! pt i.e. a G-equivariant embedding iX : X ! V into a contractible G-space V |in other words X is an invariant subset of a representation of G|then we can use Thom polynomial methods to generate OX : It is easy to see that Thom polynomials of orbits outside of X are in OX . However, we have to extend the de¯nition to the case where continuous families of orbits occur.

6.3 Thom polynomials for moduli De¯nition 6.1. A G-invariant strati¯cation ¥ of V is called a Vassiliev strati¯cation if for every S 2 ¥ the projection p : S ! S=G = M is a ¯ber bundle over the manifold M .

430


If M has positive dimension we call it moduli space. Suppose now that p : S ! S=G = M is a complex algebraic map and the manifold M admits a complex algebraic cell strati¯cation C = fCi g (i.e. the cells Ci are complex submanifolds and their closures are complex algebraic varieties). Since in this case the G-invariant subsets p¡1 (C¹i ) » V are also complex algebraic varieties they have Thom polynomials ([FR03]) and we can de¯ne a homomorphism: De¯nition 6.2. The Thom polynomial map Tp M : H ¤ (M ) ! HG¤+d (pt) is de¯ned by ¡ ¢ TpM ([Ci ]) := Tp p¡1 (Ci ) ;

where [Ci ] denotes the cohomology class de¯ned by the cell Ci in the cellular cohomology of the cell decomposition C. Remark 6.3. In general, M does not have a complex cell decomposition. In that case, take any cell decomposition C = fCi g which is a strati¯cation of M at the same time. Extend this strati¯cation into a strict Vassiliev strati¯cation ¥ of V (i.e. we additionally assume that for every S 2 ¥, the manifold S=G is contractible). Then we have a map from the group of cellular cochains C ¤ (C) to E10;¤+d of the Kazarian spectral sequence of ¥ (i.e. the cohomology spectral sequence of the induced ¯ltration, see [Kaz97]). If this map is a cochain map (see Example 6.5 for a counterexample) then it induces a map H ¤ (M ) ! E20;¤+d . Composing with the edge homomorphism we can de¯ne Tp M . Examples 6.4. In the following cases the Thom polynomial map can be de¯ned: (i) Let V = Hom(Cn ; Cn ) as in Section 4 but restrict the action to the subgroup (n). The orbit of a map ’ 2 V is determined by the image of the standard °ag. So a S natural strati¯cation of V is V = S¸ where S¸ = f’ 2 V : dim ’(Ci ) = ¸i g. The moduli space S¸ = (n) is a partial °ag manifold Fl¸ (Cn ). In particular the moduli space corresponding to the open stratum is the full °ag manifold Fl(Cn ). We can see that the orbits of (n) £ (n) correspond to the Schubert cells of Fl¸ (Cn ) so not only do they give a complex algebraic cell strati¯cation, but we can calculate their Thom polynomials by restricting the (n)£ (n) Thom polynomials to H ¤ (pt). In other words, we substitute yi = 0 into the (n) £ (n) Thom (n)

polynomials. This method can be generalized: Whenever we can enlarge the the symmetry group such that the moduli space becomes the union of ¯nitely many orbits of the larger group we have a chance to calculate TpM . In an ideal situation the orbit decomposition de¯nes a complex algebraic cell strati¯cation of M . (ii) Let V = Hom(Cd ; Cn ) as in Section 5 but restrict the action to the subgroup GL(d). The orbit of a map ’ 2 V is determined by the image Im(’). So a natural S strati¯cation of V is V = §i where §i = f’ 2 V : dim ker(’) = ig. The moduli space §i =GL(d) ¹= Grd¡i (Cn ). Again the Schubert cells give a complex algebraic cell strati¯cation and the Thom polynomial maps can be calculated by restricting the GL(d) £ (n) Thom polynomials calculated in Section 4.


431

(iii) Generalizing case (i), let V = V (Q) be a quiver representation of An -type (see [FR02]) and forget the action on the last vertex space Cdn . Then the moduli spaces are partial °ag manifolds as in (i) but the moduli space corresponding to the open stratum is also a partial °ag manifold. Now if we enlarge the symmetry group by letting (dn ) act on the last vertex space Cdn then we get a situation analogous to case (i). (Notice that if we forget the action on the last vertex space of a representation ¯nite quiver di®erent from An then the moduli spaces are not compact and don’t admit complex algebraic cell strati¯cations.) Example 6.5. This example shows how bad the situation can be even in a very simple ¡ ¢ case. Let G = GL(1) acting on V = C2 by ½(®) = ® ®¡ 1 . There are three exceptional orbits: the zero orbit, ´x = f(x; y) : x 6= 0; y = 0g and ý = f(x; y) : y 6= 0; x = 0g. And there are the orbits ć = f(x; y) : xy = cg for c 6= 0. So a Vassiliev strati¯cation can be 0; ´x ; ý ; S = [ć . The moduli space S=G ¹= C£ has a cell strati¯cation C0 = C n [0; 1); C1 = (0; 1). Though C1 is a cocycle in the cellular cochain complex and [C1 ] generates H 1 (M ) ¹= Z but ±1 p¡1 (C1 ) = ´x ¡ ý in the Vassiliev complex (E10;¤ ; ±1 ). So it cannot have a Thom polynomial. (It is easy to construct a transversal section s of the trivial C2 -bundle over the two-torus T 2 such that [s¡1 (C1 )] is not zero in H 1 (T 2 ). On the other hand for a trivial bundle any Thom polynomial has to be zero.) It might be tempting to ask what is the Thom polynomial of a circle around 0 in M , but it represents 0 2 H 1 (M ) so its Thom polynomial has to be zero for trivial reasons.

6.4 Surjectivity ¤ As we mentioned at the beginning of the section, surjectivity of the map jX : HG¤ (pt) ! HG¤ (X ) is an essential step in calculating HG¤ (X). Using the Thom polynomial map we can prove surjectivity for the zero codimensional moduli space in cases of Examples 6.4:

Proposition 6.6. Suppose that for an open and G-invariant X » V the G-action on X is free and M = X=G admits a complex algebraic cell strati¯cation. Then the map ¤ jX : HG¤ (pt) ! HG¤ (X) ¹= H ¤ (M ) is surjective. Proof 6.7. E = X £G V has a tautological section which is transversal since X is an open subset of V (notice that transversality will fail for higher codimensional moduli). It ¤ ¤ shows that TpM (®)(E) = jX (TpM (®)) = ®, i. e. TpM is a left inverse to jX . Remark 6.8. The proof of Proposition 6.6 also shows that in the Grassmannian Grn (Cn+k ), the Poincar¶e dual [¾¸ ] of the Schubert variety ¾¸ is equal to ¢¸ (c1 ; : : : ; cn ) where ci are the Chern classes of the universal Cn -bundle over Grn (Cn+k ), so we gave an independent proof of the classical Schubert calculus. Similarly, the single Schubert polynomials Schubert ¼ (x1 ; : : : ; xn )|when we substitute yi = 0 into the double Schubert polynomials|express the Poincar¶e duals of the Schubert varieties ¾¼ of the °ag manifold

432


Fl(Cn ) in terms of xi = c1 (Li ) where Li is the ith tautological line bundle over the °ag manifold.

6.5

Cohomology of the Grassmannian

According to Section 5, we can calculate the Thom polynomial maps for Example 6.4.(ii): TpMr (¾¸0 ) = ¢¸(r;¸0 ) jfxi =0g : As we mentioned, all Thom polynomials coming from the higher codimensional moduli are relations for H ¤ (M0 ). In fact, Thom polynomials of M1 already generate the ideal of relations O(M0 ): Theorem 6.9. For any 0 < d µ n H ¤ (M0 ) ¹= Im TpM0 = Im TpM1 : Proof 6.10. There are well known descriptions of the ideal I for which H ¤ (M0 ) = C[c1 ; : : : ; cd ]=I, see e.g. [FP89, p.27.]. One such is I = ©¸6½(n¡d)d Z¢¸ (1=c). Another is I = (¢(n¡d+i) (1=c)ji = 1; : : : ; d) (ideal generators). What we have to prove is that I is also equal to J := (¢¸ (1=c)j¸ » (n ¡ d +1)d ; ¸1 = n ¡ d +1). According to the ¯rst description of I clearly J » I. To prove the converse, either apply a Lagrange expansion corresponding to appropriate columns or observe that ¢n¡d+i;1j = ¢n¡d+i¡1 ¢1j + 1 ¡ ¢n¡d+i¡1;1j + 1 which, by induction, gives that ¢n¡d+i;1j (1=c) is in J for all i ¶ 1 and j ¶ 0. [Observe that this way we can obtain an even more economical ideal generator system for I, namely ¢(n¡d+1;1i ) (1=c), i = 0; : : : ; d ¡ 1.] Remark 6.11. In [Kir92, Prop.1.], Kirwan establishes a method for ¯nding generators ¤ of the relation ideal Ker jX of a moduli space X=G. So it would be tempting to use the Kirwan-basis method here to prove Theorem 6.9 instead of using a generator system for the Grassmannian. But the calculations turned out to be quite complicated. Remark 6.12. In [BF99] there is a di®erent description of the double Schubert polynomials. They use the quiver representation G(Q) = ni=1 GL(i) £ 1i=n GL(i) acting on n¡1 1 M M i i+1 n n © © V (Q) = Hom(C ; C ) Hom(C ; C ) Hom(Ci+1 ; Ci ): i=1

i=n¡1

Looking at the map V = Hom(C ; C ) ! V (Q) we can see that the two methods should give the same result. n

n

Remark 6.13. It would be interesting to ¯nd analogues of Theorem 6.9 for the °ag manifold (Example 6.4 (i)), for partial °ags (Example 6.4(iii)), for analogues where GL(n) is replaced by di®erent complex simple groups and for real Grassmannians and °ag manifolds.


433

Acknowledgements Supported by OTKA D29234 (¯rst author), OTKA T029759, FKFP 0226/99 (second author).

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[AVGL91] V.I. Arnold, V.A. Vassiliev, V.V. Goryunov, O.V. Lyashko: \Singularities. Local and global theory", Enc. Math. Sci. Dynamical Systems VI, SpringerVerlag, Berlin, 1991. [BF99]

A. Buch and W. Fulton: \Chern class formulas for quiver varieties", Inv. Math., Vol. 135, (1999), pp. 665{687.

[CLL02]

W. Chen, B. Li, J.D. Louck: \The °agged double schur function", J. Algebraic Combin., Vol. 15, (2002), pp. 7{26.

[FP89]

W. Fulton and P. Pragacz: Schubert Varieties and Degeneracy Loci, SpringerVerlag, Berlin, 1998.

[FR02]

L. Feh¶er and R. Rim¶anyi: \Classes of degeneraci loci for quivers|the Thom polynomial point of view", Duke J. Math., Vol. 114, (2002), pp. 193{213.

[FR03]

L. M. Feh¶er and R. Rim¶anyi: \Calculation of Thom polynomials and other cohomological obstructions for group actions", to appear in Sao Carlos Singularities 2002, Cont. Math. AMS, 2003.

[FRN03] L. Feh¶er, A. N¶emethi, R. Rim¶anyi: Coincident root loci of binary forms, http://www.math.ohio-state.edu/¹ rimanyi/cikkek, 2003. [Ful92]

W. Fulton: \Flags, Schubert polynomials, degeneracy loci, and determinantal formulas", Duke Math. J., Vol. 65, (1992), pp. 381{420.

[Ful98]

W. Fulton: Intersection Theory. Springer, Berlin, 1984, 1998.

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M. Kazarian: \Characteristic classes of Lagrange and Legendre singularities", Russian Math. Surv., Vol. 50, (1995), pp. 701{726.

[Kaz97]

¶ Kazarian: \Characteristic classes of singularity theory", In: V.I. M.E. Arnold et al., (Eds): The Arnold-Gelfand mathematical seminars: geometry and singularity theory, Birkhauser Boston, Boston MA, 1997, pp. 325{340.

[Kaz00]

M. Kazarian: \Thom polynomials for Lagrange, Legendre and critical point singularities", Isaac Newton Institute for Math. Sci., preprint, 2000.

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F. Kirwan: Cohomology of quotients in symplectic and algebraic geometry. No. 31 in Mathematical Notes, Princeton University Press, Princeton NY, 1984.

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F. Kirwan: \The cohomology rings of moduli spaces of bundles over Riemann surfaces", J. Amer. Math. Soc., Vol. 5, (1992), pp. 853{906.

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G. Kempf and D. Laksov: \The determinantal formula of Schubert calculus", Acta. Math., Vol. 132, (1974), pp. 153{162.

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[Las74]

Alain Lascoux: \Puissances ext¶erieures, d¶eterminants et cycles de Schubert", Bull. Soc. Math. France, Vol. 102, (1974), pp. 161{179.

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Lascoux and SchÄutzenberger: \Polynômes de Schubert", C. R. Acad. Sci. Paris, Vol. 294, (1982), pp. 447{450.

[Mac91]

I.G. MacDonald: Notes on Schubert polynomials, LACIM 6, 1991.

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¶ Piotr Pragacz: \Enumerative geometry of degeneracy loci", Ann. Sci. Ecole Norm. Sup. (4), Vol. 21, (1988), pp. 413{454.

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R. Rim¶anyi: \Thom polynomials, symmetries and incidences of singularities", Inv. Math., Vol. 143, (2001), pp. 499{521.

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E. Stiefel: \Richtungsfelder und Fernparallelismus in Mannigfaltigkeiten", Comm. Math. Helv., Vol. 8, (1936), pp. 3{51.

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A. Sz} ucs: \Analogue of the Thom space for mapping with singularity of type 1 § ", Math. Sb. (N. S.), Vol. 108, (1979), pp. 438{456.

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V.A. Vassiliev: Lagrange and Legendre Characteristic Classes, Gordon and Breach, New York, 1988.

CEJM 4 (2003) 435{440

On subadditive functions and Ã-additive mappings Janusz Matkowski¤ Institute of Mathematics, University of Zielona G¶ora, Podg¶ orna 50, PL-65-246 Zielona G¶ora, Poland

Received 7 May 2003; accepted 11 June 2003 Abstract: In [4], assuming among others subadditivity and submultiplicavity of a function Á : [0; 1) ! [0; 1), the authors proved a Hyers-Ulam type stability theorem for \Á-additive" mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function Á imply that Á = 0 and, consequently, every \Á-additive" mapping must be additive. ® c Central European Science Journals. All rights reserved. Keywords: subadditive function, submultiplicative function, Hyers-Ulam stability, Á-additive function MSC (2000): 39B72

1

Introduction

This note is motivated by some recent papers [1], [3] and [4] concerning the Hyers-Ulam type stability theorems for Ã-additive mapping where the subadditive and submultiplicative functions were used. Recall the de¯nition of Ã-additive mapping from [3]: Let Ã : [0; 1) ! [0; 1) be function, E1 and E2 normed spaces. A mapping F : E1 ! E2 is called Ã-additive if there is µ > 0 such that jF (x + y) ¡

F (x) ¡

F (y)j µ µ (Ã(kxk) + Ã(kyk))

for all x; y 2 E1 (cf. [3]). In [4] the following Hyers-Ulam type stability result is proved:

¤

Theorem 1.1. Suppose that Ã : [0; 1) ! [0; 1) satis¯es the following conditions (1) limt!1 Ã(t) =0 t E-mail: [email protected]

436

J. Matkowski / Central European Journal of Mathematics 4 (2003) 435{440

(2) Ã(st) µ Ã(s)Ã(t) for all s; t ¶ 0 (3) Ã(s + t) µ Ã(s) + Ã(t) for all s; t ¶ 0 (4) Ã is monotone increasing (5) Ã(t) < t for all t > 0: Then a mapping F : E1 ! E2 of normed space E1 into a normed space E2 is Ã-additive if, and only if, there exist a constant c > 0 and an additive mapping T : E1 ! E2 such that jF (x) ¡ T (x)j µ cÃ(kxk); for all x 2 E1 : The Ã-additive functions were also considered in [3] and in a recent paper [1]. In this note we prove some properties of subadditive and submultiplicative functions. By applying them we infer that every function Ã : [0; 1) ! [0; 1) satisfying the conditions 2 and 5 must be the zero function. Thus, every Ã-additive mapping with Ã satisfying only these two conditions is additive.

2

Some remarks on subadditive and multiplicative functions

We begin with the following Proposition 2.1. A function f : R ! R satis¯es the following two conditions: (i) f is subadditive, that is f (s + t) µ f (s) + f (t);

s; t 2 R;

(ii) there is a c 2 R such that for all t 2 R; f (t) µ ct; if, and only if, f (t) = ct for all t 2 R. Proof. Suppose that f : R ! R satis¯es condition (i) and (ii). Hence, for all s; t 2 R, f (t) = f ((t ¡

s) + s) µ f (t ¡

s) + f (s) µ c(t ¡

s) + f (s);

whence f (t) ¡

ct µ f (s) ¡

cs;

s; t 2 R.

Changing the roles of s and t, we get the converse inequality, and consequently, f (t) ¡

ct = f (s) ¡

cs;

s; t 2 R.

Taking s = 0 gives f (t) ¡

ct = f (0);

t 2 R.

Setting s = t = 0 in (i) we get f (0) ¶ 0 and from (ii) we have f (0) µ 0: Thus f (0) = 0. This shows that f (t) = ct; t 2 R.


437

Since the converse implication is obvious, the proof is complete. Remark 2.2. To show that assumption (i) cannot be replaced by the subadditivity of f t on (0; 1) or [0; 1), it is enough to observe that f : [0; 1) ! [0; 1), given by f (t) := t+1 , is subadditive and satis¯es the inequality f (t) µ t for all t ¶ 0: Corollary 2.3. A function Ã : (0; 1) ! (0; 1) satis¯es the following two conditions: (i) f is submultiplicative, that is Ã(st) µ Ã(s)Ã(t);

s; t > 0;

(ii) there is a c 2 R such that for all t > 0; Ã(t) µ tc ; if, and only if, Ã(t) = tc for all t > 0. Proof. Suppose that Ã : (0; 1) ! (0; 1) satis¯es conditions (i) and (ii). Then f : R ! R de¯ned by f := log ¯Ã ¯ exp satis¯es the conditions (i){(ii) of Proposition 2.1. Thus f (t) = ct for all t 2 R, and consequently, Ã(t) = tc for all t > 0: The converse implication is obvious. Assuming here c := 1 we obtain the following Corollary 2.4. If a function Ã : [0; 1) ! [0; 1) is submultiplicative, that is Ã(st) µ Ã(s)Ã(t);

s; t ¶ 0;

and Ã(t) < t;

t > 0;

then Ã(t) = 0 for all t ¶ 0: Proof. Suppose that Ã(s) = 0 for some s > 0: Then from the submultiplicativity of Ã; we have µ ¶ µ ¶ t t µ Ã (s) Ã Ã(t) = Ã s =0 s s for all t ¶ 0; which shows that Ã = 0 in [0; 1): ¯ Suppose that Ã(s) > 0 for all s > 0: Then with c = 1; the function Ã ¯(0;1) satis¯es all the assumptions of Corollary 2.3. Consequently, in the other case, we would obtain Ã(t) = t for all t > 0; which contradicts the assumption Ã(t) < t for all t > 0. We shall prove the following Proposition 2.5. If f : (0; 1) ! [0; 1) is subadditive, that is f (s + t) µ f (s) + f (t);

s; t > 0;

438


and lim f (t) = 0 t!0

then the limit

f (t) t!0 t ½ ¾ f (t) f (t) lim = sup :t>0 : t!0 t t lim

exists and

Proof. Set ¯ := sup

½

¾ f (t) :t>0 ; t

and assume ¯rst that ¯ < +1: Take " > 0 and a > 0 such that f (a) >¯¡ a

" : 2

For every t 2 (0; a) there is a unique positive integer n 2 N, n = n(t); such that a µ t < na , i.e. n+1 an µ nt < a; n 2 N. n+1 Now, by the de¯nition of ¯ and the subadditivity of f; we have ¯¶

f (t) nf (t) f (nt) f (a) ¡ ¶ ¶ = t nt nt

f (a ¡ nt

nt)

¶

f (a) ¡ a

f (a ¡ an n+1

nt)

:

Since n = n(t) ! 1 i® t ! 0; and 0 0 such that 0 0: This completes the proof.


439

Remark 2.6. Under the measurability assumptions the above result is proved in [2]. Corollary 2.7. Suppose that f : (0; 1) ! [0; 1) is subadditive, that is f (s + t) µ f (s) + f (t);

s; t > 0;

and lim f (t) = 0: t!0

Then

f (t) f (t) µ lim sup t!1 t t if and only if, there is a c 2 R such that f (t) = ct for all t > 0: lim inf t!0

(1)

Proof. The assumptions imply that f is bounded on every ¯nite subinterval of [0; 1): By the subadditivity of f (cf. Hille, Phillips [2, p.244, Theorem7.6.1] there exists a ¯nite limit f (t) ® := lim t!1 t and ½ ¾ f (t) ® = inf :t>0 : t>0 t Putting

¯ := sup

½

¾ f (t) :t>0 ; t

in view of Proposition 2.5, we have f (t) : t!0 t

¯ = lim

Of course we have ¯ ¶ ®: Condition (1) implies that ® ¶ ¯ and, consequently, ® = ¯: Setting c := ® we obtain f (t) = ct for all t > 0:

3

Remarks on Ã-additive maps

From Corollary 2.4 we obtain the following Remark 3.1. Suppose that Ã : [0; 1) ! [0; 1) satis¯es the conditions 2 and 5 of Theorem 1.1. Then every mapping of normed space E1 into a normed space E2 is Ãadditive if, and only if, it is additive. Applying Proposition 2.5 we get the following Remark 3.2. Let Ã : [0; 1) ! [0; 1) satisfy the conditions 2 and 3 of Theorem 1.1. Suppose that limt!0 Ã(t) = 0 and 0 is a cluster point of the set ft > 0 : Ã(t) < ctg

440


for some c < 1: Applying Proposition 2.5 we infer that Ã(t) < t for all t > 0; that is, condition 5 is satis¯ed. These remarks show that condition 5 of Theorem 1.1 should be either removed or replaced by a weaker one.

References [1] P. G~avruta: \On a problem of G. Isac and Th.M. Rassias concerning the stability of mappings", J. Math. Anal. Appl., Vol. 261, (2001), pp. 543{553. [2] E. Hille and R.S. Phillips: \Functional analysis and semi-groups", AMS, Colloquium Publications, Vol. 31, Providence, Rhode Island, 1957. [3] G. Isac and Th.M. Rassias: \On the Hyers-Ulam stability of Ã-additive mappings", J. Approx. Theory, Vol. 72, (1993), pp. 137{137. [4] G. Isac and Th.M. Rassias: \Functional inequalities for approximately additive mappings", In: Th.M. Rassias and J.Tabor, (Eds.): Stability of Mappings of HyersUlam type, Hadronic Press, Palm Harbour, Fl, 1994, pp. 117{125.

CEJM 4 (2003) 441{456

K-subanalytic rectilinearization and uniformization Artur Pi»ekosz¤ Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155 Krak¶ow, Poland

Received 31 March 2003; accepted 9 July 2003 Abstract: We prove rectilinearization and uniformization theorems for K-subanalytic (RK an de nable) sets and functions using the Lion-Rolin formula. Parallel reasoning gives standard results for the subanalytic case. c Central European Science Journals. All rights reserved. ® Keywords: K-subanalytic, preparation theorem, rectilinearization, uniformization MSC (2000): 32B20, 14P15

1

Introduction

¤

The aim of this paper is to give versions of the uniformization and rectilinearization theorems due to H. Hironaka [5] (see also E. Bierstone and P. Milman ([1]) and Parusi¶nski ([8])), for a larger class of sets which we call K-subanalytic. They can be described by compositions of globally subanalytic functions and power functions jxj¸ with ¸ 2 K, where K is a given sub¯eld of the ¯eld of real numbers R. This class of sets was earlier studied by J.-Cl. Tougeron ([11]) and by L. van den Dries and C. Miller (for example [3, 7]). In 1997, J.-M. Lion and J.-P. Rolin in a paper [6] gave an explicit formula for a globally subanalytic function using analytic functions and operations of division and taking roots. This theorem, called the preparation theorem for subanalytic functions is similar to the Weierstrass preparation theorem, and appeared in less explicit form in a paper of A. Parusi¶nski [9]. Lion and Rolin also give a preparation theorem for K-subanalytic functions (called by them x¸ -functions) and logarithmico-exponential functions (LE-functions). All these preparation theorems give more exact descriptions of de¯nable functions in the corresponding o-minimal structures, and do not use model E-mail: [email protected]

442

A. Pi¹ekosz / Central European Journal of Mathematics 4 (2003) 441{456

theory. We use the Lion-Rolin formula for K-subanalytic functions, and get uniformization and rectilinearization for K-subanalytic functions and sets. This gives a partial answer to a problem formulated by L. van den Dries and C. Miller in [4]. The main results for the K-subanalytic case are Theorem 4.7 and Theorem 4.8. In the subanalytic case parallel reasoning gives versions of standard uniformization and rectilinearization theorems (see Theorem 5.1 and Theorem 5.2). In particular, we get a version of the characterization of subanalytic continuous functions obtained by A. Parusi¶nski (Theorem 2.7 in [8]; see also Lemma 5.3 in E. Bierstone and P. Milman [1], which follows from uniformization). We do not use equidimensionality as Parusi¶nski does, but admit taking powers in several stages. (An interested reader could also look at Parusi¶nski’s exposition [10].) We give our setting of cylinders and cells in Chapter 2. Chapter 3 gives corollaries of the Lion-Rolin preparation theorem. In Chapter 4, we introduce K-subanalytic modi¯cations, and give lemmas and statements of the main results. Chapter 5 gives analogous theorems for the subanalytic case. This paper is a rewritten version of my earlier preprint \Rectilinearization and uniformization of k-subanalytic sets and functions".

2

Basic notions

2.1 General notation Let X be any set. Functions f; g : X ! R will be called equivalent (notation: f ¹ g) on X, when there exists a constant M > 0 such that for x 2 X M ¡1 ¢ jf (x)j µ jg(x)j µ M ¢ jf (x)j. A unit is a function equivalent to the constant function 1. For Z » Rn , we consider the regular part of Z (denoted reg Z) which is, by de¯nition, the set of these points of Z where Z is a (topological) manifold. The characteristic function of Z is de¯ned as 1 on Z, and 0 on the complement of Z. A compact box is a cartesian product of closed nonempty intervals (possibly degenerated to a point) in R. By P we denote the real projective line, which is assumed to contain R. Denote D n = (¡ 1; 1)n for n 2 N, and if 0 µ k < n identify D k with D k £ f0gn¡k .

2.2 Cylinders and cells Let X be any set and let F be a ¯nite family of real functions on X. An F -set is any subset of X of the form k \ l [ E= Eij ; (k; l 2 N) i=1 j=1

where Eij = fx 2 X jgij (x) = 0g or Eij = fx 2 Xjgij (x) > 0g or Eij = fx 2 Xjgij (x) < 0g and gij 2 F . Notice that F -sets form a Boolean algebra. A set C » X £ R is called an F -cylinder when it is of the form


443

(1) C = f(x; y) 2 B £ Rj g1 (x) < y < g2 (x)g, where B is an F -set in X and g1 ; g2 2 F [ f¡ 1gX [ f+1gX , with g1 (x) < g2 (x) for every x 2 B, or (2) C = f(x; y) 2 B £ Rj y = g(x)g, where B is an F -set in X and g 2 F . Then B is called the base of C and the functions g1 ; g2 (or the function g) are called the generating functions of the cylinder C. Now assume that for every n 2 N we have a family An of functions from Rn into R. S1 Denote A = n=0 A. An A-cell (more precisely: An -cell) in Rn+1 is an An -cylinder C whose base C 0 is an An¡1 -cell in Rn . (f0g is A¡1 -cell in R0 , where A¡1 = ;.) Every A-cell C » Rn+1 is associated with a sequence of cells C = C n+1 ; C 0 = C n ; : : : ; C 0 where C j is an Aj¡1 -cell in Rj (j = n + 1; : : : ; 0) and (1) C j+1 = f(x1 ; : : : ; xj+1) 2 C j £ Rj tj+1;1 (x1 ; : : : ; xj ) < xj+1 < tj+1;2 (x1 ; : : : ; xj )g; where tj+1;1 < tj+1;2 on C j ; or j+1 (2) C = f(x1 ; : : : ; xj+1) 2 C j £ Rj tj+1;1 (x1 ; : : : ; xj ) = xj+1 = tj+1;2 (x1 ; : : : ; xj )g; where tj+1;1 = tj+1;2 on C j ; j j with tj+1;i 2 Aj [ f¡ 1gR [ f+1gR (i = 1; 2), and tj+1;i ² ¡ 1 or tj+1;i ² +1 can hold only in the case (1). The sequence ftj;i gi=1;2 j=1;:::;n+1 is called the sequence of generating functions of the cell C . We say that C is full-dimensional when C 0 is full-dimensional and C is of the form (1). The cell f0g is full-dimensional in R0 . We say that C is an analytic cell, when each of the functions ftj;ijC j¡1 gi=1;2 j=1;:::;n+1 is a j¡1 n¡1 »R restriction of some analytic function in an open neighbourhood of C (or, equivalently, its sequence of generating functions consists of analytic functions on respective bases as submanifolds).

3 3.1

K-subanalytic reduction K-subanalytic sets and functions

We call a mapping g : Rn ¼ U ! Rm globally subanalytic if its graph is a subanalytic subset of the ambient Pn £ Pm or, equivalently, the graph of g is de¯nable in the structure Ran (see [2]). For totally de¯ned functions, we denote SAN

n

= ff : Rn ! Rj f is globally subanalyticg;

and SAN

=

[

n2N

SAN

n:

444


For every ¸ 2 R, we de¯ne a function

8 > > 1; x = 0; ¸ = 0 > > < j ¢ j¸ : R 3 x 7! 0; x = 0; ¸ 6= 0 > > > > : jxj¸ ; x 6= 0:

Notice that for ¸ ¶ 0 the function j ¢ j¸ is continuous, and if ¸ > p 2 N, then j ¢ j¸ is of class C p . If x 2 Rn and ¸ 2 Rn , then jxj¸ denotes jx1 j¸1 ¢ : : : ¢ jxn j¸n . Let K be a ¯xed sub¯eld of the real ¯eld R. We put n PS K n = ff : R ! Rj f is a composition of functions from SAN

and functions of the form jxj¸ (¸ 2 K)g: Here we allow combinations and functions of any arity to appear in our compositions. K K We will assume that PS K n » PS n+1 treating functions from PS n as independent on the last variable. Subsets of Rn which are PS K n -sets will be also called K-subanalytic. Mappings will be called K-subanalytic if their graphs are K-subanalytic sets. We adopt the following notation: K+ = f¸ 2 Kj¸ ¶ 0g and K+¤ = f¸ 2 Kj¸ > 0g.

3.2 Preparation on cylinders Let E » Rnx £ R1y be a PS K n+1 -set. A reducing system R on E is a system of data R = (µ; fÁ1 ; : : : ; Ás g; a; b; f¸i gri=1 , ¤ ~ f¸~k gqk=1), where s; q; r 2 N¤ , µ; Á1 ; : : : ; Ás ; a; b 2 PS K n , ¸ i ; ¸k 2 K + and for (x; y) 2 E the following conditions hold: (i) y ¡ µ(x) 6= 0 6= a(x); functions y ¡ µ(x); a(x); b(x) are of constant sign, (ii) if µ(x) 6² 0; then y ¹ µ(x); ~ b(x) ¸ j¸1 ; : : : ; j y¡µ(x) j¸r ; j y¡µ(x) j 1; : : : ; (iii) the mapping Ã(x; y) = (Á1 (x); : : : ; Ás (x); j y¡µ(x) a(x) a(x) ~

b(x) ¸q j y¡µ(x) j ) is bounded. The mapping Ã(x; y) is called the reducing morphism of R. A ¯nite family of functions ffº gº » PS K n+1 will be called R-reducible on E, when every function fº has on E the form (called R-reduction) (?) fº (x; y) = jy ¡ µ(x)j°¸ ¢ Aº (x) ¢ Vº (Ã(x; y)); where °º 2 K, Vº is an analytic, nonvanishing function of constant sign on some neighbourhood of the set Ã(E) » Rs+r+q , and Aº 2 PS K n with Aº (x) ² 0 or A º (x) 6= 0 on E. (Thus also fº ² 0 or fº 6= 0 on E.) The Lion-Rolin preparation theorem can be restated as follows:

Theorem 3.1 (Lion-Rolin [6]). For a ¯nite family of functions ffº gº » PS K n+1 , there exists a ¯nite family f(C® ; R® )g® , where: (1) fC® g® is a decomposition of Rn+1 into PS K n -cylinders, and R® is a reducing system on C® ,


445

(2) the family ffº gº is R® -reducible on C® . Remarks. (1) This statement is a little stronger than the statement in [6]. We have decompositions not coverings, get only one a(x) and only one b(x) by extracting new functions Ái (x), and can take subdivisions to obtain constant signs and boundedness of the reducing morphisms. By a careful proof of the theorem, we also have condition (ii). (2) We can additionally have the cylinders compatible with a given ¯nite family of PS K n+1 -sets by adding the characteristic functions of these sets to the considered family of functions. (3) By the above theorem, a projection of a K-subanalytic set is K-subanalytic. More generally: K-subanalytic sets are closed under ¯rst order de¯nability, and are exactly the class of de¯nable sets in the structure RK an considered by L. van den Dries and C. Miller in [3], [4], [7]. (4) We also get n PS K n = ff : R ! Rj f is K-subanalyticg; i.e. the class of functions on Rn de¯nable in the structure RK an . n (5) Q-subanalytic sets and functions in R are exactly globally subanalytic sets and functions.

3.3 Cells with reducers A reducer R on C , where C is a PS K -cell in Rn+1, is a sequence R = (R 1 ; : : : ; R n+1), where R j is a reducing system on C j » Rj . For a reducer R on C with rj ~ j;k gqj ); j = 1; : : : ; n + 1; R j = (µj ; fÁj;1 ; : : : ; Áj;sj g; aj ; bj ; f¸j;igi=1 ; f¸ k=1

we say that a ¯nite family of K-subanalytic functions ffº gº de¯ned on C is R-reducible on C when the following two conditions hold: (1) ffº gº is Rn+1 -reducible on C : fº (x; y) = jy ¡

µn+1(x)j°¸ ¢ Aº (x) ¢

j

y¡

µn+1(x) ¸n+ j an+1 (x)

y¡

µn+1 (x) ¸n+ 1;1 j ;:::; an+1 (x) bn+1 (x) ¸~ n+ 1;1 bn+1 (x) ¸~ n+ 1;r j j ;j ;:::;j y ¡ µn+1(x) y ¡ µn+1 (x)

¢Vº (Án+1;1 (x); : : : ; Án+1;sn+ 1 (x); j

1;q

);

(2) the family fµn+1 ; Án+1;1 ; : : : ; Án+1;sn+ 1 ; an+1; bn+1 g [ fAº gº is (R 1 ; : : : ; R n )-reducible on C n . (Constant functions on R0 are reducible.) A cell with a reducer in Rn+1 is a pair (C ; R), where C is a PS K n+1 -cell, and R is a reducer on C such that for j = 1; : : : ; n the family ftj+1;1 ; tj+1;2 g is (R1 ; : : : ; R j )-reducible on C j . We say that a ¯nite family ffº gº (de¯ned at least on C ) is reducible on (C ; R), when it is R-reducible on C .

446


The following proposition gives a kind of \partial desingularization". Proposition 3.2. For a ¯nite family ffº gº of K-subanalytic functions reducible on a cell with a reducer (C ; R) in Rn+1 , there exist: an open neighbourhood U of the cell C ~ of open sets in Rn+1 and a K-subanalytic and analytic isomorphism : U 3 z 7! z~ 2 U ~ » (0; +1)n+1 and for every º such that U ¡

fº ¯

¡1

¢ j (K) (~ z1 ; : : : ; z~n+1) = j~ z j°¸ ¢ hº (~ z );

where °º 2 K n+1, hº are zeroes or analytic and K-subanalytic units on U~ . Thus ffº j C g are restrictions of analytic functions on U . Proof. This goes by a straightforward induction on n due to the possibility of making an analytic and K-subanalytic shift z~n+1 = jzn+1 ¡ µ(~ z1 ; : : : ; z~n )j, where µ(~ z1 ; : : : ; z~n ) is already analytic. h Corollary 3.3. If (C ; R) is a cell with a reducer, then C is an analytic cell. Applying Theorem 3.1 recursively, we get the following Corollary 3.4. Let E » Rn+1 be a K-subanalytic set, and ffº gº a ¯nite family of Ksubanalytic functions de¯ned on E. Then there exists a ¯nite family of cells with reducers f(C ® ; R® )g® such that fC ® g® form a decomposition of E and the family ffº gº is reducible on every (C ® ; R® ). If the family ffº gº is already R-reducible on E with some reducing system R, then we can additionally get R = Rn+1 for every ®. ® Applying this to the characteristic functions of K-subanalytic sets, we get Corollary 3.5 (Miller [7], van den Dries and Miller [4]). The class of K-subanalytic sets admit analytic cell decomposition and (¯nite) analytic strati¯cation.

4

Rectilinearization and uniformization

4.1 Modi¯cations A function V : U ! R, where U is a K-subanalytic open set in Rn , will be called K-analytic if it is of the form V (x) = V~ (£(x)), where £(x) = (x1 ; : : : ; xn ; jx¾(1) j¸1 ; : : : ; jx¾(m) j¸m ), for some m 2 N, some sequence ¾ : f1; : : : ; mg ! f1; : : : ; ng, ¸i 2 K+¤ (i = 1; : : : ; m), and V~ an analytic and globally subanalytic function in some neighbourhood of £(U ) » Rn+m . Remark. K-analytic functions are continuous, analytic outside of fx 2 Rn j x1 ¢ : : : ¢ xn = 0g, and if all ¸i > p 2 N¤ then they are of class C p.


447

Example. The function z(x; y) = jjxj¼ + yj¼ is a composition of R-analytic functions which is not R-analytic. A mapping : Rn ¼ A ! B » Rm (m; n 2 N) will be called K-analytic if its coordinates are K-analytic functions. A K-analytic isomorphism is a bijective mapping which is K-analytic together with its inverse. K-analytic isomorphisms are K-subanalytic homeomorphisms of open sets. A K-power mapping is any composition of mappings of the form Pi : Rn 3 (x1 ; : : : ; xn ) 7! (x1 ; : : : ; jxi j° ; : : : ; xn ) 2 Rn ; where ° 2 K+¤ . A K-analytic function is called a K-normal crossing if it is a product of a monomial with exponents from K+ and a K-analytic unit. A mapping is called a K-normal crossing if all its coordinates are K-normal crossings. We will use the notion of a blowing-up in the most standard sense: any mappping of the form (j)

Bi;j : Rn 3 (x1 ; : : : ; xi ; : : : ; xj ; : : : ; xn ) 7! (x1 ; : : : ; xi ; : : : ; xj ¢ xi ; : : : ; xn ) 2 Rn ; where n ¶ 2, 1 µ i; j µ n and i 6= j. Remarks. (1) A composition of K-normal crossings is a K-normal crossing. Substitution of a K-normal crossing to a K-analytic mapping is a K-analytic mapping. (2) If a quotient of (functions) K-normal crossings restricted to the regular part of a compact box in Rn+ is bounded, then the resulting function is a restriction of a K-normal crossing. A K-subanalytic modi¯cation is a mapping © : U ! Rn , where U » Rn is open and K-subanalytic, and © is a composition of blowings-up, K-power mappings and K-analytic isomorphisms. Lemma 4.1. For every compact box K » Rn+ of dimension k µ n there is a ¯nite family of mappings m¹ : D n ! Rn and L » D k such that: S S (1) ¹ m¹ (D k ) = ¹ m¹ (L) = K, (2) each m¹ is a composition of taking squares (x1 ; : : : ; xn ) 7! (x21 ; : : : ; x2n ) and a linear isomorphism. Proof. As we can make cartesian products of m¹ -s, it su±ces to consider the case n = 1, when the proof is easy. h

4.2

Lemmas

In this chapter we choose and ¯x p 2 N. We consider pairs (E; ffº gº ) where E is a compact, K-subanalytic subset of Rn+1 , and ffº gº is a ¯nite family of K-subanalytic functions bounded on E. Such a pair will be called p-smooth if the following holds:

448


there exists a ¯nite family of pairs (¼® ; I® ), where ¼® : N® ! Rn+1 (N® open in Rn+1 ) \ N® , such that is a K-subanalytic modi¯cation of class C p , I® is a compact box in Rn+1 + S ® ¼® (I® ) = E and for all ®, º the function fº ¯ ¼® j reg I® is a restriction of a function f~º;® : N® ! R, which is a K-normal crossing of class C p , or the zero function. Remarks. a) Boxes I® do not have to be full-dimensional. S b) If each (Ei ; ffº gº ) is a p-smooth (i = 1; : : : ; k:) and E = ki=1 Ei , then (E; ffº gº ) is p-smooth. c) By Remark (2) from chapter 4.1, we only need to show that nonzero functions f~º;® are quotients of K-normal crossings. d) If fº0 is continuous on E, then, for every ®, f~º0 ;® jI® = fº0 ¯ ¼® jI® . Lemma 4.2. If n = 0, then every pair (E; ffº gº ) is p-smooth. Proof. We may assume that E = [t1 ; t2 ] (t1 ; t2 2 R) and ffº gº is reducible on (t1 ; t2 ): fº (y) = jy ¡

µj°¸ ¢ Aº ¢ Vº (jy ¡

µj);

where °º 2 K; µ; Aº 2 R; Vº is a K-analytic unit on an open interval containing Á(E) » R, where Á : R 3 y 7! jy ¡ µj 2 R. Let ¼ : R 3 y1 7! ² ¢ jy1 jl + µ 2 R; where ² = sgn ( 12 (t1 + t2 ) ¡ µ) and l is some su±ciently large positive even integer. 1 ~ ~ Put I = Á(E), where Á(y) = jÁ(y)j l . Take an open interval W ¼ I such that every Vº (j ¢ jl ) is de¯ned on W . Each fº ¯ ¼ is equal on the interior of I to a function y1 7! jy1 jl¢°¸ ¢ Aº ¢ Vº (jy1 jl ); which is zero or a K-normal crossing of class C p on W (for su±ciently large l), and ¼ is a polynomial mapping. We take the family f((¼jW ); I)g. h Lemma 4.3. If E is fat (i.e. dimensional boxes I® .

E = int E), then (E; ffº gº ) is p-smooth with full-

Proof. We use induction on n. The case n = 0 was proved in Lemma 4.2. Let n > 0 and let us assume that the Lemma holds for n ¡ 1. By Corollary 3.4 and Remark b) above, we can assume that E = C where (C ; R) is a full-dimensional cell with a reducer on which the family ffº gº is reducible. In particular, on C , we have µ(x)j°¸ Aº (x)Vº (Ã(x; y)), where y ¡ µ(x) ¸1 y ¡ µ(x) ¸r b(x) ¸~ 1 b(x) ¸~ q j ;:::;j j ;j j ;:::;j j ): Ã(x; y) = (Á1 (x); : : : ; Ás (x); j a(x) a(x) y ¡ µ(x) y ¡ µ(x)

fº (x; y) = jy ¡


449

Extracting some bounded functions as new functions Ái(x), we can assume that a(x) = maxfjt1 (x) ¡

µ(x)j; jt2 (x) ¡

µ(x)jg;

b(x) = minfjt1 (x) ¡

µ(x)j; jt2 (x) ¡

µ(x)jg;

where t1 and t2 are functions generating the cylinder C . (We can assume that the number of functions Ái (x) remains unchanged.) Using again Corollary 3.4, we can assume that on C for each º either Aº or its reciprocal is bounded, and denote this bounded function by A0º . Put b0 (x) = a(x) ¡ b(x). Denote by fg¹ g¹ the family fa(x); b(x); b0 (x); µ(x); Á1 (x); : : : ; Ás (x)g [ fA0 º (x)gº of bounded K-subanalytic functions on the basis C 0 of the cell C . We use the induction assumption for (C 0 ; fg¹ g¹ ). There exists a ¯nite family of pairs (©¯ ; J¯ ), where ©¯ : M¯ ! Rn (M¯ open in Rn ) is a K-subanalytic modi¯cation of class S C p , a J¯ is a full-dimensional box in Rn+ \ M¯ , such that ¯ ©¯ (J¯ ) = C 0 and for all ¯, ¹ the function g¹ ¯ ©¯ j reg J¯ is a restriction of a function g~¹;¯ , which is zero or a K-normal crossing of class C p on M¯ . The mappings ª¯ = ©¯ £ idR are K-subanalytic modi¯cations of class C p . Let S L¯ = f(u; y) 2 ( int J¯ ) £ Rj (t1 ¯ ©¯ )(u) < y < (t2 ¯ ©¯ )(u)g. Then ¯ ª¯ (L¯ ) = C , so it su±ces to prove the Lemma for each pair (L¯ ; ffº ¯ ª¯ gº ). Fix ¯ (we will drop this index). Let J be the compact box [³1 ; ´1 ] £ : : : £ [³n ; ń ] and J = fjj³j = 0g, K+J = f° 2 K+n j °j 6= 0 ) j 2 J g. Set ~b(u) = l1 (u)jujc1 ; a~(u) = l2 (u)jujc2 ; b~0 (u) = l3 (u)jujc3 ; where c1 ; c2 ; c3 2 K+J , functions l2 ; l3 are K-analytic positive units on M , and so is l1 if is not zero (in this case c1 = c2 ). Set C 1 = f(u; y1 ) 2 ( int J ) £ R+ j ~b(u) < y1 < a~(u)g ~ 2 M £ R; and ¼ : M £ R 3 (u; y1 ) 7! (u; ² ¢ y1 + µ(u)) where ² = sgn ( 12 (t1 (x) + t2 (x)) ¡ Then ¼(C 1 ) = L and

µ(x)) for x 2 C

0

(does not depend on x).

f^º (u; y1 ) = (fº ¯ ª ¯ ¼j C 1 )(u; y1 ) ~b(x) ~ y1 ¸1 y1 ¸r ~b(x) ¸~ 1 j ;:::;j j ;j j ;:::;j j¸q ); = jy1 j°¸ A~º (u)Vº (Á~1 (u); : : : ; Á~s (u); j a~(x) a~(x) y1 y1 where A~º =

8 > < A~0 º when A0 º = Aº > :

1 ~0¸ A

0

when A º =

1 A¸

:

450


By the induction assumption, A~º (u) = juj±¸ hº (u) with ±º 2 K J and hº = 0 or hº are K-analytic units on M . Extracting the units from 1~a ; ~b as new functions Á~i and joining hº with Vº , we get y1 jujc1 f^º (u; y1 ) = jy1 j°¸ juj±¸ V^º (u; c2 ; ); juj y1 where V^º are zeroes or K-analytic units. By boundedness of the function have c2 µ c1 . Put

~b(u) a(u) ~

on J, we

C 2 = f(u; y2 ) 2 ( int J ) £ Rj l1 (u) jujc1 ¡c2 < y2 < l2 (u)g and R1 : Rn+1 3 (u; y2 ) 7! (u; y2 ¢ jujc2 ) 2 Rn+1 : Then R1 (C 2 ) = C 1 and c1 ¡c2

juj (f^º ¯ R1 j C 2 )(u; y2 ) = jy2 j°¸ juj!¸ V^º (u; y2 ; y2

);

where !º = ±º + °º ¢ c2 . Case 1: ~b(u) 6= 0, c1 = c2 . Then y2 is a unit on C 2 and l2 ¡ Let us introduce the mappings

l1 = l3 jujc3 ¡c2 (c3 ¶ c2 ).

T : M £ R 3 (u; y3 ) 7! (u; y3 ¢ l3 (u) + l1 (u)) 2 M £ R R2 : Rn+1 3 (u; y4 ) 7! (u; y4 ¢ jujc3 ¡c2 ) 2 Rn+1 and cells C 3 = f(u; y3 ) 2 ( int J ) £ Rj 0 < y3 < jujc3 ¡c2 g C 4 = f(u; y4 ) 2 ( int J ) £ Rj 0 < y4 < 1g: Then T (C 3 ) = C 2 , R2 (C 4 ) = C 3 and 1 (f^º ¯ R1 ¯ T ¯ R2 j C 4 )(u; y4 ) = juj!¸ jy2 (u; y4 )j°¸ V^º (u; y2 (u; y4 ); (u; y4 )): y2 Each of the functions (V^º (u; y2 ; y12 ))(u; y4 ) is zero or a K-analytic unit on a common open ^ of a set C 4 . Now we get the result by substituting a K-subanalytic neighbourhood W K-power mapping with su±ciently large exponents. Case 2: ~b(u) = 0. Then V^º does not depend on the last variable. Applying the mapping T 0 : M £ R 3 (u; y3 ) 7! (u; y3 ¢ l2 (u)) 2 M £ R


451

we get ^ T 0 (C 3 ) = C 2 and (f^º ¯ R1 ¯ T 0 )(u; y3 ) = jy3 j°¸ juj!¸ jl2 (u)j°¸ V^ º (u; y3 ) ^ for (u; y3 ) 2 C 3 = ( int J ) £ (0; 1), where V^ º are zeroes or K-analityc units on a com^ of a set C 3 . Here also we get the result by mon open K-subanalytic neighbourhood W substituting a K-power mapping with su±ciently large exponents. Case 3: c1 6= c2 and ~b(u) 6= 0. We put C 3 = f(u; y3 ) 2 int J £ R : jujc1 ¡c2 < y3
i 6= ¾ 2 i0 or i = ¾ 2 i0 and j0 = ¾i0 ; > > :0

i = i00 and j0 = ¾i0 ; ¾ 2 i0 or i = ¾ 2 i0 and j0 6= ¾i0 ; i = i00 and j0 6= ¾i0 ; ¾ 2 i0 :

G. Bobiński, A. Skowroński / Central European Journal of Mathematics 4 (2003) 457{476

461

We left to the reader to verify that §0 meets the requirements described above. R ] by A[R] Let A be an algebra and let R be an A-module. We denote the algebra [ A0 K and call it the one-point extension of A by R. If X is an A-module then obviously it can be also considered as an A[R]-module. Moreover, by X we will denote the A[R]-module de¯ned on the vector space X © HomA (R; X) by multiplication 2 3 6a r 7 4 5 (x; f ) = (ax + f (r); ¸f ): 0¸ By an algebra with formal two-ray modules we will mean a system ¥ = hA; §; (Xi )i2I (§ ) ; (Ri )i2D¿

(§ )

i;

where A = A(¥) is an algebra, § = §(¥) is a combinatorial structure for two-ray modules, (¥) (¥) Xi = Xi , i 2 I (§) , and Ri = Ri , i 2 DÁ(§ ) , are A-modules. If i0 is an admissible index for §, then we de¯ne a system ¥0 = hA0 ; §0 ; (Xi0 )i2I (§

0)

; (R0i )i2D

0 ¿ (§ )

i;

which will be called an algebra with formal two-ray modules obtained from ¥ by extension by i0 (by abuse of language we will also say that ¥0 is obtained from ¥ by extension by Ri0 and call Ri0 an admissible module) in the following way: A0 = A[Ri0 ], §0 is obtained from § by extension by i0 , Xi0 = Xi , i 2 I (§) , Xi00 = ¿A0 XÁ0 (§ ) i0 , Ri0 = Ri , i 2 DÁ(§ ) n fi0 g, 0 and Ri0 0 is the middle term of the Auslander{Reiten sequence 0

0 ! Xi000 ! Ri0 00 ! XÁ0 (§

)i

0

! 0:

~ n , n ¶ 2. Fix a nonhomogeneous Let A be a hereditary algebra of Euclidean type A tube T in the Auslander{Reiten quiver ¡(mod A) of A. Let Xi , i 2 I = Z=mZ, m ¶ 2, be the set of modules lying on the mouth of T . We may assume in addition that ¿A Xi = Xi¡1 , i 2 I. Let Á : I ! I be given by Ái = i + 1, i 2 I, (DÁ = CÁ = I), ½ = Ã = ;, li = 1 ¡ dimK Xi¡1 , i 2 I. Let Ri be the middle term of the Auslander{Reiten sequence 0 ! Xi ! Ri ! Xi+1 ! 0; i 2 I. The system hA; hI; Á; ½; Ã; (li)i; (Xi ); (Ri )i is an algebra with formal two-ray modules. The algebras with formal two-ray modules of the above form will be called fundamental. An algebra with formal two-ray modules will be called admissible if it is obtained from the fundamental one by a sequence of extensions by admissible modules. Recall that a vertex x of a translation quiver ¡ is called a ray vertex if there is only one maximal sectional path in ¡ starting at x, this path is in¯nite, and if y ! x is an arrow in ¡, then ¿ ¡ y is on this path. A vertex x is called a two-ray vertex if x is

462


the starting vertex of an in¯nite sectional path and there is an arrow x ! y with y a ray vertex. It follows from Proposition 4.3 that if ¥ is an algebra with formal two-ray (¥ ) (¥) modules, i 2 I (§ ) is an admissible index, then Ri is a two-ray vertex of ¡(mod A(¥) ). This justi¯es the title of this paper. Before formulation of the main result of the paper we need one more observation. Let I be a ¯nite set and ’ : I ! I be an invertible partial map. If i 2 D’ , then either there exists u > 0 such that either ’u i 62 D’ or ’u i = i. In the latter case the set f’u i j u 2 Zg is a ¯nite set, which we will call a ’-cycle. Main Theorem. Let ¥ = hA; hI; Á; ½; Ã; (li)i2(D¿

[D» )nDÁ i; (X i )i2I ; (Ri )i2D¿

i;

be an admissible algebra with formal two-ray modules, and assume that ¥ is not fun(¥ ) damental. Let M = jD½ j, N = jDÃ j and L be the number of ¾ (§ )-cycles. Then A is domestic, M ¶ N ¶ L and the Auslander{Reiten quiver ¡(mod A) of A consists of the following components: ~ (1) a preprojective component of type A, (2) N + 1 families of coray tubes indexed by K, (3) M ¡ N components of 1st type, (4) N components of 2nd type, ~ (5) L preinjective components of type A, (6) countably many components of the form ZD1 , if N > 0, (7) countably many components of the form ZA1 1 , if N > L. Recall from [3] that a translation quiver is called of 1st type if its stable part is of the form ZA1 and its left and right stable parts are of the form ND1 and (¡ N)D1 , respectively. Similarly, we say that a translation quiver is of 2nd type, if its stable part is a disjoint sum of two quivers of the form ZA1 , its left stable part is of the form NA1 1 and its right stable part is a disjoint sum of two quivers of the form ND1 . Finally, we ~ if it admits a section, which is a quiver of Euclidean say that a component is of type A ~ k , for some k ¶ 1. type A

3

Vector space categories

Following [13] (see also [12]) by a vector space category we mean a pair K = (K ; j ¡ j), where K is a Krull{Schmidt category and j ¡ j : K ! mod K is a faithful functor. For a vector space category K we consider the subspace category U (K) of K. The objects of U (K) are triples V = (V0 ; V! ; °V ) with V0 2 K , V! 2 mod K and °V : V! ! jV0 j a K-linear map. If V = (V0 ; V! ; °V ) and W = (W0 ; W! ; °W ) are two objects of U (K) then a morphism f : V ! W in U (K) is a pair f = (f0 ; f! ), where f0 : V0 ! W0 is a morphism in K , f! : V! ! W! is a K-linear map and the condition jf0 j°V = °W f! is satis¯ed. If K is a Krull{Schmidt category and X is an object of K , then we denote by Hom(X; K ) the vector space category which consists of the category K modulo the ideal of


463

all maps annihilated by HomK (X; ¡ ) together with the functor induced by HomK (X; ¡ ). In particular, with a one-point extension A[R] of an algebra A by an A-module R we may associate the vector space category Hom(R; mod A). We refer to [13] for the connection of mod A[R] with mod A and U (Hom(R; mod A)). Let ° 0 = (·0 ; k 0 ) and ° 00 = (·00 ; k 00 ), where ·0 and ·00 are ordinals and k 0 and k 00 are integers. We write ° 0 µ ° 00 if either ·0 < ·00 or ·0 = ·00 and k 0 µ k 00 . Moreover, we write ° 0 < ° 00 if ° 0 µ ° 00 and ° 0 6= ° 00 . If ° = (·; k) for an ordinal · and an integer k, and m is an integer, then we denote by ° + m the pair (·; k + m). On the other hand, if ¸ is an ordinal, then we denote by ¸ + ° the pair (¸ + ·; k). Obviously, (¸ + °) + m = ¸ + (° + m). We de¯ne now vector space categories we will need in our proof. Let ¸ > 0 be an ordinal and let K (¸) be the Krull{Schmidt category whose indecomposable objects are S° , (0; 0) µ °, S·0 , ¸ µ ·, S·00 , ¸ µ ·, T° , (0; 0) µ ° µ (¸; ¡ 1), and U . Let X and Y be indecomposable objects of K (¸) . Then dimK HomK(¶ ) (X; Y ) µ 2 and HomK(¶ ) (X; Y ) 6= 0 if and only if one of the following conditions is satis¯ed: ° X = S° 0 and Y = S° 00 , ° 0 µ ° 00 ; ° X = S° and Y = S·0 , ° < (·; 0); ° X = S° and Y = S·00 , ° < (·; 0); ° X = S° 0 and Y = T° 00 , ° 0 µ ° 00 ; ° X = S° and Y = U , ° µ (¸; 0); ° X = S·0 and Y = S° , (·; 0) µ °; ° X = S·0 0 and X = S·0 00 , ·0 µ ·00 ; ° X = S·0 0 and X = S·0000 , ·0 < ·00 ; ° X = S·00 and Y = S° , (·; 0) µ °; ° X = S·000 and X = S·0 00 , ·0 < ·00 ; ° X = S·000 and X = S·0000 , ·0 µ ·00 ; ° X = S¸00 and Y = U ; ° X = T° and Y = S(¸;0) ; ° X = T° 0 and Y = T° 00 , ° 0 µ ° 00 ; ° X = T° and Y = U ; ° X = U and Y = U . Moreover, dimK HomK(¶ ) (X; Y ) = 2 if and only X = S° and Y = S(¸;0) , ° < (¸; 0). If ¸ > 0 is an ordinal then we denote by K¸ the vector space category Hom(S(0;0) ; K ¸ ), where K ¸ is the full subcategory of K (¸+1) formed by the objects S° , (0; 0) µ ° µ (¸; 0). We will also denote by L¸ the vector space category Hom(S(0;0) ; L ¸ ), where L ¸ is the full subcategory of K (¸+1) formed by the objects S° , (0; 0) µ ° µ (¸; 0), and T° , (0; 0) µ ° µ (¸; 0). Assume now that m; l ¶ 0 and ¸m > ¢ ¢ ¢ > ¸0 > 0 are ordinal numbers. By K¸0 ;:::;¸ m ;l we will denote the vector space category Hom(S(0;0) ; K ¸0 ;:::;¸m ;l ), where K ¸0 ;:::;¸ m ;l is the quotient of the full subcategory of K (¸0 ) formed by the objects S° , (0; 0) µ ° µ (¸m ; l), S·0 , · = ¸0 ; : : : ; ¸m , and S·00 , · = ¸0 ; : : : ; ¸m , modulo the ideal of all morphisms which factor through T(¸0 ;¡1) . Similarly, by L¸0 ;:::;¸m ;l we will denote the vector space category Hom(S(0;0) ; L ¸0 ;:::;¸m ;l ), where L ¸0 ;:::;¸m ;l is the full subcategory of K (¸0 ) formed by the

464


objects S° , (0; 0) µ ° µ (¸m ; l), S·0 , · = ¸0 ; : : : ; ¸m , S·00 , · = ¸0 ; : : : ; ¸m , T° , (0; 0) µ ° µ (¸0 ; ¡ 1), and U . Finally, by K0;l we will denote Hom(S(0;0) ; K 0;l ), where K 0;l is the full subcategory of K (1) formed by the objects S° , (0; 0) µ ° µ (0; l).

4

Proof of the main result

We start with some remarks concerning inductive description of forbidden indices. Let § = hI; Á; ½; Ã; (li)i2(D¿

[D» )nDÁ i

be the combinatorial structure for two-ray modules, i0 be an admissible index for §, and let §0 = hI 0 ; Á0 ; ½0 ; Ã 0 ; (li0 )i2(D¿ 0 [D» 0 )nDÁ 0 i; be the structure obtained from § by extension by i0 . We need to determine the connection 0 between Ã and Ã 0 and between ´ = ´ (§) and ´ 0 = ´ (§ ) . Recall that DÁ0 = (DÁ [fi00 g)nfi0 g, where i00 is the new element of I 0 . 0

Assume ¯rst that i0 62 C½ . Obviously we have Ã 0vi i = Ã vi i for i 2 I and Ã (§) (§ 0 ) where vi = vi and vi0 = vi . Note that ( ¾i i 6= i00 ; ¾0 i = ¾i0 i = i00 ;

0vi0 0 0 0i 0

= i00 ,

0

where ¾ = ¾ (§) and ¾ 0 = ¾ (§ ), and D´0 = D´ [ fi00 g. Consequently, we get 8 0 > > > :í i = i0 : 0

Then it follows that

0u0i

´ i= (§)

(

0

´ ui i

i 6= i00 ;

´ ui0 i0

i = i00 ;

(§ 0 )

where ui = ui and u0i = ui . Note that, in this case, C½0 = C½ [ fi00 g and C½0 n DÁ0 = (C½ n DÁ ). On the other hand, if i0 2 C½ then ( Ã vi i Ã vi i 6= j0 ; 0 Ã 0vi i = i00 Ã vi i = j0 or i = i00 ; where j0 = ´ ui0 i0 . Moreover

¾0 i =

8 > > > :i 0 0

i 6= i00 and ¾i 6= j0 ; i = i00 and ¾i0 6= j0 ; i = i00 and ¾i0 = j0 or ¾i = j0 ;


465

and D´0 = (D´ [ fi00 g) n fi0 g. As the consequence we obtain 8 0 > > > :i 0 i = i00 and ¾i0 = j0 or ¾i = j0 ; 0 and

8 > > ´ ui i ´ ui i 6= j0 or i = j0 ; > > > ¾i0 ´ ui i = j0 and 0 < ui < ui0 or i = i00 and ¾i0 6= j0 ; > > > > :i 0 i = i00 and ¾i0 = j0 : 0

In this case C½0 = C½ and C½0 n DÁ0 = (C½ n DÁ ) [ fi0 g. We may summarize the above considerations in the following lemma.

Lemma 4.1. Let § and §0 be as above. Then the following hold: (1) If i0 62 C½ , then i 2 I is forbidden for §0 if and only if one of the following conditions hold: (a) i is forbidden for §; (b) Ã vi i = i0 . Moreover, i00 is forbidden if and only if ´ i0 i0 62 DÁ . (2) If i0 2 C½ , then i 2 I is forbidden for §0 if and only if one of the following conditions hold: (a) i is forbidden for §; (b) i 2 C½ , ´ ui i = ´ ui0 i0 and ui ¶ ui0 ; (c) Ã vi i = ´ ui0 i0 and ¾i0 = ´ ui0 i0 . Moreover, i00 is forbidden if and only if ¾i0 = ´ ui0 i0 . Proof. It is an easy consequence of the de¯nition of forbidden index and the above formulas. We will call an index i 2 I potentially forbidden for § if ´ ui i 62 DÁ . Note that we may rephrase second part of the above lemma as follows: if i0 62 C½ , then i00 is forbidden if and only if i0 is potentially forbidden. Note also that ´ ui0 i0 is not potentially forbidden, if i0 2 C½ . We associate with each admissible algebra with formal two-ray modules ¥ = hA; §; (Xi )i2I (§ ) ; (Ri )i2D¿

(§ )

i;

(¥)

a sequence ¸i = ¸i , i 2 I (§) , of ordinal numbers. If ¥ is fundamental then we put ¸i = 1, i 2 I (§) . Let i0 be an admissible index for § and let ¥0 = hA0 ; §0 ; (Xi0 )i2I (§ 0 ) ; (Ri0 )i2C

0 ¿ (§ )

i;

466


be an algebra with formal two-ray modules obtained from ¥ by extension by i0 . Assume (¥) also that ¥ is admissible and the numbers ¸i = ¸i , i 2 I (§) , are de¯ned. We want to 0 0 (¥ ) de¯ne numbers ¸0i = ¸i , i 2 I (§ ). If i0 62 C½(§ ) , then we put ¸0i = ¸i , i 2 I (§) , and ¸0i0 = ¸i0 . If i0 2 C½(§ ) , then we put 0

¸0i =

8 > > < ¸ i0 ! ¸i > > :¸ ! i0

´ ui i = ´ ui0 i0 and 0 < ui < ui0 ; i = i0 ; ´ ui0 i0 or ´ ui i 6= ´ ui0 i0 or ´ ui i = ´ ui0 i0 and ui > ui0 ; i0 = i00 ;

(§) where ´ = ´ (§) and ui = ui , i 2 I (§) . (¥) Note that it is not clear from the de¯nition that the numbers ¸i do not depend on the choice of extension sequence de¯ning ¥. It will follow from Proposition 4.3 and we will not discuss this problem any more. We collect some properties of the above numbers in the following lemma.

Lemma 4.2. Let ¥ = hA; §; (Xi)i2I (§ ) ; (Ri )i2D¿ (§ ) i be an admissible algebra with formal two-ray modules. (1) For each i 2 I (§) , there exists an integer ni ¶ 0 such that ¸¥i = ! ni . (¥) (¥) (2) If i 2 DÁ(§ ) [ D½(§ ) , then ¸´(§ ) i = ¸i . (¥)

(3) If i 2 DÃ(§ ) , then ¸Ã(§

(¥)

)i

> ¸i .

Proof. An easy consequence of involved de¯nitions. The following proposition plays a crucial role in the proof of the main theorem: Proposition 4.3. Let ¥ = hA; hI; Á; ½; Ã; (li)i2(D¿

[D» )nDÁ i; (X i )i2I ; (Ri )i2D¿

i;

be an admissible algebra with formal two-ray modules. Then we have following components in the Auslander{Reiten quiver ¡(mod A): ~ (1) a preprojective component P of type A; (k) (2) a K-family T (k) = (T¸ )¸2K of tubes for each k = 0; 1; : : : ; jDÃ j. (¥) (§ (¥ ) ) Let ¸i = ¸i , i 2 I, and vi = vi . We can label the representatives of isomorphism (k) classes of indecomposable modules not in P, T¸ , ¸ 2 K, k = 0; 1; : : : ; jDÃ j: X°(i) ; (0; 0) µ ° µ (¸i ; 0); i 2 (CÁ [ CÃ ) n (D½ [ DÃ ); (Ái)

X(¸i ;1) ; i 2 DÁ \ C½ ; X°(i) ; (0; 0) µ ° µ (¸i ; ¡ 1); i 2 D½ [ C½ [ DÃ ; X 0(i) ; X 00(i) ; i 2 C½ ; (i)

Y° 0 ;° 00 ; (0; 0) < ° 0 < ° 00 < (¸i ; 0); i 2 D½ ; Y°0(i) ; Y°00(i) ; (0; 0) < ° < (¸i ; 0); i 2 D½ ;

G. Bobiński, A. Skowroński / Central European Journal of Mathematics 4 (2003) 457{476 (i)

Z° 0 ;° 00 ; (0; 0) < ° 0 = (·0 ; k 0 ); ·0 < !¸i; (0; 1) µ ° 00 µ (¸i ; ¡ 1); i 2 C½ n DÁ ; U°0(i) ; U°00(i) ; V°0(i) ; V°00(i) ; (0; 0) < ° 0 = (·0 ; k 0 ); ·0 < !¸i; i 2 C½ n DÁ ; such that they form the following Auslander{Reiten sequences (i) (Ái) 0 ! X°(i) ! X°+1 © X°¡1 ! X°(Ái) ! 0; (0; 0) µ ° = (·; k); · < ¸i ; i 2 DÁ ; (i) (Ái) (Ái) 0 ! X°(i) ! X°+1 © X°¡li ! X°¡li +1 ! 0; ° = (¸i ; k); k µ li ¡

0! 0! 0! 0! 0! 0! 0! 0!

1; i 2 DÁ ;

(Ái) (Ái) X 0(i) ! X(¸i ;0) ! X(¸i ;1) ! 0; i 2 C½ ; (i) (i) (i) (i) Y° 0 ;° 00 ! Y° 0 ;° 00 +1 © Y° 0 +1;° 00 ! Y° 0 +1;° 00 +1 ! 0; (0; 0) µ ° 0 < ° 00 < (¸i ; 0); i 2 D½ ; 00(i) (i) Y°0(i) ! Y°;°+1 ! Y°+1 ! 0; (0; 0) µ ° < (¸i ; 0); i 2 D½ ; 0(i) (i) Y°00(i) ! Y°;°+1 ! Y°+1 ! 0; (0; 0) < ° < (¸i ; 0); i 2 D½ ; (i) (i) (i) (i) Z° 0 ;° 00 ! Z° 0 ;° 00 +1 © Z° 0 +1;° 00 ¡1 ! Z° 0 +1;° 00 ! 0; (0; 0) µ ° 0 = (·0 ; k 0 ); ·0 < !¸i; (0; 0) < ° 00 < (¸i ; 0); i 2 C½ n DÁ ; 00(i) (i) U°0(i) ! Z°;(0;1) ! U°+1 ! 0; (0; 0) µ ° = (·; k); · < !¸i ; i 2 C½ n DÁ ; 0(i) (i) U°00(i) ! Z°;(0;1) ! U°+1 ! 0; (0; 0) < ° = (·; k); · < !¸i ; i 2 C½ n DÁ ; 00(i) (i) V°0(i) ! Z°+1;(¸i ;¡1) ! V°+1 ! 0;

(0; 0) µ ° = (·; k); · < !¸i ; i 2 C½ n DÁ ; 0(i)

(i) 0 ! V°00(i) ! Z°+1;(¸i ;¡1) ! V°+1 ! 0;

(0; 0) µ ° = (·; k); · < !¸i ; i 2 C½ n DÁ ; where (i)

X(0;¡1) = 0; i 2 CÁ ; (i)

(½i)

X(¸i ;0) = X(0;0) ; i 2 D½ ; (i)

X(¸i ;0) = X 0(i) © X 00(i) ; i 2 C½ ; (i)

X¸i +° = X°(Ãi) ; (0; 0) µ ° µ (¸Ãv i i ; 0); i 2 DÃ ; (i)

Y(0;0);° = X°(i) ; (0; 0) < ° µ (¸i ; 0); i 2 D½ ; (i)

Y°;(¸i ;0) = X°(½i) ; (0; 0) µ ° < (¸i ; 0); i 2 D½ ; (i) Y°;° = Y°0(i) © Y°00(i) ; (0; 0) < ° < (¸i ; 0); i 2 D½ ; 0(i)

(i)

Y(0;0) = X(0;0) ; i 2 D½ ; 0(i)

Y(¸i ;0) = X 0(½i) ; i 2 D½ ; 00(i)

Y(¸i ;0) = X 00(½i) ; i 2 D½ ; (i)

Z(0;0);° = X°(i) ; (0; 0) < ° < (¸i ; 0); i 2 C½ n DÁ ; Z°;(0;0) = U°0(i) © U°00(i) ; (0; 0) < ° = (·; k); · < !¸i ; i 2 C½ n DÁ ; Z°;(¸i ;0) = V°0(i) © V°00(i) ; (0; 0) µ ° = (·; k); · < !¸i; i 2 C½ n DÁ ; 0(i)

(i)

U(0;0) = X(0;0) ; i 2 C½ n DÁ ;

467

468

G. Bobiński, A. Skowroński / Central European Journal of Mathematics 4 (2003) 457{476 0(i)

V(0;0) = X 0(i) ; i 2 C½ n DÁ ; 00(i)

V(0;0) = X 00(i) ; i 2 C½ n DÁ : (i)

(i)

Finally, Xi = X(0;0) , i 2 I, and Ri = X(0;1) , i 2 DÁ . We ¯rst formulate some auxiliary lemmas which should be proved together with the above proposition. They describe vector space categories whose knowledge is necessary (§) to perform an inductive step. If i 2 I, then we will denote by wi = wi the biggest P (§) (§) integer w ¶ 0 such that ¾ w i is de¯ned and w . Let ui = ui , k=1 l¾k i = 0, where ¾ = ¾ i 2 I, and ´ = ´ (§) . Lemma 4.4. Let i 2 I be an index which is not forbidden. (1) If i 62 C½ [ DÃ [ ¾(C½ ), then the assignment S° 7! X°(i) ; (0; 0) µ ° µ (¸i ; 0); induces an isomorphism ©i : K¸i ! Hom(Xi ; mod A) of vector space categories. (2) If i 2 ¾(C½ ), then the assignment S° 7! X°(i) ; (0; 0) µ ° = (·; k); · < ¸i ; (i) S° 7! X°+1 ; ° = (¸i ; k); k µ 0;

induces an isomorphism ©i : K¸i ! Hom(Xi ; mod A) of vector space categories. (3) If i 2 DÃ , then the assignment S° 7! X°(i) ; (0; 0) µ ° < (¸i ; 0); S¸i +° 7! ©Ãi (S° ); (0; 0) µ ° µ (¸Ãi ; 0); where ªÃi : K¸Á i ! Hom(XÃi ; mod A) is the isomorphism de¯ned inductively, induces an isomorphism ©i : K¸Á i ! Hom(Xi ; mod A) of vector space categories. (4) If i 2 C½ and ¾ wi i 62 D½ , then the assignment S° 7! X°(i) ; (0; 0) µ ° µ (¸i ; ¡ 1); S¸0 i 7! X 0(i) ; S¸00i 7! X 00(i) ; k+ 1

¾ i S° 7! X(¸ ; ° = (¸0 ; k); 0 µ k µ wi ¡ 0 ;0)

1;

induces an isomorphism ©i : K¸i ;wi ¡1 ! Hom(Xi ; mod A) of vector space categories. (5) If i 2 C½ and ¾ wi i 2 D½ , then the assignment S° 7! X°(i) ; (0; 0) µ ° µ (¸i ; ¡ 1); S¸0 i 7! X 0(i) ; S¸00i 7! X 00(i) ; k+ 1

¾ i S° 7! X(¸ ; ° = (¸0 ; k); 0 µ k µ wi ¡ 0 ;0)

2;


469

S¸i +°+(wi ¡1) 7! ©½¾w i i (S° ); ° = (0; k); k ¶ 0;

S¸i +° 7! ©½!w i i (S° ); ° = (·; k) µ (¸0m ; l 0 ); · ¶ 1;

S¸0 i +¸0p 7! ©½!w i i (S¸0 0p ); p = 0; : : : ; m; S¸00i +¸0p 7! ©½!w i i (S¸000p ); p = 0; : : : ; m; induces an isomorphism ©i : K¸i ;¸i +¸00 ;:::;¸i +¸0m ;l ! Hom(Xi ; mod A) of vector space categories, where ©½!w i i : K¸00 ;:::;¸0m ;l ! Hom(X½!w i i ; mod A) is the isomorphism de¯ned inductively. Lemma 4.5. Let i be an admissible index. (1) If i 62 C½ , then the assignment S° 7! ©i (S°+1 ); ° = (·; k); · < ¸i ; S° 7! ©i (S° ); ° = (¸i ; k); k µ 0; T° 7! ©Ái (S° ); ° µ (¸i ; 0); induces an isomorphism L¸i ! Hom(Ri ; mod A) of vector space categories, where ©i and ©Ái are isomorphism de¯ned in Lemma 4.4. (2) If i 2 C½ , then the assignment S° 7! ©i (S°+1 ); (0; 0) µ ° = (·; k); · < ¸00 ; S° 7! ©i (S° ); ° = (·; k) µ (¸0m ; l); · ¶ ¸00 ; S¸0 0p 7! ©i (S¸0 0p ); p = 0; : : : ; m; S¸000p 7! ©i (S¸000p ); p = 0; : : : ; m; T° 7! ©Ái (S° ); (0; 0) µ ° = (·; k); · < ¸0 ; T° 7! ©Ái (S°¡1 ); ° = (¸0 ; k); k < 0; U 7! ©Ái (S(¸00 ;0) ); where ©Ái and ©i : K¸00 ;:::;¸ 0m ;l ! Hom(Xi ; mod A) are isomorphisms de¯ned in Lemma 4.4, induces an isomorphism L¸00 ;:::;¸ 0m ;l ! Hom(Ri ; mod A) of vector space categories. In order to perform the induction step we also need a description of vector space (i) categories for some injective modules. For each i 2 DÁ n(C½ [DÃ [¾(C½ )), let Ii = X(¸i ;0) . Note that all Ii are injective modules. Lemma 4.6. Let i 2 DÁ n (C½ [ DÃ [ ¾(C½ )), which is not potentially forbidden. (1) If ¾ wi 62 D½ , then the assignment S° 7! I¾k i ; ° = (0; k); 0 µ k µ wi ; induces an isomorphism K0;wi ! Hom(Ii ; mod A) of vector space categories. (2) If ¾ wi 2 D½ , then the assignment S° 7! I¾k i ; ° = (0; k); 0 µ k < wi ;

470


S°+wi 7! ©½¾w i i (S° ); ° = (0; k); k ¶ 0; S° 7! ©½¾w i i (X° ); ° = (¸; k); ¸ ¶ 1; S¸0 0p 7! ©½¾w i i (S¸0 0p ); p = 0; : : : ; m; S¸00p 7! ©½¾w i i (S¸00p ); p = 0; : : : ; m; induces an isomorphism K¸00 ;:::;¸ 0m ;l ! Hom(Ii ; mod A) of vector space categories, where ©½¾w i i : K¸00 ;:::;¸ 0m ;l ! Hom(X½¾w i i ; mod A) is the isomorphism de¯ned in Lemma 4.4. Proof (of Proposition 4.3 and Lemmas 4.4, 4.5 and 4.6). The proof is done by induction on the complexity of ¥. If ¥ is a fundamental algebra with formal two-ray modules then the assertions of Proposition 4.3 and Lemmas 4.4, 4.5 and 4.6 are easy consequences of the well-known structure of the module category for hereditary algebras ~ of type A. Assume now that we proved Proposition 4.3 and Lemmas 4.4, 4.5 and 4.6 for ¥. Let ¥0 = hA0 ; hI 0 ; Á0 ; ½0 ; Ã 0 ; (li0 )i2(D¿

0 [D» 0 )nDÁ 0

i; (Xi0 )i2I 0 ; (R0i )i2D¿ 0 i;

be obtained from ¥ by extension by an admissible index i0 . Since from Lemma 4.5 follows that Hom(Ri0 ; mod A) is either of the form K¸ or K¸00 ;:::;¸ 0m ;l , we can use the results of [3] and [4] to calculate its subspace category, and then the Auslander{Reiten quiver of A0 . In this way we verify Proposition 4.3 for ¥0 . In order to verify Lemma 4.4 in this case, it is enough to observe that Xi0 = X i , i 2 I, and Xi00 = I provided i00 is not forbidden, where I = Ii0 if i0 62 C½ , or I = Ij0 if i0 2 C½ , where j0 = ´ ui0 i0 . Thus we can calculate the vector space categories Hom(Xi0 ; mod A0 ) using the inductive information about Hom(Xi ; mod A) and Hom(Ij0 ; mod A) (if i0 2 C½ then j0 is not potentially forbidden). Similarly, we have Ri0 = R i , i 6= i00 , and Ri00 = (I © XÁi0 ; K; ¢), where ¢ : K ! HomA (Ri0 ; I © XÁi0 ) is given by ¢(¸) = (¸f; ¸g) with f 2 HomA (Ri0 ; I) and g 2 HomA (Ri0 ; XÁi0 ) chosen nonzero maps (it follows that the isomorphism class of the obtained object does not depend on the above choice). Now we can use the same method to prove Lemma 4.5. Finally, we have Ii = I i , i 6= i00 ; ¾i0 , Ii00 = I, and I¾i0 = (0; K; 0) if i0 2 C½ , and the assertions of Lemma 4.6 follow. Proof (of Main Theorem). Let ¥ = hA; hI; Á; ½; Ã; (li)i2(D¿

[D» )nDÁ i; (X i )i2I ; (Ri )i2D¿

i;

be an algebra with formal two-ray modules, which is not fundamental. If follows from [11, Theorem 3] that the vector space categories K¸ and K¸00 ;:::;¸0m ;l are of tame type, hence A is also tame. Since we know from [6] that for each dimension d all but a ¯nite number of modules of dimension d are homogeneous, we get from Proposition 4.3 that A is jDÃ jparametric, thus domestic. (¥) (§) Let § = §(¥) , ¾ = ¾ (§) , ´ = ´ (§) , ¸i = ¸i , i 2 I, and vi = vi , i 2 I. Note that there exists no Á-cycle. Thus, for each ¾-cycle J, there exists i 2 J such that


471

i 2 CÃ . Moreover, it easily follows by induction that there exists i 2 J such that i 2 CÃ and ¸i = !¸Ã¡ i . It immediately implies that N = jDÃ j = jCÃ j ¶ L, where L is the number of ¾-cycles. We also easily see that M = jD½ j ¶ jDÁj = N . Indeed, using the properties of the formal structure for two-ray modules listed in Section 2, we get jDÁ j + M = jDÁ j + jD½ j = jDÁ [ D½ j ¶ jCÁ [ CÃ j = jCÁ j + jCÃ j = jDÁ j + jDÃ j = jDÁ j + N . It also follows immediately from Proposition 4.3 that the in the Auslander{Reiten quiver ~ and N + 1 families of tubes indexed by of A we have a preprojective component of type A K. What remains is to show that the indecomposable modules listed in Proposition 4.3 form exactly the components presented in Main Theorem, points (3){(7). Let I = f(i; ¸) j i 2 I; 0 µ ¸ µ ¸i g. We de¯ne a partial invertible function ¿ : I ! I by the formula ( (Á¡ i; ¸); i 2 CÁ and 0 µ ¸ < ¸i or i 2 CÁ n DÃ and ¸ = ¸i ¿ (i; k) = ¡ vi (´ Ã i; ¸Ã¡ i + ¸); i 2 CÃ and 0 µ ¸ µ ¸i : (i)

(j)

Note that, if (i; ¸) 2 D¿ , ¿ (i; ¸) = (j; ·) and (0; 0) µ (¸; l) µ (¸i ; 0), then ¿A X(¸;l) = X(·;k) for some integer k. For (i; ¸) 2 I, put si;¸ = supfs ¶ 0 j ¿ s (i; ¸) is de¯nedg. If si;¸ < 1 then ¿ si;¶ (i; ¸) 2 f(j; ·) j j 2 C½ ; 0 µ · µ ¸j g [ f(j; ¸j ) j j 2 DÃ g. If si;¸ = 1, then there exists j 2 I which belongs to a ¾-cycle such that ¿ s (i; ¸) = (j; ·) for some s ¶ 0. For each i 2 D½ we have ¿ si ;¸i = (j; ¸j ), where j 2 C½ [DÃ . In this case, ¸j = ¸i . On the other hand, for each i 2 DÃ , there exists j 2 D½ such that (i; ¸i ) = ¿ sj;¶ j (j; ¸j ). More generally, if (i; ¸) 2 I, i 62 C½ n DÁ , then either there exists (j; ·) 2 I with j 2 D½ such that (k; ¸) = ¿ s (j; ·) for some s or i belongs to a ¾-cycle and ¸ = ¸i . (i) For each i 2 D½ and 0 µ ¸ µ ¸i , we will denote by X ¸ the full subquiver of ¡(mod A) (j) consisting of the vertices X(·;k) , (j; ·) = ¿ s (i; 0), 0 µ s µ si;¸ , (0; 0) µ (·; k) µ (¸i ; 0), (Áj)

and X 0(j) , X 00(j) X(¸j ;1) , if si;¸ < 1 and ¿ si;¶ (i; ¸) = (j; ¸j ), j 2 C½ . For each ¾-cycle J , we denote by X J the full subquiver of ¡(mod A) consisting of the (j) vertices X¸j ;k , j 2 J, k µ 0. (i)

For i 2 D½ and ordinals 0 µ ·0 µ ·00 µ ¸i , we will denote by Y·0 ;·00 the full subquiver (i)

of ¡(mod A) given by the vertices Y(·0 ;k0 );(·00 ;k 00 ) , (0; 0) µ (·0 ; k 0 ) < (·00 ; k 00 ) µ (¸i ; 0), and 0(i)

00(i)

Y(·0 ;k) , Y(·0 ;k) , (0; 0) < (·0 ; k) < (¸i ; 0), if ·0 = ·00 . (i)

For i 2 C½ n DÁ and ordinals 0 µ ·0 < !¸i , 0 µ ·00 µ ¸i , we denote by Z·0 ;·00 the (i)

full subquiver of ¡(mod A) with vertices Z(·0 ;k 0 );(·00 ;k 00 ) , (0; 0) µ (·0 ; k 0 ), (0; 1) µ (·00 ; k 00 ) µ 0(i)

00(i)

0(i)

00(i)

(¸i ; ¡ 1), and U(·0 ;k0 ) , U(·0 ;k0 ) , (0; 0) µ (·0 ; k 0 ), if ·00 = 0, or V(·0 ;k0 ) , V(·0 ;k 0 ) , (0; 0) µ (·0 ; k 0 ), if ·00 = ¸i . We describe now how the above quivers glue together and form the components of ¡(mod A). ~ Thus we (1) For each ¾-cycle J, the quiver X J is the preinjective component of type A. ~ have L preinjective components of type A. (2) For each i 2 D½ such that ¿ si;¶ i = (j; ¸j ) with j 2 DÃ , we have a component E (i) of 2nd type, constructed in the following way: Let (Ãj; 0) = ¿ s (p; 0) for p 2 D½ .

472


(a) Assume ½i 2 DÁ . Let (½i; 0) = ¿ s (q; 0) for q 2 D½ and s ¶ 0. (i) Assume sÃj;0 < 1. Let ¿ sÁ j;0 (j; ¸j ) = (r; ·). The component E (i) consists (½¡ r) (p) (i) (i) (q) (q) of the following subquivers: Y·;¸r , X 0 , X ¸i , Y0;¸i , X 0 , Y0;0 . (ii) Assume sj;¸j = 1. The component E (i) consists of the following subquivers: (i) (q) (q) X 0(p) , X ¸(i) , Y0;¸i , X 0 , Y0;0 . i (b) Assume ½i 62 DÁ . Let q = ½i. (i) Assume sÃj;0 < 1. Let ¿ sÁ j;0 (j; ¸j ) = (r; ·). The component E (i) consists (½¡ r) (p) (i) (i) (q) of the following subquivers: Y·;¸r , X 0 , X ¸i , Y0;¸i , Z0;0 . (ii) Assume sj;¸j = 1. The component E (i) consists of the following subquivers: (p) (i) (i) (q) X 0 , X ¸i , Y0;¸i , Z0;0 . Thus we have jDÃ j components of 2nd type. (3) For each i 2 D½ such that ¿ si;¶ i (i; ¸i ) = (j; ¸j ) with j 2 C½ , we have a component E (i) of 1st type. (a) Assume ½i 2 DÁ . Let (½i; 0) = ¿ s (q; 0) for q 2 D½ and s ¶ 0. The component ¡ (i) (i) (q) (q) E (i) consists of the following subquivers: Y¸(½j ;¸j) , X ¸i , Y0;¸i , X 0 , Y0;0 . j (b) Assume ½i 62 DÁ . Let q = ½i. The component E (i) consists of the following (½¡ j) (i) (i) (q) subquivers: Y¸j ;¸j , X ¸i , Y0;¸i , Z0;0 . Thus we have jD½ j ¡ jDÃ j components of 2nd type. (4) We have the following components of the form ZD1 . (a) For each i 2 C½ n DÁ and 0 < · < !¸i, Z·;0 and Z·;¸i are components of the form ZD1 . (½¡ i) (i) (b) For each i 2 C½ n DÁ and 0 < · < ¸i , the glueing of Y¸i ;¸i and Z0;¸i is a component of the form ZD1 . (i) (c) For each i 2 D½ and 0 < · < ¸i , the component Y·;· is a component of the form ZD1 . (5) We have the following components of the form ZA1 1. 0 (a) For each i 2 C½ n DÁ , 0 < · < !¸i and 0 < ·00 < ¸i , Z·0 ;·00 is a component of the form ZA1 1. (½¡ i) (i) (b) For each i 2 C½ n DÁ and 0 < · < ¸i , the glueing of Y·;¸i and Z0;· is a component of the form ZA1 1. (i) (c) For each i 2 D½ and 0 < ·0 < ·00 < ¸i , the quiver Y·0 ;·00 is a component of the form ZA1 1. (i) (d) For each i 2 D½ and 0 < · < ¸i , the quiver Y0;· is a part of a component of the form ZA1 1. (i) Assume si;· < 1. Let ¿ si;µ (i; ·) = (j; ¸). The corresponding component (½¡ j) (i) (i) consists of the subquivers Y¸;¸j , X · , Y0;· . (ii) Assume si;· = 1. The corresponding component consists of the subquivers (i) X ·(i) , Y0;· . It remains to show that the components of the form ZD1 and ZA1 1 appear exactly in the cases described in Main Theorem. For the components of the form ZD1 , the claim follows since one can easily show inductively that jDÃ j = jC½ n DÁ j, and moreover, ¸i = 1 for all i provided DÃ = ;.


473

Now the components of the form ZA1 1 appear if and only if one of the following conditions holds: either there exists i 2 C½ n DÁ such that ¸i > 1 or there exists i 2 D½ such that ¸i > 1. Note that, if the ¯rst condition holds, then ½¡ i 2 D½ and ¸½¡ i = ¸i > 1. Thus the components of the form ZA1 1 appear if and only if there exists i 2 D ½ such that ¸i > 1. The claim follows from the following observation. If §00 and §0 are two combinatorial structures for two-ray modules and §00 is obtained from §0 by extension by an admissible index, then N 00 ¡ L00 µ N 0 ¡ L0 , where N 0 = jDÃ(§ 0 ) j, N 00 = jDÃ(§ 00 ) j, L0 is the number 0 00 of ¾ (§ ) -cycles and L00 is the number of ¾ (§ ) -cycles. Using this observation it is easy to prove by induction that if N = L, then ¸i µ ! for each i, and moreover, if ¸i > 1 then i belongs to a ¾-cycle. Since i 2 D½ cannot belong to a ¾-cycle, so if N = L then there are no components of the form ZA1 1 . On the other hand, if N > L, then there exists i 2 CÃ such that i does not belong to a ¾-cycle. Obviously, then ¸i > ¸Ãi ¶ 1. Moreover, since i does not belong to a ¾-cycle, there exists u ¶ 0 such that ¾ u i 2 D½ . Then ¸¾u i = ¸i > 1, and hence components of the form ZA1 appear in ¡(mod A). We ¯nish now this section with one remark. Note that it follows from Proposition 4.3, that given an admissible algebra ¥ with formal two-ray modules, then §(¥) can be read out from ¡(mod A(¥) ). The same applies to modules Xi and Ri . Thus, by abuse of the language, we may call A(¥) an admissible algebra with formal two-ray modules, since it determines the remaining elements of the structure. However, we give no method to decide if a given algebra is an admissible algebra with formal two-ray modules di®erent from the inductive procedure.

5

Example

Let n ¶ 2. By ¢n we denote the quiver with vertices x0 , . . . , x2n+ 1 , y1 , . . . , y2n , and arrows ®i : xi¡1 ! xi , i = 1; : : : ; 2n+1, ¯i : xi¡1 ! yi , i = 1; : : : ; 2n , and °i : yi ! x2i¡1 , i = 1; : : : ; 2n . For example, ¢2 is the following quiver y4

°4

x8

®8

x7

®7

x6

®6

x5

°3 ¯4 ®5

x4

®4

y3

°2

y2 ¯3

x3

®3

x2

¯2 ®2

°1

x1

®1

y1 ¯1

x0

Let An = K¢n =In , where In is the ideal in the path algebra K¢n generated by relations ®2i+1 ®2i °i , i = 1; : : : ; 2n ¡ 1, ¯i+1®i , i = 1; : : : ; 2n ¡ 1, and ®2i °i ¯i ¡ ®2i ®2i¡1 ¢ ¢ ¢ ®i , i = 1; : : : ; 2n . In this section we investigate some of the properties of the algebras An . Recall that an algebra A is called almost sincere if there exists an indecomposable A-module such that its support contains all except one vertices of the Gabriel quiver of A. We will use the language of string modules in the proof of the proposition below. We refer the reader to [5] for more information about string modules. If ! is a string, then we denote the corresponding indecomposable string representation of ¢n by M (!).

474


Proposition 5.1. The following statements hold: (1) The algebra An is simply connected but not strongly simply connected. (2) gl: dim An = 2n. (3) An is almost sincere. Proof. (1) Since ®2i °i ¯i and ®2i ¢ ¢ ¢ ®i are homotopic in the bound quiver (¢n ; In ), °i ¯i and ®2i¡1 ¢ ¢ ¢ ®i are homotopic in (¢n ; In ). Hence the cycle ¯i¡1 ®¡1 i ®2i¡1 ¢ ¢ ¢ ®i is homotopically trivial, and An is simply connected (see also [1]). (2) Recall that gl: dim An is the maximum of the projective dimensions of simple An modules. For a vertex x of ¢n we will denote by Sx the corresponding simple An -module. For each pair (i; j) such that 0 µ i µ j µ 2i and j µ 2n+1 we de¯ne an An -module Xi;j in the following way: ( S xi i = j; Xi;j = M (®i ; : : : ; ®j¡1 ) i < j: Note that above representations of ¢n indeed de¯ne An -modules. We show now that, if j < 2n+1 , then ( pdA Xj+1;2i+1 + 1 i < 2n ; pdA Xi;j = 1 i ¶ 2n : Let pi;j : Pi;j ! Xi;j be a projective cover of Xi;j and let Yi;j = Ker pi;j . If i ¶ 2n , then Yi;j = Xj+1;2n+ 1 is a projective nonzero An -module (recall that j < 2n+1 ), hence pdA Xi;j = 1. Assume now i < 2n . Let Qi be a projective cover of Syi+ 1 . It follows that Qi is a submodule of Yi;j such that Yi;j =Qi ’ Xj+1;2i+1. Since in this case Yi;j is not projective, it follows that pdAn Xi;j = pdAn Yi;j + 1 = pdAn Xj+1;2i+1 + 1. (i) For each i ¶ 0 we de¯ne a sequence (ak ) of positive integers by the following conditions: (i)

(i)

(i)

(i)

ak+2 = 2ak + 2; a0 = i; a1 = i + 1: It can be easily veri¯ed that we have the following formulas: (i)

a2k = (2 + i)2k ¡ (i) a2k+1 (i)

= (3 + i)2k ¡

2; 2:

In particular, the sequences (ak ) are increasing. It follows from the formulas de¯ned above that, for i = 0; : : : ; 2n+1 ¡ 1, we have pdAn Sxi = k (i) + 1, where k (i) is the minimal (i) (0) (0) index k such that ak ¶ 2n . Since a2n¡2 = 2n ¡ 2 < 2n and a2n¡1 = 3 ¢ 2n¡1 ¡ 2 ¶ 2n (i) (recall that n ¶ 2), then pdAn Sx0 = 2n. On the other hand, for i > 0 we have a2n¡2 = 2n + i2n¡1 ¡ 2 ¶ 2n , thus pdAn Sxi µ 2n ¡ 1, i = 1; : : : ; 2n+1 ¡ 1. Obviously pdAn Sx2n = 0 µ 2n ¡ 1. In particular pd An Xi;j µ 2n ¡ 1 for i > 0. Finally observe that using arguments similar to the ones presented above we get pdAn Syi = pdAn X2i¡1;2i + 1 µ 2n, i = 1; : : : ; 2n . This ¯nishes the proof.


475

(3) Let A0n be the full subcategory of An consisting of all objects of An except x2n+ 1 . Let ! = (!1 ; : : : ; !n ), where !i = (¯i ; °i ; ®2i¡1 ; : : : ; ®i+1). Since both ®i+1 and ¯i+1 start at i, it follows that ! is a string. One can easily see that M (!) is a sincere A0n -module. Now we determine the representation type and the structure of the Auslander{Reiten quiver of An . Proposition 5.2. The algebra An is domestic and the Auslander{Reiten quiver of An consists of the following components: ~ 2n +2, (1) a preprojective component of type D n (2) 2 + 1 families of coray tubes indexed by K, (3) 2n components of 2nd type, ~ 2, (4) a preinjective component of type A (5) countably many components of the form ZD1 , (6) countably many components of the form ZA1 1. Proof. We ¯rst observe that A0n (where A0n was de¯ned in the proof of the previous proposition) is an admissible algebra with formal two-ray modules. Indeed, for k = 0; : : : ; 2n , let Bk be the full subcategory of A0n formed by the objects xk , . . . , x2n+ 1 ¡1 , yk+1, . . . , y2n , and let Ck be the full subcategory of A0n formed by the objects xk , . . . , x2n+ 1 ¡1 , yk , . . . , y2n . Then Ck is a one-point extension of Bk , k = 1; : : : ; 2n , and Bk is a one-point extension of Ck+1 , k = 0; : : : ; 2n ¡ 1. Note that B0 = A0n . On the other ~ 2n +1. Thus, B2n is a fundamental algebra with hand, B2n is a hereditary algebra of type A formal two-ray modules. One can verify that each extension listed above is an extension by an admissible two-ray module and we can ¯gure out the structure associated with A0n . In particular, Main Theorem implies that A0n is a domestic algebra. Moreover, the description of the Auslander{Reiten quiver ¡(mod A0n ) follows also from Main Theorem. We have that ¡(mod A0n ) consists of the following components: ~ 2n +2, (1) a preprojective component P of type A (2) 2n + 1 families of coray tubes indexed by K, (3) 2n components of 2nd type, ~ 2, (4) a preinjective component of type A (5) countably many components of the form ZD1 , (6) countably many components of the form ZA1 1. 0 Note that An is a coextension [X]An (see [12, 13]) of A0n by an indecomposable B2n -module X which belongs to a tube T , consisting of B2n -modules. Since HomAn (M; X ) 6= 0 implies X 2 P _T and P consists of B2n -modules, it is enough to consider the coextension [X]B2n ~ 2n +2, hence the corresponding vector of B2n by X. But [X]B2n is a tilted algebra of type D space category is of domestic type (see [11, Theorem 3]), and thus An is domestic. We also immediately get the required description of ¡(mod An ) using the fact that [X]B2n ~ 2n +2 , and hence admits a preprojective component of type is a tilted algebra of type D ~ 2n +2. D

476


Acknowledgments Supported by the Foundation for Polish Science and the Polish Research Grant KBN 5 PO3A 008 21.

References [1] I. Assem and A. Skowro¶nski: \On some classes of simply connected algebras", Proc. London Math. Soc., Vol. 56, (1988), pp. 417{450. [2] M. Auslander, I. Reiten, S. Smal¿: Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, 1995. [3] G. Bobi¶ nski, P. DrÄaxler, A. Skowro¶ nski: \Domestic algebras with many nonperiodic Auslander{Reiten components", Comm. Algebra, Vol. 31, (2003), pp. 1881{1926. [4] G. Bobi¶ nski and A. Skowro¶nski: \On a family of vector space categories", Cent. Eur. J. Math., Vol. 3, (2003), pp. 332{359. preprint, Toru¶n, 2002. [5] M.C.R. Butler and C.M. Ringel: \Auslander-Reiten sequences with few middle terms and applications to string algebras", Comm. Algebra, Vol. 15, (1987), pp. 145{179. [6] W. Crawley-Boevey: \Tame algebras and generic modules", Proc. London Math. Soc., Vol. 63, (1991), pp. 241{265. [7] Yu.A. Drozd: \Tame and wild matrix problems", In: Representation Theory II, Lecture Notes in Math. 832, Springer{Verlag, Berlin{New York, 1980, pp. 242{258. [8] P. Malicki, A. Skowro¶nski, B. Tom¶e: \Indecomposable modules in coils", Colloq. Math., Vol. 93, (2002), pp. 67{130. [9] J.A. de la Pe~ na: \Tame algebras with sincere directing modules", J. Algebra, Vol. 161, (1993), pp. 171{185. [10] J.A. de la Pe~ na and A. Skowro¶nski: \Geometric and homological characterizations of polynomial growth strongly simply connected algebras", Invent. Math., Vol. 126, (1996), pp. 287{296. [11] C. M. Ringel: \Tame algebras", In: Representation Theory I, Lecture Notes in Math. 831, Springer{Verlag, Berlin{New York, 1980, pp. 134{287. [12] C.M. Ringel: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer{Verlag, Berlin{New York, 1984. [13] D. Simson: Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. 4, Gordon and Breach, Montreux, 1992. [14] A. Skowro¶nski: Algebras of polynomial growth, in: Topics in Algebra, Part 1, Banach Center Publications 26, PWN, Warsaw, 1990, pp. 535{568. [15] A. Skowro¶nski: \Simply connected algebras of polynomial growth", Compositio Math., Vol. 109, (1997), pp. 99{133. [16] A. Skowro¶nski: \Tame algebras with strongly simply connected Galois coverings", Colloq. Math., Vol. 72, (1997), pp. 335{351. [17] A. Skowro¶nski and G. Zwara: \On the numbers of discrete indecomposable modules over tame algebras", Colloq. Math., Vol. 73, (1997), pp. 93{114.

CEJM 4 (2003) 477{509

Quantum scattering near the lowest Landau threshold for a Schrodinger operator with a constant magnetic ¯eld Michael Melgaard

¤

Department of Mathematics Chalmers University of Technology Eklandagatan 86, S-412 96 Gothenburg, Sweden

Received 10 June 2003; accepted 14 July 2003 Abstract: For xed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrodinger operator with a constant magnetic eld and an axisymmetrical electric potential V . In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian Hm = Hom + V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (Hm ; Hom ) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold. ® c Central European Science Journals. All rights reserved. Keywords: Near-threshold resolvent expansions, scattering matrix, auxiliary one-dimensional Schrodinger operator MSC (2000): 47N20; 35J10 35P25 47F05 81U05

1

Introduction

Spectral and scattering theory for the three-dimensional SchrÄodinger operator with a constant magnetic ¯eld H(A) = H0 (A) + V (x) = (¡ ir ¡

A)2 + V (x); A = (1=2)B £ x;

(1.1)

¤

has received substantial attention due to applications in astrophysics and solid-state physics (see the survey [25] and references therein) as well as mathematical interest. The basic mathematical aspects of the scattering theory for the pair (H (A); H0 (A)) have been E-mail: [email protected]

478

M. Melgaard / Central European Journal of Mathematics 4 (2003) 477{509

studied in [5] where the existence and completeness of the corresponding wave operators were proven for a large class of potentials V (see also [24] for a more recent extension of these results). This work concerns problems arising in the context of near-threshold scattering for the pair (H (A); H0 (A)), when the energy parameter approaches the lowest Landau threshold. A lot of work has been done in this ¯eld for SchrÄodinger operators without external ¯elds. Classic results going back to the late forties and early ¯fties treat the radially symmetric case. In the late seventies, Newton [30] was the ¯rst to give detailed results on various threshold properties of three-dimensional SchrÄodinger operators with local (noncentral) potentials. His work was followed by the monumental work by Jensen and Kato [16]. Based on a detailed analysis of the zero-energy properties of the threedimensional SchrÄodinger operator ¡ ¢ + V (x) with V satisfying an abstract short-range condition, Jensen and Kato deduce asymptotic expansions of the full resolvent as the spectral parameter tends to zero (the so-called low-energy limit). As an application they derive expansions of the scattering matrix as the energy parameter goes to zero. The closely related problem of coupling constant thresholds was studied by Klaus and Simon [21]. Threshold scattering in the three-dimensional case was then reconsidered in a very systematic way by Albeverio, Gesztesy and various co-workers [3, 2, 4], and in the two-dimensional case by Cheney [12] with a complete treatment provided later by Boll¶e, Gesztesy and Danneels [8]. The case of nonlocal interactions in three dimensions was ¯rst considered by Newton [31] and later completely resolved by Boll¶e, Gesztesy, Nessmann and Streit [10]. An excellent survey of threshold properties of SchrÄodinger operators in dimensions one, two and three can be found in [7]. Despite its obvious importance much less is known on such problems for the operator H (A) in (1.1), which is probably explained by the additional complications that arise (see below). We restrict ourselves to the case, where the electric potential is axisymmetric, i.e. V (x) = V (½; z), ½ = (x2 + y 2 )1=2 , and decays like V (x) = O(jxj¡® ) as jxj ! 1 for some ® > 2. Furthermore, we assume that the magnetic ¯eld has constant strength 2 and is aligned in the z direction. For ¯xed magnetic quantum number m the resulting Hamiltonian Hm = Hom + V on the Hilbert space H = L2 (R+ £ R; ½ d½ dz) has the structure of an in¯nite-channel operator-valued matrix. With respect to the projection P0 onto the lowest Landau level we can represent Hm in a two-channel framework 0 1 0 1 BH0 0 C B 0 V01 C Hm = @ A+@ A 0 H1 V10 0 | {z }

(1.2)

0 Hom

on H = H0 © H1 , where H0 = Ran P0 and H1 denotes its complement. By construction, H0 and H1 are self-adjoint operators in H0 and H1 , respectively. Moreover V01¤ = V10 and V01 2 B(H1 ; H0 ). 0 Due to the diagonal structure of the uncoupled Hamiltonian Hm its spectrum is the


479

union of the spectra of H0 and H1 . We have ¾ac (H0 ) = [E0 ; 1) and ¾ac (H1 ) = [E1 ; 1), where En = 2(jmj ¡ m + 1 + 2n), n = 0; 1; 2; : : :, are the Landau levels. There are several possible, mostly singular, cases to treat, e.g. the one where we assume that E0 is an 0 isolated eigenvalue of H1 . Thus Hom has an eigenvalue embedded at E0 ; the bottom of its continuous spectrum. Our aim is twofold. First we wish to derive asymptotic expansions of the resolvent R(³) = (Hm ¡ ³)¡1 as the spectral parameter tends to the lowest Landau threshold E0 . This is done in various situations, e.g. the afore-mentioned. Second we establish scattering theory for the pair (Hm ; Hom ) and, as an application of the expansions of the resolvent, we deduce asymptotic expansions of the scattering matrix as the energy parameter tends to the lowest Landau threshold. To obtain results on the resolvent of Hm we use a two-stage approach: First we develop an abstract theory for Hamiltonians, which can be represented in a two-channel framework. In various, mostly singular situations we derive asymptotic expansions for the resolvent of the abstract two-channel Hamiltonian as the spectral parameter tends to a threshold. Second we ¯t the in¯nite-channel Hamiltonian Hm into this two-channel framework and verify that the assumptions in the abstract theory are ful¯lled, which allow us to apply the abstract theory to the Hamiltonian Hm . We emphasize that it requires considerable preparation to apply the abstract theory to the Hamiltonian Hm and, consequently, most of this paper is devoted to the latter task. In order to apply the abstract theory, one of the main tools are asymptotic expansions for the resolvent R0 (³ ) of H0 . Depending on the E0 -energy properties of H0 , the expansions take di®erent forms. The operator H0 is essentially a one-dimensional SchrÄodinger operator denoted by A = ¡ d2 =dz 2 + W (z) with a local, short-range potential decaying like W (z) = O(jzj¡® ) as jzj ! 1 for some ® > 2. For such operators it is well-known that zero cannot be an eigenvalue. Hence, there are only two possible zero-energy properties of A: Case 0) Zero is a regular point of A, i.e. zero is not an eigenvalue nor a zero resonance of A. Case 1) Zero is an exceptional point (of the 1st kind) for A, i.e. zero is not an eigenvalue but zero is a resonance. In the latter case, the equation AÃ = 0 has a unique (up to multiplicative constants) solution Ã in L1 (R), but Ã 62 L2 (R). Often one refers to Ã as a half-bound state. Let j = 0; 1 distinguish between the cases 0-1. From results on one-dimensional SchrÄodinger operators [26, 27], we obtain asymptotic expansions for the resolvent of H0 on the form R0 (³) = ¡ i(³ ¡

(j)

(j)

E0 )¡1=2 G¡1 + G0 + i(³ ¡

(j)

E0 )1=2 G1 + ¢ ¢ ¢

(1.3)

in the norm of B(Hs0 ; H¡s 0 ) as ³ ! E0 , ³ 2 Cn[E0 ; 1), provided the potential V decays su±ciently rapidly at in¯nity. Here H0s denotes the weighted space associated to H0 (see (6.4)) and s is chosen appropiately. Under suitable assumptions on the decay of V at in¯nity and using the asymptotic expansions of the component Hamiltonians (as in (1.3)) we show that the abstract results can be carried over to the Hamiltonian Hm ; in fact under weaker conditions (see Remark 8.5). Let us state one of the main results (see Theorem 8.8). Assume that ® > 13,

480


13=2 < s < ®¡ 13=2, E0 is a regular point of H0 and E0 is an isolated eigenvalue of H1 with (1) ¯nite multiplicity. We denote by PE0 the eigenprojection onto the eigenspace associated (1) (0) (1) with the eigenvalue E0 of H1 . Assume, in addition, that the operator PE0 V10 G0 V01 PE0 (1) is strictly positive and invertible in B(PE0 H1 ). This is a kind of e®ective interaction assumption. Then, generically, we have in the norm of B(Hs0 © H1s ; H0¡s © H¡s 1 ) the asymptotic expansion R(³) = R0 + i(³ ¡

E0 )1=2 R1 + O(j³ ¡

E0 j)

(1.4)

as j³ ¡ E0 j ! 0, where the coe±cients R0 and R1 in (1.4) are given explicitly. Despite the singular nature of the problem, (1.4) reveals that, generically, the singularities cancel. In particular, the resolvent has a well-de¯ned limit R0 at the threshold point in the norm ¡s topology of B(H0s © H1s ; H¡s 0 © H1 ). The paper is organized as follows. In Section 2 we ¯x the notation. In Section 3 we give the abstract theory for Hamiltonians, which can be represented in a two-channel framework. In Sections 4 and 5 the Hamiltonians Hom and Hm are introduced. We ¯t the in¯nite-channel Hamiltonians Hom and Hm into a two-channel framework in Section 6. Auxiliary results on one-dimensional SchrÄodinger operators are collected in Section 7 and in Section 8 we obtain the main results on asymptotic expansions for the resolvent of the Hamiltonian Hm as the spectral parameter tends to the lowest Landau threshold. Sections 4 to 8 constitute the ¯rst part of the paper. In the second part of the paper we study scattering theory for the pair (Hm ; Hom ). In Section 9 we establish scattering theory by means of the short range scattering theory developed by Jensen, Mourre and Perry [18]. In Section 10 we deduce asymptotic expansions of the scattering matrix as the energy parameter tends to the lowest Landau threshold. These near-threshold expansions are the main results on scattering theory in this work. They are obtained by using the expansions of the resolvent of Hm . The same problem was addressed by Kostrykin, Kvitsinsky, and Merkuriev [22]. However, the authors restrict themselves to the most simple case, where E0 is a regular point of H0 and E0 2 ½(H1 ). Their method is a direct generalization of a method developed by Boll¶e, Gesztesy and Wilk [11] in a study of the one-dimensional SchrÄodinger operator. The authors write that in order to incorporate the cases where some of the thresholds are exceptional points, one can adjust the technique of [11]. In a sense, this has been one of the aims of this work, although we think that our methods go beyond what we understand as an adjustment. Tamura [39] has recently studied low-energy scattering for the SchrÄodinger operator in two dimensions with a compactly supported magnetic ¯eld. He is able to adapt the method developed by Jensen and Kato [16] to the problem. This is not possible in the present work. It is important to point out that the abstract theory found in Section 3 has already been presented as a part of [28]. However, our opinion is that it is very useful to have a paper as self-contained as possible. Herein we only treat the cases of direct relevance to the present problem.


481

It is worth while to mention that the methods in Section 3 have been extended to study perturbations of eigenvalues and half-bound states embedded at a threshold for an abstract class of two-channel Hamiltonians [17]. We refer to the paper for details.

2

Preliminaries

Let T be a self-adjoint operator on a Hilbert space H with domain D(T ). The spectrum and resolvent set are denoted by ¾(T ) and ½(T ), respectively. We use standard terminology for the various parts of the spectrum, see for example [15]. The resolvent is R(³) = (T ¡ ³ )¡1 . The spectral family associated to T is denoted by ET (¸), ¸ 2 R. For an interval ¢ » R, F (¸ 2 ¢) stands for the smoothed out characteristic function of ¢: 8 > < 1 if ¸ 2 ¢ and dist (¸; @¢) ¶ ±; ± ½ j¢j; F (¸ 2 ¢) = > : 0 if ¸ 62 ¢:

Given a self-adjoint operator T , E~T (¢) = F (T 2 ¢) will denote the smoothed-out spectral projection of T on the interval ¢. For a complex number z 2 C n [0; 1) we denote by z 1=2 the branch of the square root with positive imaginary part. Let Rd be the d-dimensional Euclidean space, denote points of Rd by x = (x1 ; : : : ; xd ) P and let jxj = ( dj=1 x2j )1=2 . For 1 µ p µ 1 let Lp (Rd ) be the space of (equivalence classes R of) complex-valued functions Ã which are measurable and satisfy R d jÃ(x)jp dx < 1 if p < 1 and kÃkL1 (R d ) = ess sup jÃj < 1 if p = 1. The measure dx is the Lebesgue R measure. For any p the Lp (Rd ) space is a Banach space with norm k ¢ kLp (R d ) = ( R d j ¢ jp dx)1=p . In the case p = 2, L2 (Rd ) is a complex and separable Hilbert space with scalar R 1=2 product (Á; Ã)L2 (R d ) = R d ÁÃ and corresponding norm kÃkL2 (R d ) = (Ã; Ã)L2 (R d ) . The space of in¯nite di®erentiable complex-valued functions with compact support will be denoted by C01 (Rd ) or D(Rd ), the space of test functions. The adjoint space of D(Rd ), D 0 (Rd ), is the space of distributions on D(Rd). The Schwarz space of rapidly decreasing functions and its adjoint space of tempered distributions are denoted by S(Rd ) and S 0 (Rd ), respectively. Let p denote the momentum operator ¡ ir and let hpi = (1 + p2 )1=2 . We use the weighted Sobolev space H m;s (Rd ) given by H m;s (Rd ) = fÃ 2 S 0 (Rd )j kÃkm;s = khxis hpim ÃkL2 < 1g: We use h¢; ¢i to denote the inner product on L2 (Rd ) and also the natural duality between 0 0 H m;s (Rd ) and H ¡m;¡s (Rd ). B(H m;s (Rd ); H m ;s (Rd )) denotes the space of bounded op0 0 erators from H m;s to H m ;s with the operator norm. The Fourier transform is given by Z ¡1=2 b (F Ã)(») = Ã(») = (2¼) e¡ix¢» Ã(x) dx Rd

and is a bounded map from H m;s (Rd ) to H s;m (Rd ).

482

3


Abstract theory

We study spectral properties at thresholds of two-channel Hamiltonians on the form H = Hdiag + Vof f , where 0 1 0 1 B Ha 0 C B 0 Vab C Hdiag = @ (3.1) A ; Voff = @ A 0 Hb Vba 0

act on the Hilbert space H = Ha © Hb . We assume that Ha and Hb are self-adjoint operators in Ha and Hb , respectively. Moreover, we assume that Vab 2 B1 (Hb ; Ha ), the space of compact operators, and require Vba = Vab¤ . The spectrum of Hdiag is the union of those of Ha and Hb . There are several possible situations to consider depending on the threshold behaviour for the resolvents of the diagonal Hamiltonians Ha and Hb . In this paper we treat only those of direct relevance to the Hamiltonian Hm . The ¯rst (nontrivial) case of interest is as follows. Assumption 3.1. Assume that ¾(Ha ) = ¾ac (Ha) = [¸; 1) and ¾ac (Hb ) = [¸1 ; 1) for some ¸1 > ¸. (i) Assume that there exists a Hilbert space K a , densely and continuously embedded in Ha , such that for some ± > 0 we have an asymptotic expansion for the resolvent Ra (³ ) of Ha in the norm of B(K a ; K a¤ ), viz. Ra (³ ) = G0 + i(³ ¡

¸)1=2 G1 ¡

(³ ¡

¸)G2 ¡

i(³ ¡

¸)3=2 G3 + O(j³ ¡

¸j2 )

(3.2)

for j³ ¡ ¸j < ±, Im ³ > 0. Assume furthermore that Gj = G¤j , j = 0; 1; 2; 3, as operators in B(K a ; K ¤a). (ii) Assume that Vab 2 B1 (Hb ; K a ). (iii) Assume that ¸ is a simple isolated eigenvalue of Hb , with normalized eigenfunction Ã. Its reduced resolvent is denoted by Cb . Under Assumption 3.1 we want to deduce an asymptotic expansion of the resolvent R(³ ) as the spectral parameter ³ tends to a threshold ¸. A key ingredient in our approach is the Feshbach formula, which gives a convenient explicit representation of the resolvent R(³ ) of H. There are two variants. We give only one of them. The other version is just an interchange of indices. De¯ne Ra (³) = (Ha ¡ Tb (³) = Hb ¡

³)¡1 ; ³¡

VbaRa (³ )Vab :

Then for Im ³ 6= 0 we have 0 1 ¡1 ¡1 B Ra (³) + Ra (³)Vab Tb (³) Vba Ra (³ ) ¡ Ra (³ )Vab Tb (³ ) C R(³ ) = @ A ¡ Tb (³)¡1 VbaRa (³ ) Tb (³ )¡1

(3.3) (3.4)

(3.5)


483

Furthermore, we use the notation Pb = h¢; ÃiÃ for the eigenprojection associated with the eigenvalue ¸ of Hb . Assumption 3.1(iii) implies that the expansion 1

X Pb Rb (³ ) = ¡ + (³ ¡ ³ ¡ ¸ n=0

¸)n Cbn+1

(3.6)

is valid for 0 < j³ ¡ ¸j < ± for some small ± > 0. See for example [20, 36]. First we consider a case simpli¯ed by assuming that the following real number, ®0 = hVba G0 Vab Ã; Ãi;

(3.7)

is di®erent from zero. Under this assumption it is convenient (and possible) to introduce the following projections in Hb : J1 = ®¡1 0 h¢; ÃiVba G 0 Vab Ã; J0 = Ib ¡

J1 :

(3.8)

Lemma 3.2. Let Assumption 3.1 hold at ¸ 2 R. Assume that ®0 6= 0. Then, we have in B(Hb ) the following asymptotic expansion Tb (³)¡1 = b0 + i(³ ¡ as j³ ¡

¸)1=2 b1 + O(j³ ¡

¸j)

(3.9)

¸j ! 0, Im ³ > 0, where the coe±cients b0 and b1 are given by b0 = Cb J0 (Ib ¡ b1 =

Vba G0 Vab Cb J0 )¡1 ;

®¡1 0 Cb J0 (Ib ¡

Vba G0 Vab Cb J0 )

(3.10) ¡1

£(®0 Vba G1 Vab Cb J0 + VbaG1 Vab Pb ) £(Ib ¡

Vba G0 Vab Cb J0 )¡1 :

(3.11)

Proof 3.3. The strategy of the proof is to factor the operator Tb (³ ) in order to show that the inverse of Tb (³ ) exists and admits an asymptotic expansion in the norm topology of B(Hb ) for j³ ¡ ¸j small enough. In the sequel we always assume that j³ ¡ ¸j < ± (with the ± from Assumption 3.1) and Im ³ > 0. We use the factorization Tb (³ ) = (Ib ¡

Vba Ra (³)Vab Rb (³ )) (Hb ¡

³):

(3.12)

The assumption gives the following asymptotic expansion in B(Hb ). Ib ¡

i Vba G1 Vab Pb ³¡ ¸ (³ ¡ ¸)1=2 ¡ (Vba G0 Vab Cb + Vba G2 Vab Pb )

Vba Ra (³)Vab Rb (³ ) = Ib +

¡ i(³ ¡ +O(j³ ¡

1

®0 J1 +

¸)1=2 (Vba G3 Vab Pb + Vba G1 Vab Cb ) ¸j):

(3.13)

We have expressed the second term on the right-hand side in terms of the projection J1 . Since µ ¶¡1 ®0 ³¡ ¸ Ib + J1 = J0 + J1 ; ³¡ ¸ ®0 + ³ ¡ ¸

484


we make the following factorization ·µ Ib ¡ Vba Ra (³)Vab Rb (³ ) =

i Vba G1 Vab Pb ¡ Vba G0 Vab Cb (³ ¡ ¸)1=2 ¡ Vba G2 Vab Pb ¡ i(³ ¡ ¸)1=2 Vba G3 Vab Pb

¸)1=2 Vba G1 Vab Cb + (³ ¡ ¸)Vba G0 Vab Cb2 µ ¶ ¸ ³¡ ¸ +O(j³ ¡ ¸j)) J0 + J1 + Ib ®0 + ³ ¡ ¸ µ ¶ ®0 £ Ib + J1 : ³¡ ¸ ¡ i(³ ¡

(3.14)

Consider [¢ ¢ ¢ ] in (3.14). Using Pb J0 = 0 we ¯nd that [¢ ¢ ¢ ](®0 + ³ ¡

¸) = ®0 (Ib ¡

Vba G0 Vab Cb J0 ) ¸)1=2 (Vba G1 Vab Pb ¡

+i(³ ¡ +O(j³ ¡

®0 VbaG1 Vab Cb J0 )

¸j):

(3.15)

From Assumption 3.1(ii) it follows that the operator Vba G0 Vab Cb J0 is a compact operator in Hb . Hence, Ib ¡ Vba G0 Vab Cb J0 is invertible and the inverse is bounded in Hb . As a consequence we are able to factor as follows. £ © Right-hand side of (3.15) = Ib + i(³ ¡ + O(j³ ¡ £®0 (Ib ¡

¸)1=2 (®0 Vba G1 Vab Cb J0 + Vba G1 Vab Pb ) ¤ ¸j)g ®0¡1 (Ib ¡ Vba G0 Vab Cb J0 )¡1

VbaG0 Vab Cb J0 )¡1 :

(3.16)

For j³ ¡ ¸j small enough, the Neumann series implies that the inverse of the right-hand side of (3.16) has the following expansion: Inverse of right-hand side of (3.16) = ®0¡1 (Ib ¡ +i(³ ¡

Vba G0 Vab Cb J0 )¡1 ¸)1=2 ®0¡2 (Ib ¡

Vba G0 Vab Cb J0 )¡1

£(®0 Vba G1 Vab Cb J0 + Vba G1 Vab Pb ) £(Ib ¡ +O(j³ ¡

Vba G0 Vab Cb J0 )¡1 ¸j):

(3.17)

In combination with (3.14), we obtain from (3.17) that (Ib ¡

Vba Ra (³ )VabRb (³))

¡1

= J0 (Ib ¡ +i(³ ¡

Vba G0 Vab Cb J0 )¡1 ¸)1=2 ®¡1 0 J0 (Ib ¡

Vba G0 Vab Cb J0 )¡1

£ (®0 Vba G1 Vab Cb J0 +Vba G1 Vab Pb ) (Ib ¡ +O(j³ ¡

¸j):

Vba G0 Vab Cb J0 )¡1 (3.18)

Finally, we obtain the desired expansion (3.9) by using the factorization (3.12), the expansion for Rb (³ ), (3.18) and the relation Pb J0 = 0.


485

Theorem 3.4. Let Assumption 3.1 hold at ¸ 2 R. Assume that ®0 6= 0. Then, we have in the norm of B(K a © Hb ; K a¤ © Hb ) the asymptotic expansion 0

1

B G0 + G0 Vab b0 Vba G0 ¡ G0 Vab b0 C R(³) = @ A ¡ b0 Vba G0 b0 0 B G1 + G0 Vab b0 VbaG1 + G1 Vab b0 Vba G0 + G0 Vab b1 Vba G0 +i(³ ¡ ¸)1=2 @ ¡ b0 Vba G1 ¡ b1 Vba G0 1 ¡ G0 Vab b1 ¡ G1 Vab b0 C (3.19) A + O(j³ ¡ ¸j) b1 as j³ ¡

¸j ! 0, Im ³ > 0.

Proof 3.5. The result follows immediately from Lemma 3.2 and the Feshbach formula (see (3.5)). We now consider the case when ¸ is an isolated eigenvalue of Hb of arbitrary multiplicity. We limit ourselves to discussing the simplest case. Assumption 3.6. Let parts (i) and (ii) of Assumption 3.1 hold at ¸ 2 R. Assume that ¸ is an isolated eigenvalue of Hb with eigenprojection Pb such that the operator Pb Vba G0 Vab Pb is strictly positive and invertible in B(Pb Hb ). Under Assumption 3.6 we de¯ne the operators L1 = (Pb Vba G0 Vab Pb )¡1 , K1 = Vba G1 Vab Pb and M1 = Vba G0 Vab Cb ¡

Vba G2 Vab Pb + Vba G0 Vab Cb VbaG0 Vab Pb L1 Pb

+Vba G2 Vab Pb Vba G0 Vab Pb L1 Pb :

(3.20)

Then we obtain the following result. Lemma 3.7. Let Assumption 3.6 hold at ¸ 2 R. Then, generically, we have in B(Hb ) the following asymptotic expansion Tb (³ )¡1 = d0 + i(³ ¡ as j³ ¡

¸)1=2 d1 + O(j³ ¡

¸j)

(3.21)

¸j ! 0, Im ³ > 0, where the coe±cients d0 and d1 are given by d0 = (Cb ¡

Pb L1 Pb )(Ib ¡

M1 )¡1 ¡

Cb Vba G0 Vab Pb L1 Pb (Ib ¡

M1 )¡1 ;

(3.22)

486


M1 )¡1 Vba G0 Vab Cb Vba G1 Vab Pb L1 Pb (Ib ¡

d1 = Pb L1 Pb (Ib ¡

M1 )¡1 Vba G0 Vab Cb Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb (Ib ¡

¡ Pb L1 Pb (Ib ¡

+Cb (Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb ¡ +Cb (Ib ¡

M1 )¡1

Vba G0 Vab Pb L1 Pb )(Ib ¡

VbaG1 Vab Pb L1 Pb )(Ib ¡

M1 )¡1

M1 )¡1

M1 )¡1

£ fVba G1 Vab Cb + Vba G0 Vab Cb Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb ¡ Vba G0 Vab Cb Vba G1 Vab Pb L1 Pb ¡ £(Ib ¡

M1 )

¡1

Vba G1 Vab Cb Vba G0 Vab Pb L1 Pb g

+ Pb L1 Pb K1 Pb L1 Pb (Ib ¡

M1 )¡1 :

(3.23)

Proof 3.8. Again we start from the factorization (3.12). This assumption gives the following asymptotic expansion in B(Hb ), Ib ¡

1

i Vba G1 Vab Pb ³¡ ¸ (³ ¡ ¸)1=2 ¡ (Vba G0 Vab Cb + Vba G2 Vab Pb )

Vba Ra (³ )VabRb (³) = Ib +

¡ i(³ ¡ +(³ ¡

Vba G0 Vab Pb +

¸)1=2 (Vba G3 Vab Pb + Vba G1 Vab Cb ) ¡ ¸) Vba G0 Vab Cb2 + Vba G2 Vab Cb

+Vba G4 Vab Pb ) + i(³ ¡ ¡ i(³ ¡

¸)3=2 Vba G5 Vab Pb

¸)3=2 Vba G1 Vab Cb2 + i(³ ¡

+O(j³ ¡

¸)3=2 Vba G3 Vab Cb

¸j2 ):

(3.24)

De¯ne Y (³) = Vba G0 Vab + i(³ ¡ and

¸)1=2 Vba G1 Vab

1

S(³) = Ib +

Y (³)Pb: (3.25) ³¡ ¸ Hence, S(³ ) consists of the identity plus the singular terms in (3.24). Let us also introduce the following, Z(³ ) = (³ ¡

¸)Pb + Pb Vba G0 Vab Pb + i(³ ¡

¸)1=2 Pb Vba G1 Vab Pb :

(3.26)

We use the following abstract result. Let P be a projection in B(H) and let X 2 B(H). Assume that the operator P + P XP is invertible in B(P H). Then the operator I + XP is invertible, and we have, with an obvious notation, (I + XP )¡1 = I ¡ XP (P + P X P )¡1 P . If, in the present situation, we assume that for some ³ the operator Z(³ ) is invertible in B(Pb Hb ) then S(³) is invertible, and the inverse is given by S(³ )¡1 = Ib ¡

Y (³)PbZ(³ )¡1 Pb :

(3.27)

Consider Z(³ ) ¯rst and bear in mind that L1 = (Pb VbaG0 Vab Pb )¡1 and K1 = Pb Vba G1 Vab Pb . Under Assumption 3.6 and for j³ ¡ ¸j small enough, the Neumann series yields that Z(³ )¡1 = Pb L1 ¡ ¡ (³ ¡

i(³ ¡

¸)1=2 Pb L1 Pb K1 Pb L1

¸)Pb L1 Pb L1 ¡ 3=2

(³ ¡

¸)Pb L1 Pb K1 Pb L1 Pb K1 Pb L1

L1 Pb L1 Pb K1 Pb L1 + i(³ ¡

¸)3=2 (Pb L1 K1 )3 Pb L1

+i(³ ¡

¸)

+i(³ ¡

¸)3=2 Pb L1 Pb K1 Pb L1 Pb L1 + O(j³ ¡

¸j2 ):

(3.28)


487

Next, we use (3.27) to obtain the following expansion for S(³ )¡1 : S(³)¡1 = Ib ¡

Vba G0 Vab Pb L1 Pb

+i(³ ¡

¸)1=2 VbaG0 Vab Pb L1 Pb K1 Pb L1 Pb

¡ i(³ ¡

¸)1=2 Vba G1 Vab Pb L1 Pb + (³ ¡

¸)Vba G0 Vab Pb L1 Pb L1 Pb

+(³ ¡

¸)Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb K1 Pb L1 Pb

¡ (³ ¡

¸)Vba G1 Vab Pb L1 Pb K1 Pb L1 Pb

¡ i(³ ¡

¸)3=2 Vba G0 Vab Pb L1 Pb L1 Pb K1 Pb L1 Pb

¡ i(³ ¡

¸)3=2 Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb K1 Pb K1 Pb L1 Pb

¡ i(³ ¡

¸)3=2 Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb L1 Pb

+i(³ ¡

¸)3=2 VbaG1 Vab Pb L1 Pb K1 Pb L1 Pb K1 Pb L1 Pb

+i(³ ¡

¸)3=2 VbaG1 Vab Pb L1 Pb L1 Pb + O(j³ ¡

¸j2 ):

Next, we consider U (³) de¯ned by ¡ U (³ ) = Ib ¡ Vba G0 Vab Cb + Vba G2 Vab Pb + ¢ ¢ ¢ + O(j³ ¡

(3.29)

¢ ¸j2 S(³ )¡1 :

Using the de¯nition of M1 in (3.20) we ¯nd the following expression for U (³ ) up to an error term: U (³ ) = Ib ¡

M1 ¡

i(³ ¡

¸)1=2 Vba G3 Vab Pb

¸)1=2 Vba G1 Vab Cb ¡

¡ i(³ ¡

i(³ ¡

¸)1=2 £

£ (Vba G0 Vab Cb + Vba G2 Vab Pb ) Vba G0 Vab Pb L1 Pb K1 Pb +i(³ ¡

¸)1=2 (Vba G0 Vab Cb + Vba G2 Vab Pb )Vba G1 Vab Pb L1 Pb

+i(³ ¡

¸)1=2 (Vba G3 Vab Pb + Vba G1 Vab Cb )Vba G0 Vab Pb L1 Pb

+(³ ¡

¸)(VbaG0 Vab Cb2 + Vba G2 Vab Cb + Vba G4 Vab Pb )

¡ (³ ¡

¸)(VbaG0 Vab Cb + Vba G2 Vab Pb )Vba G0 Vab Pb L1 Pb L1 Pb

¡ (³ ¡

¸)(VbaG0 Vab Cb + Vba G2 Vab Pb )Vba G0 Vab Pb L1 Pb

£Pb K1 Pb L1 Pb K1 Pb L1 Pb +(³ ¡

¸)(VbaG0 Vab Cb + Vba G2 Vab Pb )Vba G1 Vab Pb L1 Pb

£Pb K1 Pb L1 Pb ¡

(³ ¡

¸)(Vba G0 Vab Cb2 + Vba G2 Vab Cb + Vba G4 Vab Pb )

£Vba G0 Vab Pb L1 Pb + (³ ¡

¸)(VbaG3 Vab Pb + Vba G1 Vab Cb )

£ [Vba G0 Vab Pb L1 Pb K1 Pb L1 Pb ¡ +O(j³ ¡

¸j

3=2

):

Vba G1 Vab Pb L1 Pb ] (3.30)

The operator M1 is compact. Hence, the operator Ib ¡ M1 is invertible. Therefore we factorize U (³) in the following way: © U (³ ) = (Ib ¡ M1 ) Ib + (Ib ¡ M1 )¡1 ¡ ¢ª £ i(³ ¡ ¸)Vba G3 Vab Pb + ¢ ¢ ¢ + O(j³ ¡ ¸j3=2 ) : (3.31)

From (3.30), (3.31) and the Neumann series we obtain an expansion for the inverse of U (³ ) up to O(j³ ¡ ¸j3=2 ). Finally, the expansion (3.21) is obtained via the factorization

488


Tb (³ )¡1 = Rb (³ )S(³)¡1U (³ )¡1 in conjunction with the expansions (3.6), (3.29) and the expansion of the inverse of U (³ ). In view of Lemma 3.7 and the Feshbach formula we obtain the following theorem. Theorem 3.9. Let Assumption 3.6 hold at ¸ 2 R. Then, we have in the norm of B(K a © Hb ; K a¤ © Hb ) the asymptotic expansion 0 1 B G0 + G0 Vab d0 Vba G0 ¡ G0 Vab d0 C R(³) = @ A+ ¡ d0 Vba G0 d0 0

B G1 + G0 Vab d0 Vba G1 + G1 Vab d0 VbaG0 + G0 Vab d1 Vba G0 ¸)1=2 @ ¡ d0 Vba G1 ¡ d1 Vba G0 1 ¡ G0 Vab d1 ¡ G1 Vab d0 C (3.32) A + O(j³ ¡ ¸j) d1

+i(³ ¡

as j³ ¡

¸j ! 0, Im ³ > 0.

Finally we consider the case when Ha has a so-called half-bound state (or resonance) at ¸, i.e. there exists a Hilbert space K a , densely and continuously embedded in Ha , and a solution Á to Ha Á = ¸Á, where Á 2 K ¤a but Á 62 Ha . Motivated by the known results for SchrÄodinger operators in dimensions one and three, see e. g. [7, 16], we assume a particular form of the singularity of the resolvent. Assumption 3.10. Assume that ¾(Ha ) = ¾ac (Ha ) = [¸; 1) and ¾ac (Hb ) = [¸1 ; 1) for some ¸1 > ¸. (i) Assume that there exists a Hilbert space K a, densely and continuously embedded in Ha , such that for some ± > 0 we have an asymptotic expansion Ra (³) =

i Qa + G0 + i(³ ¡ (³ ¡ ¸)1=2

¸)1=2 G1 + O(j³ ¡

¸j)

(3.33)

for j³ ¡ ¸j < ±, Im ³ > 0, in norm in B(K a ; K a¤ ). Assume that G0 = G¤0 for G0 2 B(K a ; K a¤ ). Assume Qa = h¢; ’i’ for some ’ 2 K a¤ . (ii) Assume that Vab 2 B1 (Hb ; K a ). (iii) Assume that ¸ 2 ½(Hb ). Under this assumption we have that Rb (³ ) =

1 X

(³ ¡

¸)n Cn

(3.34)

n=0

for j³ ¡ ¸j su±ciently small. The series converges in B(Hb ), and we have Cn = Rb (¸)n+1. We introduce the real constant µ0 = hVab C0 Vba ’; ’i. Assuming that µ0 6= 0, we may


489

introduce the projections J~1 = µ0¡1 h¢; ÁiVab C0 Vba Á; J~0 = Ia ¡

J~1 :

Moreover, it is convenient to introduce the operator E = Vab C0 Vba G0 J~0 . Then we obtain the following result. Lemma 3.11. Let Assumption 3.10 hold at ¸ 2 R. Assume that µ0 6= 0. Then we have in B(K a ; K a¤ ) the following asymptotic expansion Ta(³ )¡1 = f0 + i(³ ¡ as j³ ¡

¸)1=2 f1 + O(j³ ¡

¸j)

(3.35)

¸j ! 0, Im ³ > 0, where the coe±cients f0 and f1 are given by f0 = G0 J~0 (Ia ¡

E)¡1 ¡

µ0¡1 Qa (Ia ¡

E)¡1 ;

f1 = µ0¡1 G0 J~0 (Ia ¡ E)¡1 Vab C0 VbaG0 (Ia ¡ E)¡1 + +G0 J~0 (Ia ¡ E)¡1 Vab C0 Vba G1 J~0 (Ia ¡ E)¡1 + µ0¡1 G0 (Ia ¡ ¡ µ ¡1 G0 J~0 (Ia ¡ E)¡2 + G1 J~0 (Ia ¡ E)¡1 : 0

(3.36)

E)¡1 (3.37)

Proof 3.12. Having introduced the projections J~0 , J~1 , the proof follows the pattern of the proof of Lemma 3.2. The details are omitted. As usual we immediately obtain an expansion of the resolvent R(³ ) via the Feshbach formula. Theorem 3.13. Let Assumption 3.10 hold at ¸ 2 R. Then we have in the norm of B(K a © Hb ; K a¤ © Hb ) the asymptotic expansion 0 1 ¡ f0 Vab C0 f0 B C R(³ ) = @ A+ ¡ C0 Vba f0 C0 + C0 Vba f0 Vab C0 0 1 ¡ f1 Vab C0 C f1 B +i(³ ¡ ¸)1=2 @ (3.38) A + O(j³ ¡ ¸j) ¡ C0 Vba f1 C0 Vba f1 Vab C0 as j³ ¡

¸j ! 0, Im ³ > 0.

Remark 3.14. (i) In [28, Theorem 3.3] we show how to treat the case where Assumption 3.1 holds but ®0 = 0. (ii) In [28] we treat other abstract situations. However, these cases are not relevant for the present work.

490

4


The free Hamiltonian Hom

In R3 we consider a charged, spinless particle in a homogeneous magnetic ¯eld with no other forces present. Assume that the mass of the particle is 1=2 and its electric charge is 1, and that the magnetic ¯eld B has constant strength 2 and is aligned in the z direction: B = (0; 0; 2). The Hamiltonian of the particle is H0 (A) = (p ¡ A)2 , where p = ¡ ir is the momentum operator and A is the vector potential associated with the ¯eld, viz. B = r £ A, and de¯ned up to a gauge transformation. We choose the gauge in which A = 12 (B £ r) and denote the Hamiltonian by H0 (B) to emphasize that the magnetic ¯eld is constant. The Hamiltonian H0 (B) is essentially self-adjoint on C01 (R3 ) [35]. For convenience, we use the same notation H0 (B) for its closure. With an appropiate choice of the system of units, H0 (B) may be written in cylindrical coordinates (½; Á; z) in the form @ H0 (B) = ¡ ¢ ¡ 2i + ½2 : (4.1) @Á We rewrite (4.1) as H0 (B) = p2z + Hosc ¡ where Hosc

1 @ =¡ ½ @½

µ ¶ @ ¡ ½ @½

2Lz ;

1 @2 @ + ½2 ; L z = ¡ i : 2 2 ½ @Á @Á

(4.2) (4.3)

It is well-known (see [5] for details), that Hosc and Lz acting on L2 (R2 ) have a complete, joint set of eigenfunctions ffmn gm=0;§1;§2;:::;n=0;1;2;::: : Lz fmn = mfmn ;

Hosc fmn = 2(jmj + 1 + 2n)fmn :

The m and n are called the magnetic and radial quantum numbers, respectively. If we denote H? = Hosc ¡ 2Lz then H? has eigenfunctions fmn with corresponding eigenvalues Emn = 2(jmj ¡ m + 1) + 4n. The functions fmn are known explicitly: fmn (½; Á) = (2¼)¡1=2 e¡imÁ ªmn (½); Z (jmj)

1 0

ªmn (½) = 21=2 fn!(n + jmj)!g¡1=2 ½jmj exp(¡ ½2 =2)L(jmj) (½2 ); n ª2mn (½)½d½ = 1; m = 0; §1; §2; : : : ; n = 0; 1; 2; : : :

Here Ln are the generalized Laguerre polynomials (see [1]). It follows from the prop(jmj) erties of Ln that for any ¯xed m the set fªmn g1 n=0 forms an orthonormal basis in 2 L ((0; 1); ½d½). Often the eigenfunctions fmn are referred to as Landau orbits, and the corresponding eigenvalues Emn of H? are called the Landau energy levels. It is well-known from [5] that i) inf spec(H0 (B)) = 2, ii) ¾(H0 (B)) = ¾ac (H0 (B)) = [2; 1). Let fPmn g, m 2 Z, n ¶ 0 be the set of orthonormal one-dimensional projections onto P 2 the corresponding eigenspaces of H? . Hence, H? = mn Emn Pmn in L (½d½dÁ). Let Hmn = Ran Pmn « L2 (R). Decompose the space L2 (R3 ) into the orthogonal sum of the subspaces Hmn corresponding to ¯xed magnetic and radial quantum numbers. Then we


491

P can express the free, magnetic Hamiltonian H0 (B) as H0 (B) = I «p2z + mn Emn Pmn «I in L is a reducing subspace of H0 (B) and, in addition, mn Hmn . One easily shows that HmnL 1 for ¯xed m the orthogonal sum Hm = n=0 Hmn is a reducing subspace of H0 (B). Thus, the restriction of H0 (B), denoted by Hom , to a ¯xed, magnetic quantum number m is a self-adjoint operator in Hm with domain D(Hom ) = D(H0 (B)) \ Hm .

5

The full Hamiltonian Hm

We make the following assumption on the potential V . Assumption 5.1. Let V = V (x) be a real-valued, measurable function on R3 . (i) Let V satisfy the estimate jV (x)j µ C(1 + jxj)¡®

(5.1)

for some constants C > 0 and ® > 1. (ii) Let V be axisymmetric, i.e. V (x) = V (½; z) with ½2 = x2 + y 2 . We refer to ® as the decay parameter. Note that under Assumption 5.1 with ® > 1 we 0 0 have that V is a compact map from H 1;0 (R3 ) to H ¡1;® (R3 ) for all 1 < ® < ®. Let us state here as a lemma the following consequences of the Kato-Simon inequality je¡tH0 (A) Ãj µ je¡t¢ jjÃj and a result of Dodds-Fremlin [14] and Pitt [34]: Lemma 5.2. Let W be a multiplication operator and let A 2 L2loc (Rn ). Then (i) if W is ¢-bounded with relative bound a, W is H0 (A)-bounded with relative bound at most a; (ii) if W is ¢-compact, it is H0 (A)- compact. We have the following lemma. Lemma 5.3. In L2 (R3 ) let V satisfy Assumption 5.1(i) with ® chosen as below. (i) For ® > 0 V is H0 (A)-compact. (ii) For ® > 1 jxjV is H0 (A)-compact. (iii) For ® > 2 jxj2 V is H0 (A)-compact. Proof 5.4. The assertions are easy to show for H0 (A) replaced by ¢. Then the assertions for H0 (A) follow by Lemma 5.2. Let V satisfy Assumption 5.1(i) with ® > 0 and let B be de¯ned as in Section 4. It follows from Lemma 5.3 that the SchrÄodinger operator H(B) = H0 (B) + V is self-adjoint in L2 (R3 ) on the domain D(H0 (B)) and, due to Weyl’s essential spectrum theorem, ¾ess(H (B)) = ¾ess(H0 (B)) = [2; 1). In the sequel we assume that the electric potential V is axisymmetric. In particular the projection Qm onto Hm commutes with V and, consequently, Hm reduces H(B).

492


Therefore, if Assumption 5.1 holds with ® > 0 then the operator Hm = H (B)j m = Hom +V is a self-adjoint operator in Hm with domain D(Hm ) = D(H (B))\Hm . Moreover, ¾ess(Hm ) = ¾ess(Hom ) = [E0 ; 1), where En := 2(jmj ¡ m + 1 + 2n). If Assumption 5.1 holds with ® > 2 then the number of eigenvalues of Hm below E0 is ¯nite [38]. In particular, the eigenvalues of Hm below E0 cannot accumulate at E0 . In general, for the full Hamiltonian H (B) this is not necessarily the case [5]. t;s 3 For later purposes, we introduce the spaces Ht;s m := Q m H (R ). Then it follows from the mapping properties of V between weighted Sobolev spaces that V is a compact 0 0 ¡s operator from H1;¡s to H¡1;® for all 1 < ® < ®. m m We close this section by noting that the spaces Hm and H = L2 (R+ £ R; ½d½dz) are isomorphic. Let U denote the isomorphism between Hm and H. We shall use the same notation for Hom , V and Hm , when we regard them as operators in L2 (R+ £ R; ½d½dz). Furthermore, we shall use the spaces H t;s := U Ht;s m.

6

Two-channel framework

The basic Hilbert space is H = L2 (R+ £ R; ½ d½ dz). To represent the magnetic Hamiltonians Hom and Hm as two-channel Hamiltonians we need two additional Hilbert spaces de¯ned via the following projections. For any Á 2 H de¯ne the projection (P0 Á)(½; z) = hÁ; ª0 iL2 (R + ;½d½) ª0 (½), where ª0 (½) := ªm0 (½) is de¯ned in Section 4. Let the complement of P0 in H be denoted by P1 . Introduce the Hilbert spaces H0 = Ran P0 , H1 = Ran P1 b 0 = L2 (R). Then we have the basic decomposition H = H0 © H1 . As for the and H adjoints we have that Pj¤ = Pj , j = 0; 1. We want to ¯t the magnetic Hamiltonian Hm into a two-channel Hamiltonian via the decomposition H = H0 © H1 . For this purpose, we de¯ne the following matrix elements, where i; j = 0; 1: Hij = Pi Hom Pj¤ : Hj ¡ ! Hi ;

Vij = Pi V Pj¤ : Hj ¡ ! Hi :

(6.1)

With respect to this decomposition the magnetic Hamiltonian Hom is represented as a 2 £ 2 matrix given by 0

1

0

1

B H00 0 C B V00 V01 C Hm = Hom + V = @ A+@ A: 0 H11 V10 V11

(6.2)

in H0 ©H1 .The operators P0 and P1 are spectral projections and therefore they commute with Hom . Since P0 and P1 are orthogonal, the elements H01 and H10 vanish in the representation of Hom . We have the following results. Lemma 6.1. Let Assumption 5.1 be satis¯ed with ® > 0. Then V0j and V1j are Hjj compact, j = 0; 1. Proof 6.2. We prove the assertions for j = 0. Under Assumption 5.1 with ® > 0 the potential V is Hom -compact, i.e. V (Hom + i)¡1 is compact in H. Since P0 commutes with


493

Hom , P0 (Hom + i)¡1 P0¤ = (H00 + i)¡1 and P0 is bounded from H to H0 , it follows that V00 (H00 + i)¡1 is compact in H0 . Similarly, we prove that V10 is H00 -compact from H0 to H1 . Let H0 = H00 + V00 and H1 = H11 + V11 . Under Assumption 5.1 with ® > 0 the operators H0 and H1 are self-adjoint in H0 and H1 , respectively. Moreover, we have that ¾ess(Hj ) = ¾ess (Hjj ) = [Ej ; 1), j = 0; 1. For later purposes we note that the decomposition (1.2) of Hm in terms of H0 and H1 is valid. We wish to give conditions, which quarantees that ¾d (H0 ) consists of ¯nitely many eigenvalues below E0 . For this purpose we begin by expressing V00 in an explicit way. If f 2 H0 then there exists g(z) 2 L2 (R) such that f (z; ½) = g(z)ª0 (½). It is easily seen that V00 f (z; ½) = V~00 (z)f (z; ½), where Z V~00 (z) = V (z; ½0 )jª0 (½0 )j2 ½0 d½0 : (6.3) R+

0 The representation H0 = (p2z + E0 + V~00 ) « P0 implies that ¾d (H0 ) = ¾d (p2z + E0 + V~00 ), hence we need to put conditions on V such that ¾d (p2z + E0 + V~00 ) consists of ¯nitely many eigenvalues below E0 . If Assumption 5.1 holds with ® > 2 then it follows from the estimate jV~00 (z)j µ C(1 + jzj)¡® and [32, Equation (5)] that the number of eigenvalues below E0 for the self-adjoint operator p2z + E0 + V~00 in L2 (R) is ¯nite. De¯ne ¾0 (f (z)ª0 ) = f (z) for any element f (z)ª0 2 H0 . Then ¾0 is an isomorphism b 0 . Moreover, ^ 0 and its adjoint acts as ¾ ¤ f (z) = f (z)ª0 (½) for any f (z) 2 H from H0 onto H 0 b 1 = H1 and ¾1 = I. We de¯ne the projections P0 = ¾0 P0 and P1 = ¾1 P1 . let H We have the following result.

Lemma 6.3. Let Assumption 5.1 be satis¯ed with ® > 1. Then the operator P0 Hm P0¤ = b 0 = L2 (R) has no eigenvalues larger than E0 . p2z + E0 + V~00 in H

Proof 6.4. It follows immediately from the estimate jV~00 (z)j µ C(1 + jzj)¡® and [36, Theorem XIII.56]. In the sequel we also need the following spaces H0t;s = P0 Ht;s ;

t;s Ht;s 1 = P1 H :

(6.4)

We will use the notation Hsj = Hj0;s , j = 0; 1.

7

Auxiliary results for the resolvent of one-dimensional Schrodinger operators

It is well-known that zero cannot be an eigenvalue of a one-dimensional SchrÄodinger operator A = ¡ d2 =dz 2 + W (z) in L2 (R) if the (local) potential W (z) decays like W (z) = O(jzj¡® ) at in¯nity for some ® > 2. Hence, there are only two possible zero-energy

494


properties of A: Case 0) Zero is a regular point of A, i.e. zero is not an eigenvalue nor a zero resonance of A. Case 1) Zero is an exceptional point (of the 1st kind) for A, i.e. zero is not an eigenvalue but zero is a resonance. In the latter case, the equation AÃ = 0 has a unique (up to multiplicative constants) solution Ã in L1 , but Ã 62 L2 . We need the following results. Theorem 7.1. Suppose zero is a regular point of A. Assume ® > 9 and let s satisfy 9=2 < s < ® ¡ 9=2. For some ± > 0 we have in the norm of B(H ¡1;s (R); H 1;¡s (R)) the asymptotic expansion (0)

(0)

R(A; ³) = B0 + i³ 1=2 B1 ¡ (0)

(0)

³ B2 + O(³ 3=2 )

(7.1)

(0)

for j³ j < ±, Im ³ 1=2 > 0, where (Bj )¤ = Bj , j = 0; 1; 2, as operators in B(H ¡1;s (R); H 1;¡s (R)). Theorem 7.2. Suppose zero is an exceptional point (of the 1st kind) for A. Assume ® > 13 and let s satisfy 13=2 < s < ® ¡ 13=2. For some ± > 0 we have in the norm of B(H ¡1;s (R); H 1;¡s (R)) the asymptotic expansion (1) (1) (1) R(A; ³) = ¡ i³ ¡1=2 B¡1 + B0 + i³ 1=2 B1 ¡ (1)

(1)

³ B2 + O(³ 3=2 )

(7.2)

(1)

for j³ j < ±, Im ³ 1=2 > 0, where (Bj )¤ = Bj , j = ¡ 1; 0; 1; 2, as operators in B(H ¡1;s (R); H 1;¡s (R)). R 6 0. Theorems 7.1 and 7.2 are found in [26] under the assumption that W (z)dz = R Expansions with a similar structure can be derived if W (z)dz = 0 but in this case the coe±cients are di®erent (see [27] for details). By stating the theorems as above, we do not di®erentiate between the two situations. The proofs of Theorems 3.1 and 3.2 are based on a combination of the methods in [16] and [11, 9]. If the potential W decays exponentially at in¯nity, (7.1) and (7.2) are norm-convergent Taylor, respectively Laurent series [11, 9]. b 0 = p2 + E0 + The results in Theorems 7.1 and 7.2 hold for the SchrÄodinger operator H z ¤ b 2 b ~ V00 in H0 = L (R) and since H0 = ¾0 H0 ¾0 we obtain the following asymptotic expansion for the resolvent of H0 ,

R0 (³) = ¡ i(³ ¡ (j)

(j)

E0 )¡1=2 G¡1 +

+G0 + i(³ ¡

(j)

E0 )1=2 G1 ¡

(³ ¡

(j)

E0 )G2 + O(j³ ¡

E0 j3=2 );

(7.3)

(j) (j) in the norm of B(H0s ; H0¡s ) as ³ ! E0 , ³ 2 Cn[E0 ; 1), and Gk = ¾0¤ Bk ¾0 , j = 0; 1, k = ¡ 1; 0; 1; 2:


8

495

Asymptotic expansions of the resolvent at the lowest Landau threshold

In this section we carry over the abstract results in Section 3 to the Hamiltonian Hm decomposed as in (1.2). We show how to apply one of the abstract theorems in Section 3, namely Theorem 3.4. The remaining theorems in Section 3 are carried over in a similar way and stated without further explanation. Let Assumption 5.1 be satis¯ed throughout. The component Hamiltonians H0 and H1 from Section 6 play the roles of Ha and Hb in Section 3. We consider the situation, where Assumption 3.1 in Section 3 hold for H1 at the threshold E0 . In the present context Assumption 3.1[(i) and (iii)] can be formulated as follows. Assumption 8.1. Let E0 2 ¾(H0 ). (i) Suppose that E0 is a regular point of H0 . (ii) Assume that E0 is a simple isolated eigenvalue of H1 , with normalized eigenfunction Â0 . Its reduced resolvent is denoted by C1 . Assumption 8.1(i) implies that the asymptotic expansion (7.3) holds with j = 0 when ® > 9 and s satis¯es 9=2 < s < ®¡ 9=2. The expansion (7.3) corresponds to the expansion under Assumption 3.1(i). (1) We use the notation PE0 = h¢; Â0 iÂ0 for the eigenprojection. Let 5=2 < s < ® ¡ 5=2. The following real number is needed to state the result. (0)

®0 = hV10 G0 V01 Â0 ; Â0 i:

(8.1)

De¯ne, formally, the operators (0) (1) F¡2 = ¡ V10 G0 V01 PE0 ;

(0) (1) F¡1 = ¡ V10 G1 V01 PE0 ;

and, for j ¶ 0, 8 j¡2 > k < ¡ V10 G(0) V01 P (1) + Pj¡ 2 (¡ 1)k V10 G(0) j+2 E0 k=1 j+2¡2k V 01 C1 ; j = 0; 2; 4; : : : Fj = Pj¡ j ¡2 1 > (1) (0) k k : ¡ V10 G(0) j+2 V 01 PE0 + k=1 (¡ 1) V 10 G j+2¡2k V 01 C1 ; j = 1; 3; 5; : : :

We have the following elementary lemma.

Lemma 8.2. Let Assumption 5.1 and Assumption 8.1 be satis¯ed. Then the operators Fj , j = ¡ 2; ¡ 1, are rank one operators in Hs1 for ® > 2j + 9 and j + 9=2 < s < ® ¡ j ¡ 9=2 and the operators Fj , j = 0; 1; 2; : : :, are compact operators in H1s for ® > 2j + 9 and j + 9=2 < s < ® ¡ j ¡ 9=2. (1)

Proof 8.3. The projection PE0 extends to a bounded operator from Hs1 to H¡s 1 , s > 0, and under Assumption 5.1 with ® > 1 we have that Vij is a compact map from H1;0 to j

496

M. Melgaard / Central European Journal of Mathematics 4 (2003) 477{509 0

Hi¡1;® for all 1 < ® < ®. The assertions follow from these observations in conjunction (0) with the mapping properties of Gj ; the latter imposes the restrictions on ® and s. 0

Assume that ®0 6= 0. Under this assumption the projections (0)

J1 = ®¡1 0 h¢; Â0 iV 01 G0 V 01 Â0 ;

J0 = I ¡

J1 :

are well-de¯ned in H1s , 5=2 < s < ® ¡ 5=2. They correspond to the projections in (3.8). From Lemma 8.2, Lemma 3.2 and Theorem 3.4 we obtain the following result. Theorem 8.4. Let Assumption 5.1 with ® > 11 hold and let 11=2 < s < ® ¡ 11=2. Let Assumption 8.1 hold. Assume that ®0 6= 0. Then, we have in the norm of ¡s B(H0s © H1s ; H¡s 0 © H1 ) the asymptotic expansion 0

(0) B G0

R(³) = @

as j³ ¡

0

+ ¡ (0)

(0) (0) G0 V01 b0 V10 G0

¡

(0) G0 V01 b0

(0) b0 V10 G0 (0)

b0 (0)

(0)

1

C A + i(³ ¡

(0)

E0 )1=2 (0)

(0)

B G1 + G0 V01 b0 V10 G1 + G1 V01 b0 V10 G0 + G0 V01 b1 V10 G0 £@ (0) ¡ b0 V10 G(0) 1 ¡ b1 V10 G0 1 (0) (0) ¡ G0 V01 b1 ¡ G1 V01 b0 C A + O(j³ ¡ E0 j) b1

(8.2)

E0 j ! 0, where the coe±cients b0 and b1 are given by b0 = (1 ¡

(0)

J0 C1 V10 G0 V01 )¡1 J0 C1 ;

(8.3)

and b1 = ®0¡1 (1 ¡

(0)

J0 C1 V10 G0 V01 )¡1

£(®0 J0 E1 ¡

E¡1 )(1 ¡

(0)

J0 C1 V10 G0 V01 )¡1 J0 C1 :

(8.4)

Remark 8.5. Observe that Assumption 3.1(ii) is not necessary to establish Theorem 8.4. Su±cient decay of Vij at in¯nity is su±cient. The following, more simple case can be established by imitating the proofs of Lemma 3.2 and Theorem 3.4. Bear in mind that R1 (0) := R1 (E0 ) = (H1 ¡ E0 )¡1 . Theorem 8.6. Let Assumption 5.1 with ® > 9 hold and let 9=2 < s < ® ¡ 9=2. Let Assumption 8.1(i) be ful¯lled. Assume that E0 2 ½(H1 ). Then we have in the norm of


497

¡s B(H0s © H1s ; H¡s 0 © H1 ) the asymptotic expansion

0

(0) B G0

R(³ ) = @

0

+ ¡

(0) (0) G0 V01 a0 V10 G0

(0) G0 V01 a0

¡

(0) a0 V10 G0

(0)

a0

(0)

(0)

(0)

1

C A + i(³ ¡

(0)

E0 )1=2 (0)

(0)

B G1 + G0 V01 a0 V10 G1 + G1 V01 a0 V10 G0 + G0 V01 a1 V10 G0 £@ (0) ¡ a1 V01 G(0) 0 ¡ a0 V10 G1 1 (0) (0) ¡ G0 V01 a1 ¡ G1 V01 a0 C A + O(j³ ¡ E0 j) a1

as j³ ¡

(8.5)

E0 j ! 0, Im ³ > 0, where

a0 = L0 ; a2 =

and L0 = (I ¡

(0)

a1 = L0 V10 G1 V01 R1 (0)L0 ;

(0) L0 V10 G2 V01 R1 (0)L0 (0) ¡ L0 V10 G0 V01 R1 (0)2 L0

(8.6) (0)

+ L0 (V10 G1 V01 R1 (0)L0 )2

(8.7)

(0)

V10 G0 V01 R1 (0))¡1 .

We now consider the case when E0 is an isolated eigenvalue of H1 of arbitrary multiplicity. We limit ourselves to discussing the simplest case. Assumption 8.7. Let part (i) of Assumption 8.1 hold at E0 . Assume that E0 is an iso(1) (1) (0) (1) lated eigenvalue of H1 with eigenprojection PE0 such that the operator PE0 V10 G0 V01 PE0 (1) is strictly positive and invertible in B(PE0 H1 ). Under Assumption 8.7 (1) (1) K1 = V10 G1 V01 PE0 and (0)

M1 = V10 G0 V01 C1 ¡ (0)

we

de¯ne

the

operators

(0)

(0)

(1)

(0)

(1)

L1 = (PE0 V10 G0 V01 PE0 )¡1 ,

(0)

(1)

(1)

V10 G2 V01 PE (1) + V10 G0 V01 C1 V10 G0 V01 PE0 L1 PE0 0

(1)

(0)

(1)

(1)

+V10 G2 V01 PE0 V10 G0 V01 PE0 L1 PE0 :

(8.8)

Then we obtain the following result from Theorem 3.9. Theorem 8.8. Let Assumption 8.7 hold at E0 2 R. Then, generically, we have in the ¡s norm of B(H0s © H1s ; H¡s 0 © H1 ) the asymptotic expansion

498


0

(0) B G0

R(³) = @

as j³ ¡

0

+ ¡

(0) (0) G0 V01 d0 V10 G0

(0) G0 V01 d0

¡

(0) d0 V10 G0

(0)

d0

(0)

(0)

(0)

1

C A + i(³ ¡

E0 )1=2

(0)

(0)

(0)

B G1 + G0 V01 d0 V10 G1 + G1 V01 d0 V10 G0 + G0 V01 d1 V10 G0 £@ (0) (0) ¡ d0 V10 G1 ¡ d1 V10 G0 1 (0) (0) ¡ G0 V01 d1 ¡ G1 Vab d0 C A + O(j³ ¡ E0 j) d1

(8.9)

E0 j ! 0, Im ³ > 0, where the coe±cients d0 and d1 are given by (1)

(1)

(0)

(1)

(1)

d0 = (C1 ¡ PE0 L1 PE0 )(I ¡ M1 )¡1 ¡ C1 V10 G0 V01 PE0 L1 PE0 (I ¡ n (1) (1) (0) (0) (1) (1) d1 = PE0 L1 PE0 (I ¡ M1 )¡1 V10 G0 V01 C1 V10 G1 V01 PE0 L1 PE0 (1)

M1 )¡1 V10 G0 V01 C1 V10 G0 V01 PE0 L1 PE0 K1 PE0 L1 PE0

(0)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(1)

(1)

¡ V10 G0 V01 C1 V10 G1 V01 PE0 L1 PE0 ¡ (1)

(1)

(1)

(0)

(0)

(1)

(1)

+C1 (V10 G0 V01 PE0 L1 PE0 K1 PE0 L1 PE0 ¡ V10 G1 V01 PE0 L1 PE0 ) ª (0) (1) (1) £(I ¡ M1 )¡1 + C1 (I ¡ V10 G0 V01 PE0 L1 PE0 )(I ¡ M1 )¡1 n (0) (0) (1) (1) (1) (1) £ V10 G(0) 1 V 01 C1 + V10 G0 V 01 C1 V10 G0 V 01 PE0 L1 P E0 K1 PE0 L1 P E0 £(I ¡

(8.10)

(1)

¡ PE0 L1 PE0 (I ¡

(0)

M1 )¡1 ;

(1)

(1)

V10 G1 V01 C1 V10 G0 V01 PE0 L1 PE0

(1)

M1 )¡1 + PE0 L1 PE0 K1 PE0 L1 PE0 (I ¡

M1 )¡1 :

o

(8.11)

Remark 8.9. The setting considered in [28, Theorem 3.5] does not occur for the Hamiltonian Hm because E0 cannot be an eigenvalue of H0 under Assumption 5.1 with ® > 2. Finally we consider the case when H0 has a half-bound state at E0 (or E0 -resonance). In the present context Assumption 3.10 of Section 3 can be formulated as follows. Assumption 8.10. Let E0 2 ¾(H0 ). (i) Suppose that E0 is an exceptional point of H0 (1st kind), i.e. there exists a solution Á to the equation H0 Á = E0 Á, where Á 2 H¡s 5=2, but Á 62 H0 . 0 , 5=2 < s < ¯ ¡ (ii) Assume that E0 2 ½(H1 ). Assumption 8.10(i) implies that the asymptotic expansion (7.3) hold with j = 1 when ® > 13 and s satis¯es 13=2 < s < ® ¡ 13=2. The expansion (7.3) corresponds to the expansion (3.33) valid under Assumption 3.10 in Section 3.


Let 5=2 < s < ® ¡

499

5=2 and introduce the real constant µ0 = hV01 R1 (0)V10 Á; Ái:

Assuming that µ0 6= 0 we may introduce the projections Jb1 = µ0¡1 h¢; ÁiV01 R1 (0)V10 Á; Jb0 = I ¡

Jb1 :

in Hs0 , 7=2 < s < ® ¡ 7=2. Moreover, it is convenient to introduce the operator E = (1) V01 R1 (0)V10 G0 Jb0 . It is well-de¯ned when 9=2 < s < ® ¡ 9=2. Then we obtain the following result from Theorem 3.13. Theorem 8.11. Let Assumption 5.1 with ® > 13 hold and let 13=2 < s < ® ¡ 13=2. Let Assumption 8.10 be ful¯lled. Assume that µ0 6= 0. Then we have in ¡s B(H0s © H1s ; H¡s 0 © H1 ) the asymptotic expansion 0 1 ¡ f0 V01 R1 (0) f0 B C R(³) = @ A ¡ R1 (0)V10 f0 R1 (0) + R1 (0)V10 f0 V01 R1 (0) 0 1 ¡ f1 V01 R1 (0) f1 B C +i(³ ¡ E0 )1=2 @ A + O(j³ ¡ E0 j) (8.12) ¡ R1 (0)V01 f1 R1 (0)V10 f1 V01 R1 (0) as j³ ¡

E0 j ! 0, Im ³ > 0, where the coe±cients f0 and f1 are given by

(1) (1) f0 = G0 Jb0 (I ¡ E)¡1 + µ0¡1 G¡1 (I ¡ E)¡1 ; (1) (1) f1 = µ0¡1 G0 Jb0 (I ¡ E)¡1 V01 R1 (0)V10 G0 (I ¡ E)¡1 (1) (1) (1) +G0 Jb0 (I ¡ E)¡1 V01 R1 (0)V10 G1 Jb0 (I ¡ E)¡1 + µ0¡1 G0 (I ¡ ¡ µ ¡1 G(1) Jb0 (I ¡ E)¡2 + G(1) Jb0 (I ¡ E)¡1 : 0

9

0

(8.13) E)¡1

1

(8.14)

Scattering theory for the pair (Hm ; Hom )

We establish the scattering theory for the pair (Hm ; Hom ) by means of the abstract short range scattering theory developed by Jensen, Mourre, and Perry [18] (See also [19]) which is based on the following two de¯nitions: De¯nition 9.1. Let I0 be an open interval. Let A be a self-adjoint operator in H. We say that H0 satis¯es propagation estimates with respect to A on I0 if there exist real numbers s > s0 > 1 such that for all g 2 C01 (I0 ) the following two estimates hold: k(1 + A2 )¡s=2 e¡itH0 g(H0 )(1 + A2 )¡s=2 k µ c(1 + jtj)¡s k(1 + A2 )¡s=2 e¡itH0 g(H0 )PA§ k µ c(1 + jtj)

0

¡s0

for all t 2 R;

(9.1)

for all § t > 0:

(9.2)

500


Here PA+ = EA ((0; 1)) and PA¡ = 1 ¡

PA+ .

De¯nition 9.2. Let A be a self-adjoint operator on H. The potential V is said to be a short range perturbation of H0 with respect to A, if (H + i)¡1 ¡

(H0 + i)¡1 is a compact operator on H;

(9.3)

and if there exist a real number ¹ > 1 and integers j; k ¶ 0 such that the operator (H + i)¡j V (H0 + i)¡k (1 + A2 )¹=2

(9.4)

extends to a bounded operator on H. The main theorem is: Theorem 9.3 (Jensen-Mourre-Perry). Let H0 , V and H be as above. Assume that there exists a self-adjoint operator A such that H0 satis¯es the propagation estimates with respect to A and such that the potential V is a short range perturbation of H0 with respect to A. Then the wave operators W§ (H; H0 ; I0 ) exist and are strongly asymptotically complete. Furthermore, ¾s (H ) \ I0 is discrete in I0 . To complete the abstract theory one can give other conditions which are simpler to verify than the propagation estimates in De¯nition 9.1. One such method is the Mourre theory. We state the result (see [18, 19]) in the following form: Theorem 9.4 (Jensen-Mourre-Perry). Let H0 and A be self-adjoint operators on a Hilbert space H. Let ¸0 2 R. Suppose: (a) D(A) \ D(H0 ) is a core of H0 . (b) ei°A D(H0 ) » D(H0 ) and for each Ã 2 D(H0 ) we have supj°j·1 kH0 ei°A k < 1. (c) The commutator [H0 ; iA], de¯ned as a form on D(A) \ D(H0 ), is bounded below and closable. The self-adjoint operator associated with its closure is denoted iB1 . Assume D(H0 ) » D(B1 ). Assume inductively for j = 2; 3; : : : that the form i[iBj¡1 ; A] is bounded below and closable. The associated operator is denoted iBj . Assume D(H0 ) » D(Bj ). (d) There exist ® > 0, ± > 0, and a compact operator K such that with J = (¸0 ¡ ±; ¸0 +±) the Mourre estimate EH0 (J)iB1 EH0 ¶ ®EH0 (J) + K (9.5) holds. Then we have: (i) ¾s (H0 ) \ J is discrete in J. (ii) The propagation estimates in (9.1) and (9.2) hold with I0 = Jn¾s (H0 ) for all s > s0 > 0. To establish the scattering theory for the pair of Hamiltonians (Hm ; Hom ) we need the following preparations. The essential spectrum of Hom is the union of in¯nitely


501

many semilines starting at En , n = 0; 1; 2; : : :, respectively. The points En constitute the threshold set m = fEn : n = 0; 1; 2; : : : g which underpins the de¯nition of the intervals In = (En ; En+1 ); n = 0; 1; 2; : : : : We consider scattering theory for the pair (Hm ; Hom ) localized to the intervals In . We will use the conjugate operator Az = I « 12 (zpz + pz z). Note that Az commutes with P1 0 Hom;? = n=0 En Pn . Moreover, we denote by Hnom (n 2 Z) the Hilbert space de¯ned in the following way: De¯ne H+2 om = D(Hom ) with the graph norm kÁk +om2 = k(Hom +1)Ák . ¡1 +1 Similarly, de¯ne Hom = D(jHom j1=2 ) with its graph norm. De¯ne H¡2 om and Hom to be +2 the dual spaces of Hom and H+1 om , respectively, which may be thought of as the closure ¡j ¡ j = k(jH om j + 1) of Hom in the norm kÁk om Ák . The spaces Hjom , j = ¡ 2; ¡ 1; 0; 1; 2, constitute the scale of spaces in the Mourre theory. We are ready to verify the conditions in Theorem 9.4. Lemma 9.5. The self-adjoint operators Hom and Az satisfy conditions (a)-(c) in Theorem 9.4. Moreover, for any ¸0 2 Rn m , condition (d) in Theorem 9.4 is satis¯ed. Proof 9.6. We verify the hypotheses in Theorem 9.4. (a) C01 is a common core of Hom and Az . 0 (b) Let Pn denote the projection in L2 (R+ ; ½d½) onto the eigenspace associated with the radial quantum number n. Then we can represent the free Hamiltonian Hom P L 0 as Hom = I « p2z + n En Pn « I in L2 (R+ £ R; ½d½dz) = n Hn . The resolvent ¡1 Rom (³) = (Hom ¡ ³) , ³ 2 C+ , can be written as X 0 Rom (³) = Pn « (p2z + En ¡ ³ )¡1 (9.6) n

L in the weak sense on L2 (R+ £ R; ½d½dz) = n Hn [6]. Via Pythagoras’ formula and 0 the orthogonality of the projections Pn we obtain (9.6) in norm sense. The explicit formula (via (9.6)) X 0 eiAz a (Hom + i)¡1 = Pn « (e¡2a p2z + En ¡ ³)¡1 eiAz a (9.7) n

shows that eiAz a leaves D(Hom ) invariant. (c) We apply Proposition II.1 in [29]. For this purpose, we verify several conditions. By (a)-(b), the set C01 » D(Az ) \ D(Hom ) is a core of both Az and Hom and ei°Az C01 » C01 . In addition, the form [Hom ; iAz ] on C01 satis¯es [Hom ; iAz ] = 2p2z in the sense of forms. Clearly, the form p2z de¯ned on C01 (R+ £ R) is bounded below and closable in H £ H. The closed form generates a unique self-adjoint operator iB1;C01 with domain D(iB1;C01 ) ¼ D(Hom ). Therefore, Proposition II.1 in [29] asserts that the form [Hom ; iAz ] de¯ned on D(Az ) \ D(Hom ) is bounded below and closable, and the associated self-adjoint operator, denoted by iB1 , satis¯es iB1 = iB1;C01 on +2 C01 . Moreover, it is clear that iB1 extends to a bounded operator from Hom into H. The multiple commutators are treated similarly.

502

M. Melgaard / Central European Journal of Mathematics 4 (2003) 477{509 0

(d) Let J » R be an interval away from m and let 0 < ± < dist(J; m ). Let J 0 be an interval slightly larger than J such that EHom (J )EHom (J ) = EHom (J) and 0 0 0 ± < dist(J ; m ) < dist(J; m ). Using the fact that E~Hom (J ) = F (Hom 2 J ) = P 0 0 2 En ) it follows immediately that En s0 > 0. In order to establish Theorem 9.3, it remains to verify that V is a short range perturbation of Hom with respect to Az . Below we do this in the proof of Proposition 9.7. However, some preparations are necessary before we can give the details. We introduce the following weighted Sobolev spaces n o 0 Hzm;t (R+ £ R) = Á(½; z) 2 S j (1 + z 2 )t=2 (1 + p2z )m=2 Á 2 H :

Note that the weights (1 + z 2 )t=2 and (1 + p2z )m=2 include the variable z and its dual variable pz but not the variable ½. Furthermore, we use the following properties: (P1) H2 » Hz2;0 (R+ £ R). (P2) If V (½; z) satis¯es jV (½; z)j µ C(1 + ½2 + z 2 )¡® (9.8) for some C > 0 and some ® > 1=2 then the multiplication operator induced by V (½; z) is a compact operator from Hz2;0 (R+ £ R) to Hz0;® (R+ £ R). Property (P1) follows from (9.6). Property (P2) is veri¯ed by imitating the reasoning in Schechter [37]. We omit the details. Proposition 9.7. Let Assumption 5.1 hold with ® > 1. Then V is a short range perturbation of Hom with respect to Az . Proof 9.8. In order to apply Theorem 9.3, we need to verify (9.3) and (9.4) for Hom and Az . It follows immediately from the second resolvent equation and the Hom -compactness of V that the condition (9.3) is satis¯ed. For any ® > 1=2 we show that (9.4) is satis¯ed with ¹ > 1, j = 1, and k = 2. First, for any 0 µ ¹ µ 2 we show that T (¹) := (1 + z 2 )¡¹=2 (Hom + i)¡2 (1 + A2z )¹=2 extends to a bounded operator on H. For ¹ = 0 + i¿ , ¿ 2 R, T (0 + i¿ ) = (1 + z 2 )¡i¿ =2 (Hom + i)¡2 (1 + A2z )i¿ =2 is clearly a bounded operator on H since the operators (1 + z 2 )¡i¿ =2 and (1 + A2z )i¿ =2 are both unitary operators in H. Next, consider the case ¹ = 2 + i¿ , ¿ 2 R. For Á; Ã 2 C01 , we show that the form h(1 + z 2 )¡(2+i¿ )=2 (Hom + i)¡2 (1 + A2z )(2+i¿ )=2 Á; Ãi


extends to a bounded form on H £ H. Using twice pz z = zpz ¡ (3=4) ¡ z 2 p2z . Thus, we have to show that the form

503

i, we ¯nd that 1 + A2z =

h(1 + z 2 )¡i¿ =2 (1 + z 2 )¡1 (Hom + i)¡2 z 2 p2z (1 + A2z )i¿ =2 Á; Ãi on C01 £ C01 extends to a bounded form on H £ H. Let (Á; Ã) 2 C01 £ C01 . Using that [z 2 ; (Hom + i)¡2 ] = ¡ (Hom + i)¡2 4i(Hom + i)¡2 we get that h(1 + z 2 )¡i¿ =2 (1 + z 2 )¡1 (Hom + i)¡2 z 2 p2z (1 + A2z )i¿ =2 Á; Ãi = = h(1 + z 2 )¡i¿ =2 (1 + z 2 )¡1 z 2 (Hom + i)¡2 p2z (1 + A2z )i¿ =2 Á; Ãi +h(1 + z 2 )¡i¿ =2 (1 + z 2 )¡1 (Hom + i)¡2 4i(Hom + i)¡2 p2z (1 + A2z )i¿ =2 Á; Ãi and this form extends to a bounded form on H£H. From Hadamard’s three line theorem we conclude that T (¹) extends to a bounded operator on H for any 0 µ ¹ µ 2. Next, consider the operator (1 + z 2 )¹=2 V (Hm + i)¡1 = V (1 + z 2 )¹=2 (Hm + i)¡1 : Using the properties (P1){(P2) we infer that H

(Hm +i)¡ 1

¡ !

H 2;0 » Hz2;0

(1+z 2 )·

¡ !

=2

V

Hz2;¡¹ ¡ ! Hz0;®¡¹ » H;

where all maps are bounded. Hence, the operator (1 + z 2 )¹=2 V (Hm + i)¡1 is bounded on H and therefore its adjoint (Hm + i)¡1 V (1 + z 2 )¹=2 is likewise. In conclusion, for ¹ > 1, j = 1, and k = 2, (Hm + i)¡1 V (1 + z 2 )¹=2 T (¹) extends to a bounded operator on H. This proves (9.4). Lemma 9.5 and Proposition 9.7 in conjunction with Theorem 9.4 and Theorem 9.3 imply that the wave operator W§ (Hm ; Hom ; In ) exist and are strongly asymptotically complete. Moreover, ¾s (Hm ) \ In is discrete in In .

9.1 Representation of the scattering matrix on I0 It follows from Theorem 9.3 that the local scattering operator Sj , de¯ned by Sj = W+¤ (Hm ; Hom ; Ij )W¡ (Hm ; Hom ; Ij ); j = 0; 1; 2; : : : is a unitary operator on EIj (Hom )Pac (Hom )H. Let S~0 denote the unitary representation of S0 in L2 (I0 ; C2 ). There is a general theorem asserting that S~0 admits a diagonal representation (S~0 Ã)(¸) = S~0 (¸)Ã(¸) (see, e.g., [23, Theorem 6.2]). Here ¸ denotes the energy parameter. We restrict ourselves to S~0 , since we are only interested in the lowest Landau threshold E0 . In this section we give an explicit representation of S~0 (in the sequel we suppress the tilde character and the lower index). We need several de¯nitions. For j = 0; 1; 2; : : :, we de¯ne kj (¸) = (¸ ¡ Ej )1=2 and ºj (¸) = (2kj (¸))¡1 . In addition, de¯ne j(¸) as the largest integer satisfying 2(m + jmj +

504


2j(¸)) µ ¸, i.e. j(¸) is the number of Landau thresholds open at the energy ¸. Introduce b the layer H(¸) as the direct sum of a ¯nite number of two-dimensional linear spaces C2 with elements 0 1 (+) j(¸) M B gj C b 3g= H(¸) ºj (¸)gj ; gj = @ A: (¡) j=0 gj Via the functions

e¡i»z F~ (j; »; ½; z) = p ªj (½); j = 0; 1; 2; : : : ; 2¼

we can introduce the trace operator for any Ã 2 EHom (I0 )H = EH00 (I0 )H0 as p °~ (¸)Ã = º0 (¸) £8 0 1 0 19 > Z Z > < 1C 0 C= B B £ F~ (0; k0 (¸); ½; z) @ A + F~ (0; ¡ k0 (¸); ½; z) @ A Ã(½; z)½d½dz > R R+ > : 0 1 ; 0 1 1=2 \ B ¾0 P0 Ã((¸ ¡ E0 ) ) C = 2¡1=2 (¸ ¡ E0 )¡1=4 @ (9.9) A ; ¸ 2 I0 ; 1=2 P ¡ ¾\ Ã(¡ (¸ E ) ) 0 0 0 where the widehat symbol denotes the one-dimensional Fourier transform with respect b to the z variable. The trace operator maps from H 0;s © H0;s to H(¸) for s > 1=2. Having introduced the necessary objects, we are ready to formulate the result. Since the proof imitates (but is more involved than) the proof of the representation of the scattering operator associated with SchrÄodinger operators found in, e.g. Kuroda [23], we omit the details. Proposition 9.9. Let Assumption 5.1 hold with ® > 1 and let 1=2 < s < ® ¡ 1=2. Then the operator S is represented as (SÃ)(¸) = S(¸)Ã(¸); a.e. ¸ 2 I0 n¾pp+ (Hm ); Ã 2 L2 (I0 ; C2 ); where S(¸) = 1 ¡

2¼i~ ° (¸)(1 ¡

V R(¸ + i0))V °~ (¸)¤ ; ¸ 2 I0 n¾pp+ (Hm ):

(9.10)

The operator S(¸) is unitary for all ¸ 2 I0 n¾pp+ (Hm ). The operator S(¸) ¡ 1 is compact for all ¸ 2 I0 n¾pp+ (Hm ).

10

The scattering matrix near the lowest Landau threshold

We collect our main results on scattering for the pair (Hm ; Hom ) in the low-energy limit, i.e. near the lowest Landau threshold E0 . The scattering matrix has the diagonal representation in (9.10), which can be rewritten as S(¸) = 1 ¡

¼i(¸ ¡

E0 )¡1=2 °0 (¸)(1 ¡

R(¸ + i0)V )V °0 (¸)¤ ;

(10.1)


505

where the de¯nition of °0 (¸) is obvious from (9.10) and (10.1). To derive asymptotic expansions for S(¸) as ¸ # E0 we need expansions for the operators °0 (¸) and °0 (¸)¤ . Formally, we have 1 X °0 (¸) = ij (¸ ¡ E0 )j=2 ¡j ; (10.2) j=0

where

0

1

j B(¡ z) ª0 (½)C ¡j : (2¼)¡1=2 (j!)¡1 @ A: j z ª0 (½)

(10.3)

This follows from a formal expansion of 0 1 1=2 Bexp(¡ i(¸ ¡ E0 ) z)ª0 (½)C °0 (¸) : (2¼)¡1=2 @ A: 1=2 exp(i(¸ ¡ E0 ) z)ª0 (½) From ª0 2 L2 (R+ ; ½d½) and HÄolder’s inequality we see that ¡j 2 B(H0;s ; C2 ); s > j + 1=2:

(10.4)

Moreover, it follows from Taylor’s formula with remainder that the expansion (10.2) is valid as ¸ # E0 in the sense that if °0 (¸) is approximated by a ¯nite series up to j = k, k being the largest integer satisfying s > k + 1=2, then the remainder is o((¸ ¡ E0 )k=2 ) in the norm of B(H 0;s ; C2 ). In various, mostly singular, settings we establish the following leading order behaviour of the scattering matrix in the limit ¸ ! E0 . The results are based on (10.1) and the asymptotic expansions of the resolvent R(³) in Section 8. Theorem 10.1. Assume that one of the following hypotheses is ful¯lled. (a) Let Assumption 5.1 hold with ® > 9. Let Assumption 8.1(i) hold. Assume that E0 2 ½(H1 ). (b) Let Assumption 5.1 hold with ® > 11. Let Assumption 8.1 be satis¯ed. Assume that ®0 6= 0. (c) Let Assumption 5.1 hold with ® > 13. Let Assumption 8.7 be satis¯ed. Then we have in the norm of B(C2 ) the following leading order expansion 0 1 B 0 ¡ 1C S(¸) = @ (10.5) A + o(1): ¡ 1 0 as ¸ # E0 .

Proof 10.2. We prove (a). The remaining assertions are shown in a similar way. Let s (0) satisfy 9=2 < s < ¯ ¡ 9=2 and let Rj , j = 0; 1, denote the coe±cients in (8.5). From

506

M. Melgaard / Central European Journal of Mathematics 4 (2003) 477{509 (0)

(0)

(10.1), (10.2) and Theorem 8.6 we have the expansion S(¸) = ¡ i(¸ ¡ E0 )¡1=2 S¡1 + S0 + o(1) in B(C2 ), where (0) S¡1 = ¼¡0 (V ¡ (0)

S0

(0)

V R0 V )¡¤0 ;

= 1¸ + ¼¡1 (V ¡

(0)

V R0 V )¡¤0 ¡

(0)

¼¡0 V R1 V ¡¤0 ¡

¼¡0 (V ¡

(0)

V R0 V )¡¤1 :

Using 1 = S(¸)S(¸)¤ and the simple fact that T 2 = 0 implies that T = 0 for any self(0) (0) (0) adjoint operator T , we obtain that ¡0 (V ¡ V R0 V )¡¤0 = 0. Thus S¡1 = 0. As for S0 (0) we begin by rewriting the term ¼¡0 V R1 V ¡¤0 via the expression for ¡0 in (10.3) and the (0) expression for R1 given in Theorem 8.6. For any (z1 ; z2 ) 2 C2 , the operator acts as 0 1 0 10 1 Bz 1 C B1 1C Bz1 C (0) ¼¡0 V R1 V ¡¤0 @ A = c @ A @ A ; z2 11 z2 where

c=

1 (0) (0) hV00 G(0) 1 V 00 ª0 + V00 G1 V 01 a0 V 10 G 1 V00 ª0 2 (0) (0) (0) (0) +V00 G0 V01 a0 V10 G0 V00 ª0 + V00 G0 V01 a1 V10 G0 V00 ª0 (0)

¡ V00 G0 V01 a1 V10 ª0 ¡

(0)

(0)

V00 G1 V01 a0 V10 ª0 ¡

(0)

V01 a1 V10 G0 V00 ª0 (0)

(0)

¡ V01 a0 V10 G1 V00 ª0 + V01 a1 V10 ª0 + V00 G0 V01 a0 V10 G1 V01 ª0 (0)

(0)

(0)

(0)

+V00 G0 V01 a0 V10 G0 V01 ª0 + V00 G0 V01 a1 V10 G0 V10 ª0 ¡ V00 G(0) 0 V 01 a1 V11 ª0 ¡

(0)

V00 G1 V01 a0 V11 ª0 ¡

(0)

V01 a1 V10 G0 V01 ª0

¡ V00 a0 V10 G(0) 1 V01 ª0 + V 01 a1 V 11 ª0 ; ª0 i:

(10.6)

(0)

The operator ¡1 (V ¡ V R0 V )¡¤0 can be written as a matrix with real elements. Therefore, for some real number a we ¯nd that 0 1 B0 ¡ aC (0) (0) ¼¡1 (V ¡ V R0 V )¡¤0 ¡ ¼¡0 (V ¡ V R0 V )¡¤1 = @ A; a 0 since the terms on the left-hand side are each other adjoints. Hence, 0 1 0 1 0 1 B1 0 C B 1 1C B 0 ¡ a C (0) S0 = @ A ¡ c @ A + @ A: 01 11 a 0 (0)

By the unitarity of S0 , we infer that a = 0 and either c = 0 or c = 1. We show that c = 1. First, we observe that c depends continuously on V00 , V01 , V10 and V11 , hence it su±ces to consider the case where V01 = V10 = 0. Moreover, only the ¯rst term on the (0) right-hand side of (10.6) remains. In order to compute c we need the expression for B1 (0) (0) (bear in mind that G1 = sigma¤0 B1 ¾0 ) which is given in [26, Theorem 4]. Using this (0) expression we ¯nd that hV00 G1 V00 ª0 ; ª0 i = 2. Hence c = 1 as desired.


507

Remark 10.3. In principle, the method allows us to derive an asymptotic expansion of the scattering matrix under Assumption 8.10. However, we would need to derive an expansion of the resolvent of the one-dimensional SchrÄodinger operator H0 up to an error term of order O(j³ ¡ E0 j) via the method used in [26]. In practice this turns out to be extremely tedious and complicated, and, as a consequence, we have not succeeded in doing so.

Acknowledgements The author thanks Professor Grigori Rozenblum for several valuable discussions.

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Handbook of Mathematical Functions, Dover

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CEJM 4 (2003) 510{560

Variation of the reduction type of elliptic curves under small base change with wild rami¯cation Masanari Kida¤ Department of Mathematics The University of Electro-Communications Chofu, Tokyo 182-8585, Japan

Received 8 May 2003; accepted 10 July 2003 Abstract: We study the variation of the reduction type of elliptic curves under base change. A complete description of the variation is given when the base eld is the p-adic eld and the base change is of small degree. c Central European Science Journals. All rights reserved. ® Keywords: elliptic curve, conductor, reduction type, base change, rami¯cation group. MSC (2000): 11G07; 11S15, 14G20

Contents Introduction Theorems Basic de¯nitions and properties Variation of the conductor 4.1 Tamely rami¯ed extension 4.2 Cubic base change over Q3 4.3 Quadratic base change over Q2 5 Variation of the reduction type 5.1 Tamely rami¯ed extension (Proof of Theorem 2.1) 5.2 Cubic base change over Q3 (Proof of Theorem 2.4) 5.3 Quadratic base change over Q2 (Proof of Theorem 2.5) References Tables ¤

1 2 3 4

E-mail: [email protected]

511 511 514 519 519 522 527 542 543 545 546 554 556

M. Kida / Central European Journal of Mathematics 4 (2003) 510{560

1

511

Introduction

Let E be an elliptic curve de¯ned over a local ¯eld k with positive residue characteristic p and Type(E) the reduction type of E given by Kodaira symbol. In this paper, we study the variation of the reduction type under a ¯nite extension K=k. If p is greater than 3, then the description of the variation is essentially known. In particular, the reduction type of the base change EK is determined only by Type(E) and the rami¯cation index of the extension K=k (see Theorem 2.1). On the other hand, if p = 2 or 3, this is no longer true. For example, consider the elliptic curve C=Q2 : y 2 = x3 + x2 + 3x + 3: Tate’s algorithm shows Type(C) = I¤0 . By rami¯ed quadratic extensions, we have Type(CQ2 (p2) ) = I¤0 and Type(CQ2 (p6) ) = I¤1 : As this example shows, the reduction type is not determined even by the exponent of the discriminant of the extension. The variation is thus complicated and no general result is known. In this paper, we give a complete description of the variation of the reduction type for the cases where the base change is of small degree over the 2-adic and 3-adic ¯elds (Theorems 2.4 and 2.5). Though we deal with these easiest non-trivial cases, our result also indicates how complicated it is to describe the variation in the general case and what we need to extend the result. Throughout this paper, we ¯x an algebraic closure Qp of the p-adic ¯eld Qp for each p and assume that all algebraic extensions containing Qp sit inside the ¯xed algebraic closure.

2

Theorems

Let E be an elliptic curve de¯ned over a local ¯eld k. Let p be the residue characteristic of k. For a ¯nite extension K=k, let EK be the base change of E to K, i.e., EK = E £Spec(k) Spec(K). The reduction type of EK is given by Kodaira symbol that refers to the type of the special ¯ber of the minimal proper regular model of EK . As in the introduction, we denote it by Type(EK ). Let f (E=K) be the exponent of the conductor of EK whose precise de¯nition will be given in the following sections. We ¯rst recall the result for the case of p ¶ 5 and extend it a little. Theorem 2.1. Let E be an elliptic curve de¯ned over a local ¯eld k containing Qp . Let K=k be a ¯nite extension and e(K=k) the rami¯cation index of K=k. Assume that one of the following condition holds: (i) p is greater than 3; (ii) the extension K=k is unrami¯ed or tamely rami¯ed; (iii) for a prime number ` di®erent from p, the `-division ¯eld of E is unrami¯ed or tamely rami¯ed over k.

512


Then Type(EK ) and f (E=K) are determined only by Type(E) and e(K=k) as in Table 1. Our assumptions in Theorem 2.1 are not independent. In fact, if p ¶ 5, then `-division ¯elds are always unrami¯ed or tamely rami¯ed (see [23]). Using our previous result, we can make our third assumption precise. Proposition 2.2 ([8, 1.1, 1.4]). Let E be an elliptic curve de¯ned over a local ¯eld k containing Qp . Assume that ` is a prime number which is di®erent from p. (i) When ` ¶ 3, the extension k(E[`])=k is unrami¯ed if and only if Type(E) = I0 or In (` divides n): (ii) When ` = 2, the extension k(E[2])=k is unrami¯ed if and only if Type(E) = I0 ; In or I¤n (n is even). (iii) The extension k(E[`])=k is tamely rami¯ed if and only if either p = 3 and Type(E) = In (` 6 jn); III; I¤0 ; I¤n or III¤ ; or p = 2 and Type(E) = In (n is odd); IV or IV¤ holds. By Theorem 2.1 and Proposition 2.2, if the base change is tamely rami¯ed or unrami¯ed, or if the reduction type is good or multiplicative, then the variation of the reduction type is easily described. Thus the problem is to give a description when the base change involves a wild extension and the reduction of E is additive. Our main result gives an unabridged description of the variation for the cases where K is either cubic extensions over Q3 or quadratic extension over Q2 . We begin with our result on elliptic curves de¯ned over Q3 . To state our theorem, we ¯rst classify totally rami¯ed cubic extensions over Q3 . A cubic extension K=Q3 is either b over Q3 has a Galois a cyclic extension or a non-Galois extension whose Galois closure K b contains a group isomorphic to the symmetric group S3 of order 6. In the latter case, K p p p unique quadratic sub¯eld, which coincides with one of Q3 ( ¡ 1); Q3 ( 3); Q3 ( ¡ 3). The following classi¯cation by the exponent of the di®erent DK=Q3 is a classical result due to Hasse [6]. For an elementary proof, see [11]. b the Galois closure Lemma 2.3. Let K=Q3 be a totally rami¯ed cubic extension and K of K over Q3 and DK=Q3 the di®erent of the extension. We denote by vK the valuation of K normalized so that vK (3) = 3 holds. Then we have vK (D p K=Q3 ) = 3; 4 or 5 and p b b (C1 ) if vK (D K=Q3 ) = 3, then K ¼ Q3 ( ¡ 3) or K ¼ Q3 ( 3). p b or K b ¼ Q3 ( ¡ 1). (C2 ) if vK (D K=Q3 ) = 4, then K = K p b ¼ Q3 ( ¡ 3). (C3 ) if vK (D K=Q3 ) = 5, then K


513

For convenience, we write K 2 Ci to specify that K is of type Ci . Our second theorem is as follows. Theorem 2.4. Let E be an elliptic curve de¯ned over Q3 with additive reduction. Let K=Q3 be a wildly rami¯ed cubic extension and D K=Q3 the di®erent of the extension. Then the reduction type Type(EK ) and the conductor f (E=K) are determined by Type(E) and the conductor f (E=Q3 ) and the valuation vK (D K=Q3 ) as in Table 2 except for the cases with K 2 Cf (E=Q3 )¡2 . If K 2 Cf (E=Q3 )¡2 , then the reduction type also depends on the isomorphism class of K (see Theorem 4.6 for detail). It is worth noting here that the conductor of an elliptic curve is not determined solely by the reduction type when the residue characteristic is 2 or 3. Next we state our result for p = 2. There are seven quadratic extensions over Q2 . p The unique unrami¯ed extension is Q2 ( ¡ 3). We divide the rami¯ed six ¯elds into three groups: p p p p p p Q1 = fQ2 ( ¡ 1); Q2 ( 3)g; Q2 = fQ2 ( 2); Q2 ( 10)g; Q3 = fQ2 ( ¡ 2); Q2 ( 6)g: (2.1) For the case p = 2, we need more information to determine the conductors and the reduction types. Theorem 2.5. Let E be an elliptic curve de¯ned over Q2 with additive reduction. Let K=Q2 be a rami¯ed quadratic extension. (i) The reduction type Type(EK ) and the conductor f (E=K) are determined as in Table 3 by ° the reduction type Type(E), ° the conductor f (E=Q2 ), ° to which Qi the ¯eld k belongs, ° the Galois group of the 3-division ¯eld over Q2 except for the cases with `?’ and `??’. (ii) Cases with ?: Additionally some information on the 3-division ¯eld Q2 (E[3]) is necessary to determine the reduction types and the conductor (see Theorem 4.16 for detail). (iii) Cases with ??: Let v2 be the normalized dyadic valuation. Applying Tate’s algorithm over Q2 , we can take a minimal model of E over Z2 satisfying v2 (a1 ) ¶ 1; v2 (a3 ) ¶ 2; v2 (a4 ) ¶ 1; v2 (a6 ) = 1: By taking this model, Type(E) = III¤ if and only if v2 (a4 ) ¶ 2. If an elliptic curve is given by a Weierstrass equation y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ;

(2.2)

514


then the 3-division polynomial of the elliptic curve is de¯ned by ©3 (X) = 3X 4 + b2 X 3 + 3b4 X 2 + 3b6 X + b8

(2.3)

and the Galois group of the 3-division ¯eld is determined by that of ©3 (X ) (see Remark 4.17). In Tables 2 and 3, we supply the valuation of the minimal discriminant ¢Qp (p = 2; 3) of elliptic curves only for computational convenience. According to Ogg’s formula (2.4) below, giving vp (¢Qp ) is equivalent to giving f (E=Qp ), if the reduction type is speci¯ed. Also in these tables, we give examples of elliptic curves with prescribed variation of reduction type. These examples are found by means of TECC [9] and Cremona’s elliptic curve data [3]. Our proofs of these theorems are organized as follows. After recalling some basic de¯nitions and facts in Section 3, we compute the variation of the conductor in Section 4. This determination involves the computation of the higher rami¯cation groups of certain division ¯elds of elliptic curves. Having calculated the conductor f (E=K), the reduction type can be almost determined by Ogg’s formula ([25, IV-11]) f (E=K) = v(¢K ) + 1 ¡

mK ;

(2.4)

where mK is the number of irreducible components counted without multiplicity on the special ¯ber of the minimal proper regular model of EK and ¢K is, as above, the minimal discriminant of EK . In fact, we can deduce a congruence for mK from f (E=K). Then the reduction type and mK correspond as follows when EK has additive reduction: Type(EK )

II

III

IV

I¤0

I¤n

IV¤

mK

1

2

3

5

5+n

7

III¤ II¤ 8

(2.5)

9

This shows that mK determines the reduction type in most cases. The ambiguous cases can be eliminated by the applications of Tate’s algorithm. This ¯nal computation will be done in Section 5. We should note here that applying Tate’s algorithm directly to a model over Qp causes di±culty in evaluating the coe±cients of models under various variable changes. Thus, instead, we make the above detour. By doing so, ¯nal application of Tate’s algorithm becomes much easier. Also, as byproducts, we can obtain the description of the higher rami¯cation groups of certain division ¯elds of elliptic curves (Propositions 4.5 and 4.11).

3

Basic de¯nitions and properties

In this section, we recall some basic de¯nitions and properties of the conductor of an elliptic curve. We are naturally concerned with the higher rami¯cation groups of certain division ¯elds of an elliptic curve. The standard reference for the higher rami¯cation groups is [21, IV].


515

We start with a general situation. Let k be a ¯nite extension of Qp . We assume that all algebraic extensions of k are also sitting inside the ¯xed algebraic closure Qp . Let L=k be a ¯nite Galois extension. The Galois group G = Gal(L=k) of the extension has a natural descending ¯ltration by the higher rami¯cation groups Gi = Gi (L=k) = f¾ 2 G j vL (¾(¼L ) ¡

¼L ) ¶ i + 1g

where i ¶ ¡ 1 is an integer and vL is the discrete valuation of L normalized so that vL (L£ ) = Z and ¼L is a prime element in L. We have a sequence G = G¡1 ¼ G0 ¼ G1 ¼ G2 ¼ : : : consisting of normal subgroups of G. Here G0 agrees with the inertia group of G. We denote the rami¯cation index of L=k by e(L=k), which is equal to #G0 . We collect the basic properties of Gi . In what follows, we use the notation gi = gi (L=k) = #Gi . Lemma (i) (ii) (iii) (iv) (v)

3.1. The groups Gi are normal subgroups of G. For a su±ciently large i, we have Gi = f1g. The group G1 is a p-Sylow subgroup of G0 . The quotient Gi =Gi+1 is an elementary abelian group for i ¶ 1. Let D L=k be the di®erent of the extension L=k. Then vL (D L=k ) =

1 X

(gi ¡

1):

(3.1)

i=0

(vi) Let L0 be an intermediate ¯eld of L=k and H = Gal(L=L0 ). Then we have Hi = Gi \ H for all i ¶ ¡ 1:

(3.2)

(vii) In (vi), if moreover L0 is a Galois extension over k, then Hi is a normal subgroup of G. Proof. All statements except (vii) are proved in [21, IV]. Here we prove (vii). Since Gi C G by (i) and H C G by our assumption, our claim follows immediately from (vi). For a real number x ¶ ¡ 1, we de¯ne Gx = Gi where i is the least integer greater than or equal to x. We also de¯ne the function ’L=k : [¡ 1; 1) ¡ ! [¡ 1; 1) by Z s dx t = ’L=k (s) = 0 (G 0 : G x ) where (G0 : Gx ) = 1 for all x 2 (¡ 1; 0]. An explicit form of ’L=k is given by ’L=k (s) =

1 (g1 + g2 + ¢ ¢ ¢ + gm + (s ¡ g0

m)gm+1 )

(3.3)

516


for s lying in the interval [m; m + 1] with a positive integer m. Since ’L=k is a strictly increasing function, there exists an inverse function ÃL=k called the Hasse-Herbrand function. The higher rami¯cation groups in the upper numbering are then de¯ned by Gt = GÃL=k (t) or equivalently G’L=k (s) = Gs . The proof of the following lemma is also found in [21, IV]. Lemma 3.2. Let L0 =k be a Galois subextension of L=k and H = Gal(L=L0 ). Then we have the following equalities. Transitivity formulas: Herbrand’s theorem:

’L=k = ’L0 =k ¯ ’L=L0 ; ÃL=k = ÃL=L0 ¯ ÃL0 =k : t

t

0

G (L=k)H=H = G (L =k):

(3.4) (3.5)

We also need the following de¯nition. De¯nition 3.3. We de¯ne L(L=k) = fs 2 [¡ 1; 1) j Gs 6= Gs+" for every " > 0g; U(L=k) = ft 2 [¡ 1; 1) j Gt 6= Gt+" for every " > 0g: We also de¯ne L0 (L=k) = L(L=k) ¡

f0g; U0 (L=k) = U(L=k) ¡

f0g

and l(L=k) = max L(L=k); u(L=k) = max U(L=k): The elements in L(L=k) (resp. U(L=k)) are called the rami¯cation breaks (resp. the upper rami¯cation breaks) of L=k. From the above de¯nition and (3.3), it follows that l(L=k)

u(L=k) =

X gi : g 0 i=1

(3.6)

For our purpose, we are interested in the behavior of the upper rami¯cation breaks in the compositum of ¯elds. The description of the behavior is rather di±cult in general. The following lemma gives a bound for the largest upper break. Lemma 3.4 (Maus [13, (2.12)]). Let K1 and K2 be linearly disjoint ¯nite Galois extensions of a local ¯eld k and K = K1 K2 . Then we have u(K=K1 ) µ ÃK1 =k (u(K2 =k)): For a special case called the arithmetically disjoint case, there is a more precise description due also to Maus.


517

Lemma 3.5 (Maus [13] (see also [28])). Let K1 and K2 be ¯nite totally and wildly rami¯ed linearly disjoint Galois extensions of a local ¯eld k and K = K1 K2 . Assume that U(K1 =k) \ U(K2 =k) = ;. Then we have U(K=K1 ) = fÃK1 =k (t) j t 2 U(K2 =k)g; U(K=K2 ) = fÃK2 =k (t) j t 2 U(K1 =k)g: If the assumption is satis¯ed, then K1 and K2 are said to be arithmetically disjoint over k. The following lemma is also useful to determine the rami¯cation breaks. Lemma 3.6. Let K be a composite ¯eld of two linearly disjoint wildly rami¯ed cyclic extensions of degree n over a local ¯eld k. Let F be any sub¯eld of K of degree n over k. (i) Let K1 and K2 be di®erent sub¯elds of K of degree n over k. If u = u(K1 =k) = u(K2 =k) holds, then we have u(F=k) µ u. (ii) Suppose that there is a higher rami¯cation group of Gal(K=k) of order n. Let F 0 be the ¯xed ¯eld of the group. Then we have vK (DK=F 0 ) ¶ vK (DK=F ): The equality holds if and only if F = F 0 . (iii) Suppose that there is no higher rami¯cation group of Gal(K=k) of order n. Let u be the unique rami¯cation break in K=k. Then we have u(K=F ) = u(F=k) = u: Proof. (i) Suppose to the contrary that u(F =k) > u. Then F and K1 are arithmetically disjoint over k and by Lemma 3.5 we have u(K=K1 ) = ÃK1 =k (u(F =k)) = u + n(u(F =k) ¡

u) > u:

On the other hand, from Lemma 3.4, it follows u(K=K1 ) µ ÃK1 =k (u(K2 =k)) = u. This is a contradiction. (ii) By our assumption, there are two rami¯cation breaks in K=k. Let L(K=k) = fl1 ; l2 g and assume l1 < l2 . If F = F 0 , then the desired equality obviously holds. If F 6= F 0 , then from Lemma 3.1 (v) and (vi), we have vK (DK=F 0 ) = (n ¡

1)(l2 + 1) > (n ¡

1)(l1 + 1) = vK (DK=F ):

(iii) This part follows immediately from Lemma 3.1 (iv) and (3.5). We now consider the `-division ¯eld of an elliptic curve. Let E be an elliptic curve de¯ned over k. For a prime number ` di®erent from p, let E[`] be the kernel of the ¡ ¢ multiplication-by-p map: E[`] = ker [`] : E(Qp ) ¡ ! E(Qp ) . The `-division ¯eld D =

518


k(E[`]) » Qp is a Galois extension over k. We denote the Galois group Gal(D=k) by G(`). De¯nition 3.7. Let G = G(`). The wild part of the conductor of E is de¯ned by 1 X ¡ ¢ gi ±(E=k) = dimZ=`Z E[`]=E[`]Gi : g i=1 0

It is known that ±(E=k) is an integer and independent of the choice of ` (see [25, IV.10.2(c)]). By Lemma 3.1 (ii), the sum is indeed a ¯nite sum and using l(k(E[`])=k) in De¯nition 3.3, we can express it as a ¯nite sum explicitly: l(k(E[`])=k)

±(E=k) =

X i=1

¡ ¢ gi dimZ=`Z E[`]=E[`]Gi : g0

(3.7)

In particular, if k(E[`])=k is tamely rami¯ed, then ±(E=k) = 0 . We shall further rewrite it using u(k(E[`])=k) for later use. Proposition 3.8. Let p = 2 or 3 and k a local ¯eld containing Qp . For an elliptic curve E de¯ned over k, let D = k(E[`]) where ` = 3, if p = 2 and ` = 2 otherwise. Then we have ±(E=k) = 2u(D=k): (3.8) By taking account of (3.6) and (3.7), our formula (3.8) immediately follows from the following lemma. Lemma 3.9. Under the same notation and assumptions of Proposition 3.8, let Gi (i ¶ 1) be a non-trivial i-th higher rami¯cation group of G(3) = Gal(D=k). Then the Gi -invariant subspace of E[`] is trivial, namely we have dimZ=`Z E[`]Gi = 0: Proof. We shall prove the case where p = 2 and ` = 3. The case p = 3 can be proved in a similar manner with a simpler argument. Since E[3] is a free Z=3Z-module of rank 2, the Galois group G(3) is isomorphic to a subgroup of GL2 (Z=3Z). Suppose that E[3]Gi 6= 0. By choosing an appropriate basis of E[3], we may assume that 8 2 39 2 3 2 3 * + > > < 1¤ = 6 7 61 1 7 61 1 7 Gi » 4 5 = 4 5 ; 4 5 > : 0¤ > ; 01 02 and consider G(3) as a subgroup of GL2 (Z=3Z) with respect to this basis. Basic properties of the Weil paring [24, III.8.1] imply that D contains the ¯eld of third roots of unity


519

p k( ¡ 3), which is unrami¯ed over k and that the elements in GL2 (Z=3Z) operate on p the third roots of unity by the determinant. Therefore k( ¡ 3) corresponds to G(3) \ SL2 (Z=3Z). By the assumption i ¶ 1, it follows that 82 > < 1 B 6 Gi » @G(3) \ SL2 (Z=3Z) \ 4 > : 0 0

2

61 where 4 0

3

39 1 2 * > ¤ 7= C 61 5 A = G(3) \ 4 > ¤ ; 0

3

17 5 1

+

;

17 5 is of order 3 in GL2 (Z=3Z). On the other hand, from Lemma 3.1 (iii), the 1

order of Gi is a power of 2. Thus it follows Gi = 1. This contradicts our assumption.

4

Variation of the conductor

Let E be an elliptic curve de¯ned over a local ¯eld k. In this section, we study the variation of the exponent of the conductor of E under base change. The exponent f (E=k) of the conductor is de¯ned by the sum of the tame part "(E=k) and the wild part ±(E=k): f (E=k) = "(E=k) + ±(E=k): Here the tame part "(E=k) is 0; 1 or 2 according as E has good, multiplicative or additive reduction, respectively (see [25, IV.10.2]). Therefore the variation of "(E=k) under base change relates to the problem of potential good or multiplicative reduction. Kraus [10] studies the problem of potential good reduction and Papadopoulos [18] obtained conditions on potential multiplicative reduction. We will mention their results later in this section. The wild part is subtler to deal with. For example, let us consider the case p = 2 or 3. By Proposition 3.8, giving the description of the variation of ±(E=k) under base change is equivalent to computing the largest upper rami¯cation break of some composite ¯eld. This is a di±cult problem in algebraic number theory and no simple general method of computing it is known. We solve this problem for certain division ¯elds of elliptic curves in this section. We begin with the tame case.

4.1 Tamely rami¯ed extension Let k be a ¯nite extension of Qp and E an elliptic curve de¯ned over k. Let K=k be a ¯nite extension. We want to compute ±(E=K). If K or a certain division ¯eld of E is unrami¯ed or tamely rami¯ed over k, this is easily accomplished by the following proposition.

520


Proposition 4.1. Let E be an elliptic curve de¯ned over a local ¯eld k whose residue ¯eld is of characteristic p > 0. Let D = k(E[`]) be the `-division ¯eld of E where ` is a prime number di®erent from p. Let K be a ¯nite extension of k. (i) If D=k is unrami¯ed or tamely rami¯ed, then ±(E=K) = ±(E=k) = 0: (ii) If K=k is unrami¯ed or tamely rami¯ed, then ±(E=K) = e(K=k) ¢ ±(E=k): Our proof of the proposition is based on the following two general lemmas. Lemma 4.2. Let K1 and K2 be linearly disjoint Galois extensions of a local ¯eld k. Let p be the characteristic of the residue ¯eld of k. Assume that K1 =k is unrami¯ed or tamely rami¯ed. Then we have L0 (K1 K2 =K1 ) = e(K1 =k) ¢ L0 (K2 =k); 0

(4.1)

0

U (K1 K2 =K1 ) = e(K1 =k) ¢ U (K2 =k):

(4.2)

Proof. We write e = e(K1 =k) for short in this proof. If K2 =k is also at most tamely rami¯ed, then so is K1 K2 =K1 . Therefore the result trivially holds. Thus we may assume that K2 =k is wildly rami¯ed. Let H = Gal(K1 K2 =K2 ). Then by (3.5), we have Gt(K1 K2 =k)H=H = Gt (K2 =k) for all t ¶ ¡ 1. Since K1 K2 =K2 is at most tamely rami¯ed, the order of H is prime to p. It follows from Lemma 3.1 (iii) that #Gt (K1 K2 =k) = #Gt (K2 =k) for t ¶ 1. In particular, we obtain U0 (K1 K2 =k) = U0 (K2 =k): (4.3) By the de¯nition of the upper numbering, we have Gt (K1 K2 =k) = Gs (K1 K2 =k) with s = ÃK1 K2 =k (t) for all t ¶ ¡ 1. Since K1 K2 =K2 is at most tamely rami¯ed, Gs (K1 K2 =k) is contained in Gal(K1 K2 =K1 ) if s ¶ 1. By Lemma 3.1 (iv), it yields Gs (K1 K2 =K1 ) = Gs (K1 K2 =k) for s ¶ 1. Thus we obtain L0 (K1 K2 =K1 ) = L0 (K1 K2 =k):

(4.4)

Now we compute L0 (K1 K2 =K1 ) = L0 (K1 K2 =k)

by (4.4) 0

= fÃK1 K2 =k (u) j u 2 U (K1 K2 =k)g = fÃK1 K2 =K2 (ÃK2 =k (u)) j u 2 U0 (K1 K2 =k)g 0

= fÃK1 K2 =K2 (ÃK2 =k (u)) j u 2 U (K2 =k)g 0

0

0

= fÃK1 K2 =K2 (l ) j l 2 L (K2 =k)g = fe ¢ l 0 j l0 2 L0 (K2 =k)g:

by (3.4) by (4.3)


521

Here the last equality follows from (3.3) and g0 (K1 K2 =K2 ) = e; gi (K1 K2 =K2 ) = 1 for i ¶ 1 ; since K1 K2 =K2 is at most tamely rami¯ed. Similarly we obtain U0 (K2 =k) = U0 (K1 K2 =k)

by (4.3)

= f’K1 K2 =k (l) j l 2 L0 (K1 K2 =k)g = f’K1 =k (’K1 K2 =K1 (l)) j l 2 L0 (K1 K2 =k)g 0

= f’K1 =k (’K1 K2 =K1 (l)) j l 2 L (K1 K2 =K1 )g 0

by (3.4) by (4.4)

0

= f’K1 =k (u ) j u 2 U(K1 K2 =K1 )g ½ ¾ 1 0 0 ¢ u j u 2 U(K1 K2 =K1 ) : = e This completes the proof of the lemma. Lemma 4.3. Let K1 =k be a Galois extension of local ¯elds. Let K2 be an intermediate ¯eld of K1 =k. We assume that K2 =k is unrami¯ed or tamely rami¯ed. Then g0 (K1 =k) = e(K2 =k)g0 (K1 =K2 ) and gi (K1 =k) = gi (K1 =K2 ) for i ¶ 1: In particular, we have u(K1 =K2 ) = e(K2 =k)u(K1 =k):

(4.5)

Proof. The formula for g0 readily follows from the transitive formula for the rami¯cation indices. Since K2 =k is at most tamely rami¯ed, we have Gi (K1 =k) » Gi (K1 =K2 ). Thus the formula for gi follows from (3.2). Substituting these formulas into (3.6), it is easy to obtain (4.5). Now we can give a proof of our proposition. Proof (of Proposition 4.1). If D=k is unrami¯ed or tamely rami¯ed, then so is DK=K. This means ±(E=K) = 0. This is the ¯rst assertion. We shall show the second half. If p ¶ 5, the wild part ±(E=k) is always 0 by [25, IV, Theorem 10.2 (b)]. Therefore we assume p = 2 or 3. Since K=k is at most tamely rami¯ed, so is the subextension K \ D over k. By Lemma 3.1 (iii), we have L(D=k) = L(D=K \ D). Applying (4.5) to D » (K \ D) » K and using (3.8), we obtain ±(E=D \ K) = e(D \ K=K)±(E=K):

(4.6)

On the other hand, by using Lemma 4.2 over D \ K, it follows from (3.8) ±(E=K) = e(K=D \ K)±(E=k): Combining (4.6) and (4.7), we complete the proof.

(4.7)

522


Note again that the condition of the ¯rst part of Proposition 4.1 can be stated in terms of the reduction type Type(E) (see Proposition 2.2). If p ¶ 5, the variation of the tame part "(E=k) is well-known. Namely we have the following proposition. Proposition 4.4. Let E=k be an elliptic curve with additive reduction. Assume that the residue characteristic is greater than 3. Let K be a ¯nite extension of k. Then the following equivalences hold: (i) "(E=K) = 0 if and only if e(K=k) divides the denominator of vk (¢)=12 where ¢ is the minimal discriminant of E. (ii) "(E=K) = 1 if and only if 2 divides e(K=k) and Type(E) = I¤n for some n ¶ 1. For the proof of the ¯rst part of the proposition, see, for example, [10, Proposition 1]. The second part follows from the theory of Tate curves [25, V]. Alternatively, we can verify this proposition by case-by-case computation. Actually it is well-known that if p ¶ 5, the reduction types are determined only by the triple of the valuations of c4 ; c6 and ¢ associated to a minimal model of E (for example, see [18, Tableau 1]). And the minimality of a model can be checked by the condition v(¢) < 12 or v(c4 ) < 4 (see [24, VII, 1.1]). In contrast with this, the situation is much more complicated in the cases of p = 2 and 3. We will study these cases in the remainder of this section assuming that the base change is of small degree over Qp .

4.2 Cubic base change over Q3 In this section, let E be an elliptic curve de¯ned over Q3 and K=Q3 a totally rami¯ed cubic extension. We want to compute the wild part ±(E=K). For this purpose, we take ` = 2 and consider the 2-division ¯eld D = Q3 (E[2]) of E, which is the splitting ¯eld of the 2-division polynomial ©2 (X) = 4X 3 + b2 X 2 + 2b4 X + b6 :

(4.8)

We have an injection G(2) = Gal(D=Q3 ) ,! GL2 (Z=2Z) ¹= S3 . Since we have settled the case where D=Q3 is tamely rami¯ed, we assume that E has additive reduction, therefore "(E=Q3 ) = 2 and that the degree of D over Q3 is divisible by 3. The following proposition asserts that the wild part ±(E=Q3 ) determines the rami¯cation breaks in D=Q3 . Proposition 4.5. Let Ci denote the type of totally rami¯ed cubic ¯elds over Q3 de¯ned in Lemma 2.3. Let E=Q3 be an elliptic curve with additive reduction and M the cubic


523

¯eld de¯ned by the 2-division polynomial (4.8) of E and D the splitting ¯eld of ©2 . Assume that M=Q3 is totally rami¯ed. Then we have the following equivalences: ½

¾ 1 () M 2 C1 ; ±(E=Q3 ) = 1 () U(D=Q3 ) = 0; 2 ±(E=Q3 ) = 2 () U(D=Q3 ) = f1g () M 2 C2 ; ½ ¾ 3 () M 2 C3 : ±(E=Q3 ) = 3 () U(D=Q3 ) = 0; 2 Proof. We begin the proof with several remarks. Since we assume D=Q3 is wildly rami¯ed, ±(E=Q3 ) ¶ 1 always holds. By Proposition 2.2 and (2.5), it is veri¯ed that mQ3 in the Ogg’s formula (2.4) is always odd. Thus the parity of f (E=Q3 ) and v3 (¢E ) coincides. Further U(D=Q3 ) contains 0 if and only if the extension contains a non-trivial tamely rami¯ed subextension. Thus this happens if and only if 2 divides e(D=Q3 ). Since D p contains Q3 ( ¢E ), it implies that 2 divides e(D=Q3 ) if and only if f (E=Q3 ) is odd. Now in each equivalence, the left equivalence follows from (3.8). For the right equivalence, the transitivity law of the di®erent DD=Q3 = DD=M D M=Q3 implies

vD (D D=Q3 ) = e(D=M ) ¡

1 + e(D=M )vM (D M=Q3 );

since D=M is tame. Knowing U(D=Q3 ), we can compute vD (DD=Q3 ) by (3.1) and thus vM (D M=Q3 ). Conversely, if we know vD (D D=Q3 ) from the above formula, we can recover U(D=Q3 ) since there is at most one rami¯cation break other than 0. This completes the proof. Before computing ±(E=K), we ¯x the names of various ¯elds appearing in the computation and use this notation until the end of Section 4.2. K

a totally rami¯ed cubic extension of Q3 .

b K

the Galois closure of K over Q3 .

the 2-division ¯eld of E.

p e = Q3 ( dK ) K D = Q3 (E[2])

b where dK is the discriminant of K. a sub¯eld of K

M

a cubic sub¯eld of D de¯ned by ©2 (X).

p F = Q3 ( ¢E ) a sub¯eld of D where ¢E is the minimal discriminant of E. The variation of the wild part is given by the following theorem. Theorem 4.6. Let E=Q3 be an elliptic curve with additive reduction. We assume that D=Q3 is wildly rami¯ed. Then we have ±(E=Q3 ) = 1; 2 or 3 and if K 2 Ci , then the wild

524


part of the conductor ±(E=K) is given by the following. ±(E=Q3 )

C1

C2

C3

1

0 or 1

1

1

2

4

0 or 2

2

3

7

5

0 or 1

In the cases where ±(E=K) is not determined only by the type Ci , we have ±(E=K) = 0 if and only if K is isomorphic to M over Q3 . Remark 4.7. Let f be a polynomial over Q3 de¯ning the ¯eld K. To verify if K and M are isomorphic, it is enough to examine whether f has a root in M . This can be done by either the polynomial factorization algorithm due to Pauli [19] or Panayi’s root ¯nding algorithm [20, Section 8]. Moreover if ±(E=Q3 ) = 1 or 2, then we have a direct criterion for ±(E=K) = 0. b contain Namely in the case of ±(E=Q3 ) = 1, we have ±(E=K) = 0 if and only if D and K e Also in the case of ±(E=Q3 ) = 2, we have the same quadratic sub¯eld, that is F = K. ±(E=K) = 0 if and only if the order of Gal(D=Q3 ) agrees with the order of the Galois b over Q3 . group of K These criteria are also shown in the following proof. Proof. In this proof, we use (4.2) and (4.5) repeatedly without mentioning them explicitly. We want to calculate ±(E=K) = 2u(KD=K) (see (3.8)). For this purpose, it is enough b b ), because they are related by to know u(KD= KF b KF b ) = e(KF b =K)u( b KD= b b u(KD= K) b =K)e( b K=K)u(KD=K) b = e(KF

We ¯rst compute

e =Q3 )u(KD=K): = e(KF

b KF e ) = e(K e F=K)u( e K= b K) e u(KF= e F=K)e( e K=Q e 3 )u(K=Q b 3) = e(K Similarly we get

e F=Q3 )u(K=Q b 3 ): = e(K

e KF e ) = e(KF e =Q3 )u(D=Q3 ): u(D K=

b F=KF e and D K= e KF e are totally and wildly rami¯ed cyclic cubic Since the extensions K extensions, their U’s consist of only one upper break u. Therefore, these two equalities


525

b and D K e are arithmetically disjoint over KF e if and only if u(K=Q b 3 ) 6= show that KF b b ) can be computed by u(D=Q3 ). If this is the case, by virtue of Lemma 3.5, u(KD= KF b b ) = Ã b e (u(D K= e KF e )) u(KD= KF KF =KF e =Q3 )u(D=Q3 )): = Ã b e (e(KF KF =KF

Calculating the Hasse-Herbrand function explicitly, we obtain u(DK=K) =

(

u(D=Q3 ) 3u(D=Q3 ) ¡

b 3) 2u(K=Q

b 3 ); if u(D=Q3 ) µ u(K=Q otherwise.

By using Proposition 4.5, these values can be computed easily. b 3 ) = u(D=Q3 ). We need the following The remaining cases are those with u(K=Q lemma on the number of cubic extensions over Q3 . Lemma 4.8. Let Ci (i = 1; 2; 3) be the classes of totally rami¯ed cubic extensions over b the Galois closure of K Q3 de¯ned in Lemma 2.3. For a cubic ¯eld K, we denote by K over Q3 . (i) We have p p b ¼ Q3 ( ¡ 3)g = #fK 2 C1 j K b ¼ Q3 ( 3)g = 3: #fK 2 C1 j K

All ¯elds in each set are isomorphic over Q3 . (ii) We have b = Kg = #fK 2 C2 j K b = 6 Kg = 3: #fK 2 C2 j K

The three ¯elds in the latter set are isomorphic over Q3 . (iii) We have #C3 = 9. The nine ¯elds in C are divided into three isomorphism classes over Q3 . Proof. A classical formula due to Krasner (see [20, Theorem 6.1]) enables us to compute the number of local ¯elds with given degree and discriminant. By the formula, we obtain #C1 = 6, #C2 = 6 and #C3 = 9. Further, there are three cubic cyclic ¯elds by the Galois extension counting formula by Yamagishi [27]. These cyclic ¯elds belong to C2 by Lemma 2.3. The Galois groups of other ¯elds are isomorphic to S3 . For those ¯elds having S3 Galois group, three cubic sub¯elds are isomorphic over Q3 . It remains to prove (i). For that purpose, we have only to ¯nd a polynomial that generates ¯elds in each set. Let f1 (X) = X 3 + 3X 2 + 6X + 3 and f2 (X) = X 3 + 3X 2 + 3X + 3. Then these polynomials satisfy our requirement. In fact, the discriminants of 2 these polynomials are 33 5 and 22 33 respectively. Noting that ¡ 5 2 (Q£ 3 ) , we ¯nd that the p splitting ¯eld of f1 (X ) contains Q3 ( ¡ 3), whereas the splitting ¯eld of f2 (X) obviously p contains Q3 ( 3). The polynomials f1 and f2 in the above proof are found by the algorithm in [20].

526


Now we return to the proof of Theorem 4.6. We continue to use the notation introduced just before the statement of the theorem. First note that if K is isomorphic to M , then we have DK = D and the extension DK=K is at most tamely rami¯ed and we thus have ±(E=K) = 0. b 3 ) = u(D=Q3 ) = 1=2. Namely, by Proposition 4.5, K Next we assume that u(K=Q and M belong top C1 . From Lemma 2.3, it follows that their Galois closures contain either p ¡ Q3 ( 3) or Q3 ( 3). e = F . If D 6= K, b then there are four cubic extensions over F Suppose ¯rst that K b contained in KD. They are all Galois extensions over Q3 with Galois group isomorphic e Then it follows from Lemma 3.6 (i) to S3 . Suppose that all these ¯elds ramify over K. that all these ¯elds must belong to C1 . This contradicts Lemma 4.8 (i). Thus one of the e This implies that ±(E=K) = 0. If D = K, b then M and K are ¯elds is unrami¯ed over K. isomorphic by Lemma 4.8 (i). Thus by the above remark, we have ±(E=K) = 0 again. e 6= F . In this case, we consider the rami¯cation in KD= b KF e . We next assume that K e contained in KD b that is di®erent from KF b and Let L be a cubic extension over KF e KD. Then it can be observed that L=Q2 is not a Galois extension over Q3 . Therefore, by Proposition 2.1 (vii), if there exists a higher rami¯cation group of order 3, then it b or KD. e corresponds to either KF But since it is easy to see that vKD b ) = b (D KD= b KF vKD e ), neither of them can correspond to a higher rami¯cation group by Lemma b (DKD= b KD b KF e . Since we have 3.6 (ii). Hence there exists only one rami¯cation break in KD= b KF b ) = 1. u(D=F ) = 1 by Lemma 4.2, it follows from Lemma 3.6 (iii) that u(KD= Consequently, we obtain u(DK=K) = 1=2 and ±(E=K) = 1. b 3 ) = u(D=Q3 ) = 1. By Proposition 4.5, K We next consider the case where u(K=Q b and D are of the same degree, then these two ¯elds must be and M belong to C2 . If K isomorphic over Q3 by Lemma 4.8 (ii). Hence ±(E=K) = 0. Assume now that one of K and M is normal over Q3 and the other is not. Since the argument proceeds symmetrically, we assume that K=Q3 is a cyclic extension. In this case, we consider the rami¯cation of DK=F . We notice that any cubic extension L over F di®erent from KF and D is not normal over Q3 . Thus, as before, there is only one rami¯cation break in DK=F . We compute u(DK=F ) = 1 and ±(E=K) = 2u(DK=K) = 2 by Lemma 3.6 (iii). b 3 ) = u(D=Q3 ) = 3=2. By Proposition 4.5, We ¯nally consider the case where u(K=Q p b and D have a common quadratic sub¯eld F = Q3 ( ¡ 3) K and M belong to C3 . Then K by Lemma 2.3. We may assume that K and M are not isomorphic. Let Li (i = 1; 2) be b which are di®erent from K b and D. It is readily cubic extensions of F contained in KD seen that each Li =Q3 is a Galois extension with Galois group isomorphic to S3 . Let ki be a cubic sub¯eld of Li . Suppose that both k1 and k2 also belong to C3 . Then at least two of k1 ; k2 ; K; M are in the same isomorphism class by Lemma 4.8 (iii). This is impossible since their Galois closures are all di®erent. Therefore we may assume k1 62 C3 . Taking b are the quadratic sub¯eld of L1 into consideration, we ¯nd k1 2 C1 . Then L1 and K b K) b = 1. It yields u(KD=K) = 1=2 arithmetically disjoint over F and we obtain u(KD= by Lemma 4.2. Thus we conclude ±(E=K) = 1. This completes the proof of Theorem 4.6.


527

Now we mention the tame part. By the theory of Tate curves, elliptic curves with non-integral j-invariant attain multiplicative reduction in a quadratic extension (see [25, Exercise 5.11]). Hence we have only to consider the case of potential good reduction. The following result follows from [10, p.362 Corollary]. Proposition 4.9. An elliptic curve E=Q3 having additive reduction attains good reduction over K if and only if D=Q3 is a cyclic cubic extension and v3 (¢E ) ² 0 (mod 4). Looking up Table 2 in [18], we ¯nd that the conditions in the above proposition hold for the cases Type(E) = II; v3 (c4 ) = 2; v3 (c6 ) = 3; f (E=Q3 ) = 4; Type(E) = II¤ ; v3 (c4 ) = 5; v3 (c6 ) = 8; f (E=Q3 ) = 12: If E attains good reduction over a cubic ¯eld K, then we have "(E=K) = 0. And this exactly happens when ±(E=K) = 0. This completes the proof on the assertion on the conductor in Theorem 2.4. and we are able to ¯ll out f (E=K) in Table 2.

4.3 Quadratic base change over Q2 Let E be an elliptic curve de¯ed over Q2 . We study the variation of the wild part of the conductor in rami¯ed quadratic extensions of Q2 . We take ` = 3 and consider the 3-division ¯eld D = Q2 (E[3]). There is a faithful representation G(3) = Gal(D=Q2 ) ,! GL2 (Z=3Z). Hence G(3) can be considered as a subgroup of GL2 (Z=3Z), which is of p order 48. Also the ¯eld D contains the ¯eld of third roots of unity Q2 ( ¡ 3). Since the elements of GL2 (Z=3Z) acts on the the third roots of unity by the determinant, the inclusion G(3) » SL2 (Z=3Z) never happens. The transitive subgroups of GL2 (Z=3Z) that satisfy these conditions are isomorphic to one of the following eight abstract groups: GL2 (Z=3Z)

the general linear group of order 48,

SD16

the semi-dihedral group of order 16,

Dm

the dihedral group of order m = 12; 8,

Cm

the cyclic group of order m = 8; 2,

S3

the symmetric group on three letters of order 6,

E4 the elementary abelian group of order 4. Therefore G(3) is isomorphic to one of the above groups. Remark 4.10. The subgroups of GL2 (Z=3Z) that appear as a Galois group of the 3division ¯eld of an elliptic curves over local ¯elds are determined by Naito [16] (see also

528


the review of [15]). Bayer and Rio [2] determined all Galois extensions over Q2 whose Galois groups are isomorphic to these subgroups of GL2 (Z=3Z). We use a part of their results later. We shall show the following proposition corresponding to Proposition 4.5. Proposition 4.11. Let E be an elliptic curve de¯ned over Q2 with additive reduction. Assume that the 3-division ¯eld D is wildly rami¯ed over Q2 . Let M be the ¯xed ¯eld of the ¯rst rami¯cation group G1 (D=Q2 ). Then we have the following correspondence of the conductor and the Galois group. U(D=M ) ½ ¾ 3 1; 2

±(E=Q2 )

Gal(D=Q2 )

Gal(D=M )

1

GL2 (Z=3Z)

Q8

2

GL2 (Z=3Z)

Q8

f1; 3g

D12

C2

f3g

C8

C2

f1g

E4

C2

3

SD16

Q8

f1g ½ ¾ 3 1; 2

4

GL2 (Z=3Z)

Q8

f1; 6g

D12

C2

f6g

SD16

Q8

f1; 2g

C8

C2

f2g

E4

C2

5

GL2 (Z=3Z)

Q8

f2g ½ ¾ 15 5; 2

6

SD16

Q8

f1; 2; 3g

C8

C4

f2; 3g

D8

C4

f2; 3g

In the third column, the isomorphism class of the Galois group Gal(D=M ) is noted, where Ci denotes the cyclic group of order i and Q8 denotes the quaternion group of order 8. For the proof of this proposition, we need the following lemmas. We do not intend to consider the full generality of these lemmas. The ¯rst lemma giving some restriction on the upper rami¯cation breaks is due to Fontaine.


529

Lemma 4.12 (Fontaine [5, Section4]). Let k be a local ¯eld containing Q2 and e = e(k=Q2 ) the absolute rami¯cation index. Let L=k be a totally rami¯ed Galois extension with Galois group G. (i) Suppose that G is isomorphic to an elementary abelian group (Z=2Z)©m . Let u 2 U(L=k). Then either of the following holds. (a) 0 < u < 2e and u 6² 0 (mod 2). (b) u = 2e. Moreover, we have #(G2e ) µ 2. (ii) Suppose that G is isomorphic to a cyclic group of order 2m . Let u1 < u2 < : : : < um be the elements of U(L=k). Then either of the following holds. (a) u1 = 2e or 0 < u1 < 2e and u1 6² 0 (mod p). (b) for j = 1; 2; : : : ; m ¡ 1, (1) if uj ¶ e, then uj+1 = uj + e. (2) if uj < e, then uj+1 = 2uj or uj+1 = 2e or 2uj < uj+1 < 2e and uj+1 6² 0 (mod 2). (iii) Suppose that G is isomorphic to Q8 . Then either of the following holds. (a) All the elements in U(L=k) are integers. ½ ¾ 3 (b) There exists an odd integer u such that U(L=k) = u; u . If this is 2 the case, we have G = Gu and Gu+1 is the center of G. Here we recall that the famous Hasse-Arf theorem ([21, V, Theorem 1]) claims that, if G is an abelian group, then U(L=k) » Z. The next lemma gives upper bounds for the exponent of the di®erent. Lemma 4.13. Let k be a local ¯eld containing Q2 . Let L=k be a Galois extension with Galois group G. Then we have vL (D L=k ) µ vL (e(L=k)) + e(L=k) ¡

1:

(4.9)

Further if G is isomorphic to Q8 , then 5 vL (D L=k ) µ e(L=Q2 ) + 4: 2

(4.10)

Proof. The inequality (4.9) follows from [21, Remark 1 following III, Proposition 13]. The latter equality is proven in [12, Proposition 6.1]. We can now give the proof of Proposition 4.11. Proof (of Proposition 4.11). We compute the possible rami¯cation types for each isomorphism class of the Galois group G(3) = Gal(D=Q2 ). We ¯rst show that G(3) is isomorphic to neither C2 nor S3 . Suppose to the contrary that G(3) is isomorphic to one of these groups. Then the extension D=Q2 is unrami¯ed p or tamely rami¯ed, because D ¼ Q2 ( ¡ 3). This is irrelevant to our assumption.

530


p p Since the 3-division ¯eld D contains F = Q2 ( 3 ¢E ; ¡ 3), it is easy to see that 3 divides e(D=Q2 ), equivalently the degree [D : Q2 ], if and only if v2 (¢E ) 6² 0 (mod 3). (A) The case G(3) ¹= GL2 (Z=3Z). It is known that the normal subgroups of GL2 (Z=3Z) are 2 3 2 3 * + 6 2 17 6 0 17 GL2 (Z=3Z) ¼ SL2 (Z=3Z) ¼ Q = 4 5 ; 4 5 ¼ (Z=3Z)£ ¼ 1 11 20 (see, for example, [1, Chapter 5]). The group Q is isomorphic to Q8 . The corresponding ¯elds are p Q2 » Q2 ( ¡ 3) » F » Dx » D;

where Dx = Q2 (x(E[3])) is the splitting ¯eld of ©3 (X ) over Q2 . It is readily seen that p the inertia ¯eld is Q2 ( ¡ 3). We also observe that D=F is a wildly and totally rami¯ed extension whose Galois group is isomorphic to Q8 . We thus have M = F . It is easily shown that Dx =F is a Galois extension whose Galois group is isomorphic to (Z=2Z)©2 . Since there is no normal subgroup of G(3) between Q and (Z=3Z)£ , there cannot be higher rami¯cation groups of Gal(D=F ) of order 4 by Lemma 3.1 (vii). This implies that #U(D=F ) = 2. Let U(D=F ) = fu1 ; u2 g. It follows from Lemma 4.12 (i) that the unique upper break u(Dx=F ) is less than 6. Applying Herbrand’s theorem (3.5) to this situation, we obtain u1 = 1; 3 or 5: (4.11) Using (3.1), we compute vD (D D=F ) = 7 + 7u1 + 4(u2 ¡

u1 ) µ 64;

(4.12)

where the last inequality follows from (4.10). By Lemma 3.1 (vi), the unique upper break in D=Dx is l(D=F ) = u1 + 4(u2 ¡ u1 ). Thus it follows from (4.9) that vD (DD=Dx ) = 1 + u1 + 4(u2 ¡

u1 ) µ 25:

(4.13)

By Proposition 4.12 (iii), we have 3 U(D=F ) » Z or u2 = u1 : 2

(4.14)

Furthermore, using Proposition 3.8 and Lemma 4.3, we obtain 2 ±(E=Q2 ) = 2u(D=Q2 ) = u2 2 Z: 3

(4.15)

To restrict the possibility further, we need the following result due to Weil. Lemma 4.14. (Weil [26]) There are eight Galois extensions over Q2 whose Galois groups are isomorphic to GL2 (Z=3Z). They are obtained as the 3-division ¯elds of the following elliptic curves de¯ned over Q2 : Wr(1) : ry 2 = x3 + 3x + 2;

r = §1; §2;

Wr(2) : ry 2 = x3 ¡

r = §1; §2:

3x + 1;


531

Applying Tate’s algorithm over Q2 , we compute ¡

¢ (1)

± Wr

¡

¢ (2)

= 5 (r = §1; §2); ± Wr

=

8 > > > :4

(4.16)

if r = §2:

With (4.15), this computation yields a restriction on u2 . Namely we have u2 =

15 3 ; 3; or 6: 2 2

(4.17)

By taking the obvious inequality u1 < u2 into account and using (4.11), (4.12), (4.13), (4.14), (4.15) and (4.17), the possible pairs for (u1 ; u2 ) are µ ¶ µ ¶ 3 15 (u1 ; u2 ) = (1; 3); (1; 6); (3; 6); 1; ; 5; : 2 2

(4.18)

We now show that, when ±(E=Q2 ) = 4, only (u1 ; u2 ) = (1; 6) is possible. In this (2) case, we have D = Q2 (W§2 [3]) by (4.16). We compute the valuation of D Dx =Q2 in two di®erent ways. Let T be a quartic ¯eld de¯ned by the 3-division polynomial ©3 (see (2.3)). By looking at the corresponding Galois group, it appears that T does p not contain Q2 ( ¡ 3). Thus T =Q2 is totally rami¯ed and Dx =T is tamely rami¯ed. Therefore we have vDx (DDx =Q2 ) = vDx (D Dx =T ) + vDx (D T =Q2 ) = 2 + 3vT (DT =Q2 ): (2)

Since the 3-division polynomial of W§2 is 3X 4 ¡ 18X 2 +12X ¡ 9, we have vT (DT =Q2 ) = 4 by an explicit computation and obtain vDx (DDx =Q2 ) = 14. On the other hand, from (4.12) and (4.13), it follows that 1 vDx (D Dx =Q2 ) = (vD (D D=F ) + vD (D F=Q2 ) ¡ 2 = 11 + 3u1

vD (D D=Dx ))

Therefore the only possible value of u1 is 3. (B) The case G(3) ¹= SD16 . Fixing a basis of E[3], we may consider

G(3) =

*

2

62 a=4 1

3

07 5;b = 1

We have the following subgroup lattice of G(3).

2

61 4 1

3

17 5 0

+

:

532


G(3)

H7

H6

H3

H8

H4

H2

H9

H5

H1 = f1g

Here all two groups joined with a vertex are of index 2. We have H2 = (Z=3Z)£ and H9 = G(3) \ SL2 (Z=3Z). The non-trivial normal subgroups of G(3) are H2 ; H4 ; H7 ; H8 and H9 . The ¯eld corresponding to H9 is the unique unrami¯ed quadratic ¯eld p Q2 ( ¡ 3). Considering the rami¯cation in the biquadratic ¯eld ¯xed by H4 , we ¯nd p p M = Q2 ( ¡ 3). Namely D=Q2 ( ¡ 3) is totally and wildly rami¯ed. It is easy to see that the group H9 = hab; b2 i is isomorphic to Q8 . Let ki be the sub¯eld of D ¯xed by Hi . We now divide into two cases according to the number of the upper breaks. p First assume that U(D=Q2 ( ¡ 3)) = fu1 ; u2 g. Then from (4.10) it follows vD (D D=Q2 (p¡3) ) = 7 + 7u1 + 4(u2 ¡

u1 ) µ 24:

(4.19)

By Lemma 3.1 (vi) and Herbrand’s theorem (3.5), it is easily seen that U(k2 =k9 ) = fu1 g and U(D=k2 ) = fl(D=k9 )g = fu1 + 4(u2 ¡ u1 )g. Applying (4.9) to the extensions k2 =k9 and D=k2 , we obtain vk2 (D k2 =k9 ) = 3 + 3u1 µ 11; vD (DD=k2 ) = 1 + u1 + 4(u2 ¡

u1 ) µ 7: ½

¾ 3 From these three estimates, we have two possibilities: U(D=M ) = 1; and f1; 2g: 2 p Next assume that U(D=Q2 ( ¡ 3)) = fu1 ; u2 ; u3 g. In this case, by (4.10) we see vD (DD=Q2 (p¡3) ) = 7 + 7u1 + 2 ¢ 3(u2 ¡

u1 ) + 4(u3 ¡

u2 ) µ 24:

By noticing that u2 ¡ u1 ¶ 1 and u3 ¡ u2 ¶ 1, it follows that the only possibility is U(D=M ) = f1; 2; 3g. For each case, the wild part is computed by ±(E=Q2 ) = 2u(D=M ) since M=Q2 is unrami¯ed. (C) The case G(3) ¹= D12 . Since there are three subgroups of G(3) of index 2, the ¯eld D contains a biquadratic ¯eld. In particular e(D=Q2 ) is divisible by 2. Moreover G(3) has no normal subgroup of order 4. This implies that there is no cubic Galois sub¯eld in D. Thus e(D=Q2 ) is divisible by 3. Consequently we have e(D=Q2 ) = 6. p p Since D=Q2 ( ¡ 3) is totally rami¯ed, Gal(D=Q2 ( ¡ 3)) cannot be isomorphic to S3 . p Therefore the extension D=Q2 ( ¡ 3) is cyclic. Now let M be the sub¯eld of D that is p of degree 3 over Q2 ( ¡ 3). There is only one integral upper rami¯cation break u(D=M ) in this case. By (4.9), we have vD (D D=M ) = 1 + u(D=M ) µ 7:


533

Also from (4.5) and (3.8), we obtain 2 ±(E=Q) = 2u(D=Q2 ) = u(D=M ); 3 which is an integer. Thus we have u(D=M ) = 3 or 6. (D) The case G(3) ¹= D8 . By ¯xing a basis of E[3], we may consider 2 3 2 3 * + 1 2 2 0 6 7 6 7 G(3) = a = 4 5 ; b = 4 5 : 22 11

The order of a is 4 and that of b is 2. It is obvious that hai » SL2 (Z=3Z). This shows p that the ¯xed ¯eld by hai is Q2 ( ¡ 3). Since D contains a biquadratic ¯eld, it is easy p to see e(D=Q2 ) = 4 and M = Q2 ( ¡ 3). Let U(D=M ) = fu1 ; u2 g. Naito observes in [14] that p p the biquadratic ¯eld contained in D is Q2 ( ¡ 3; ¡ 2) (4.20)

in this situation. Herbrand’s theorem (3.5) implies u1 = 2. We apply Lemma 4.12 (ii) p to the cyclic extension D=M and obtain u2 = u1 + e(Q2 ( ¡ 3)=Q2 ) = 3. (E) The case G(3) ¹= C8 . In this case, the corresponding ¯eld tower is p D ¼ Dx ¼ Q2 ( ¡ 3) ¼ Q2 : There are two cases to be considered depending on the rami¯cation index e(D=Q2 ). Assume ¯rst that e(D=Q2 ) = 2. Then it follows M = Dx . We have only one upper break u(D=M ). Applying (4.9) to D=Dx , we obtain vD (D D=M ) = 1 + u(D=M ) µ 3: This implies u(D=M ) = 1 or 2. p If e(D=Q2 ) = 4, then we have M = Q2 ( ¡ 3). Let u1 and u2 be the upper rami¯cation breaks in D=M . From (4.9), it follows that vD (DD=M ) = 3 + 3u1 + 2(u2 ¡

u1 ) µ 11:

By Lemma 4.12 (ii), we have u2 = u1 + 1. Therefore there are two possibilities: U(D=M ) = f1; 2g and f2; 3g. We shall show the former is impossible. By the local Kronecker-Weber theorem (see [17, Chapter V, (1.9)]), the ¯eld D is contained in a local cyclotomic ¯eld. Considering the degree and the rami¯cation bound above, we see that D is contained in Q2 (³(28 ¡1)24 ). From now on, we denote by Kn the n-th cyclotomic ¯eld Q2 (³n ), where ³n is a primitive n-th root of unity. Let ½ be a generator of Gal(K28 ¡1 =Q2 ) and ¾ and ¿ the generators of Gal(K24 =Q2 ) ¡1 5 such that ¾(³16 ) = ³16 and ¿ (³16 ) = ³16 . Then we have ord(¾) = 4 and ord(¿ ) = 2. Then the Galois group Gal(K(28 ¡1)24 =Q2 ) is canonically isomorphic to h½; ¾; ¿ i. If D is a cyclic extension of degree 8 over Q2 , then an easy computation shows that G = Gal(K(28 ¡1)24 =D) is isomorphic to one of the following groups: If e(D=Q2 ) = 1 then G1 = h¾; ¿ i:

534


If e(D=Q2 ) = 2 then G2 = h½4 ¾; ¿ i; G3 = h½4 ¿; ¾i; G4 = h½4 ¿; ½4 ¾i:

(4.21)

If e(D=Q2 ) = 4 then G5 = h½2 ¾; ¿ i; G6 = h½2 ¾; ½4 ¿ i; G7 = h½6 ¾; ½4 ¿ i; G8 = h½6 ¾; ¿ i:

(4.22)

In particular, it follows that, if e(D=Q2 ) = 4, then D is not contained in K(28 ¡1)23 , which corresponds to h¾ 2 i. Thus the conductor of D=Q2 is 4, thus by [17, V. Satz 6.2] we have u(D=M ) = 3 and U(D=M ) = f2; 3g. (F) The case G(3) ¹= E4 . The ¯eld D is a biquadratic ¯eld over Q2 containing the unrami¯ed quadratic ¯eld, which clearly coincides with M . Thus the unique rami¯cation break in D=M is determined by the rami¯ed quadratic ¯elds contained in D. Therefore we have u(D=M ) = 1 or 2. Remark 4.15. By the part (A) of the above proof, we can compute the discriminant of (i) the 3-division ¯elds of Weil’s curves Wr in Lemma 4.14using v2 (the discriminant of D=Q2 ) = 2vD (D D=Q2 ) = 2f(7 + 7u1 + 4(u2 ¡

u1 )) + 16)g

by (4.12)

= 23 + 3u1 + 4u2 : This recovers the computation in [2], where the details are omitted. The following theorem describes the variation of the wild part of the conductor in quadratic extensions. Theorem 4.16. Let E=Q2 be an elliptic curve with additive reduction. We assume that the 3-division ¯eld D of E is wildly rami¯ed over Q2 . We write G(3) for Gal(D=Q2 ) as before. Let Qi be the classi¯cation of the quadratic ¯elds over Q2 given in (2.1). If K 2 Qi , then the wild part of the conductor ±(E=K) is given by the following. ±(E=Q2 )

G(3)

Q1

Q2

Q3

1

1

1

GL2 (Z=3Z)

1

2

2

otherwise

0

2

2

4

3 8 > :1

1 2


535

±(E=Q2 )

G(3)

Q1

Q2

Q3

4

SD 16

6

3

otherwise

6

3 8 > :0 6

6

6

10

8

8

2

The cases where there are two possibilities for ±(E=K) can be determined as follows. (2) (i) When G(3) is isomorphic to GL2 (Z=3Z), the former occurs if D = Q2 (W2 [3]) (2) (2) and the latter occurs if D = Q2 (W¡2 [3]), where Wr are Weil’s elliptic curves de¯ned in Lemma 4.3. (ii) When G(3) is isomorphic to either D12 or E4 , then the smaller value occurs exactly when K » D. (iii) When G(3) is isomorphic to C8 , ±(E=K) = 0 if and only if the twist E K of E corresponding to K=Q2 has an unrami¯ed 3-division ¯eld. Remark 4.17. The condition on G(3) in Theorem 4.16 can be translated to that on the 3-division polynomial ©3 (X ). In fact, it is easy to show the following one-to-one correspondence between G(3) and Gal(Dx =Q2 ). G(3)

GL2 (Z=3Z)

Gal(Dx =Q2 )

S4

SD16 D12

D8

C8

D8

E4

C4

S3

Since ©3 (X ) is a quartic polynomial, its Galois group is easy to compute by the method in [7]. Proof (of Theorem 4.16). Let K be a rami¯ed quadratic ¯eld. The unique upper break u(K=Q2 ) equals 1 if K 2 Q1 , and equals 2 otherwise. As in Proposition 4.11, let M denote the ¯xed ¯eld of G1 (D=Q2 ). Since the extension M=Q2 is at most tamely rami¯ed, u(KM=M ) = e(M=Q2 )u(K=Q2 ) is the unique upper break of KM=M by Lemma 4.2. First we shall compute u(DK=M K). We use the arithmetically disjointness of KM and D over M if possible. After that, other cases are computed by the knowledge on extensions of small degree over Q2 . If we have computed u(DK=M K), then the wild part is given by 2 u(DK=M K); ±(E=K) = (4.23) e(M K=K) which follows from (3.8) and Proposition 4.1. We divide into six cases from (A) to (F) according to the isomorphism class of the Galois group G(3) = Gal(D=Q2 ). We use the notation in the corresponding parts in the proof of Proposition 4.11.

536


p p (A) The case G(3) ¹= GL2 (Z=3Z). We have M = Q2 ( 3 ¢E ; ¡ 3) and e(M=Q2 ) = 3. Thus we obtain u(KM=M ) = 3u(K=Q2 ). By Proposition 4.11, the ¯elds KM and D are arithmetically disjoint in the following cases: (AD1) ±(E=Q2 ) = 1; (AD2) ±(E=Q2 ) = 2 and u(K=Q2 ) = 2 (that is K 2 Q2 or Q3 ); (AD3) ±(E=Q2 ) = 4 and u(K=Q2 ) = 1 (that is K 2 Q1 ); (AD4) ±(E=Q2 ) = 5. Lemma 3.5 enables us to compute u(DK=M K) for these cases. Calculating the HasseHerbrand function explicitly, we have u(DK=M K) =

(

u(D=M ) 2u(D=M ) ¡

if u(D=M ) µ u(KM=M ); u(KM=M )

otherwise.

From this formula and (4.23), it follows that 3 (AD1) u(DK=M K) = and ±(E=K) = 1; 2 (AD2) u(DK=M K) = 3 and ±(E=K) = 2; (AD3) u(DK=M K) = 9 and ±(E=K) = 6; (AD4) if u(K=Q2 ) = 1, then u(DK=M K) = 12 and ±(E=K) = 8; if u(K=Q2 ) = 2, then u(DK=M K) = 9 and ±(E=K) = 6. We now consider non-arithmetically disjoint cases: (NAD1) ±(E=Q2 ) = 2 and u(K=Q2 ) = 1; (NAD2) ±(E=Q2 ) = 4 and u(K=Q2 ) = 2. We need the following easy but useful lemma. In this lemma, we do not have to make any assumption on G(3). Lemma 4.18 Let K=Q2 be a quadratic extension. Let E be an elliptic curve de¯ned over Q2 and E K the quadratic twist of E corresponding to the quadratic extension K=Q2 . Then we have Q2 (x(E[3])) = Q2 (x(E K [3])) and K(E[3]) = Q2 (E[3]; E K [3]): Proof (of Lemma 4.18) Noting that the division ¯eld is independent of the choice of the model of E, we may assume E is given by a short Weierstrass equation y 2 = p x3 + a4 x + a6 . The quadratic twist E K corresponding to K = Q2 ( m) is then given by my 2 = x3 + a4 x + a6 . There is an isomorphism between E and E K de¯ned over p K sending (x; y) to (x; y= m). From this, Q2 (x(E[3])) = Q2 (x(E K [3])) follows. We also have K(E[3]) = K(E K [3]). Thus K(E[3]) contains both Q2 (E[3]) and Q2 (E K [3]). If K is a sub¯eld of Q2 (E[3]), then the lemma is trivially true. Now suppose that p Q2 (E[3]) = Q2 (E K [3]). Then it is readily seen that m 2 Q2 (E[3]). In other words, if K 6» Q2 (E[3]), then Q2 (E[3]) 6= Q2 (E K [3]). Since [K(E[3]) : Q2 (E[3])] = 2, we obtain K(E[3]) = Q2 (E[3]; E K [3]). The proof of the lemma is now complete. We return to the proof of the case G(3) ¹= GL2 (Z=3Z).


537

(i)

By Lemma 4.3, we have Dx = Q2 (x(Wr [3])); (i = 1 or 2). It follows from (4.16) (2) that we have Dx = Q2 (x(Wr )) for the cases (NAD1) and (NAD2). For simplicity, we (2) write Wr instead of Wr from now on in this proof. First we consider the case (NAD1). Since ±(E=Q2 ) = 2, we have D = Q2 (W1 [3]) by (4.16) and U(D=M ) = f1; 3g by (4.15) and (4.18). From Lemma 4.18, it follows that KD = Q2 (W1 [3]; W1K [3]) with K 2 Q1 . Now we compute the upper rami¯cation breaks. By Lemma 3.1 (vi), we obtain U(D=Dx ) = f9g. On the other hand, the wild part of the conductor of W1K for K 2 Q1 is computed by Tate’s algorithm: ±(W1K =Q2 ) = ±(Wr =Q2 ) = 1; where we can take r = ¡ 1; 3. Using Proposition 4.11 and Lemma 3.1 (vi) again, we obtain U(Q2 (Wr [3])=Dx ) = f3g. This shows that Q2 (Wr [3]) and D are arithmetically disjoint over Dx . By Lemma 3.5, we ¯nd U(KD=KDx ) = f3g since KD = D(Wr [3]); (r = ¡ 1; 3) by Lemma 4.18. In a similar manner, we can verify that Dx and M K are arithmetically disjoint and U(KDx =M K) = f1g. From these computation, we ¯nally have U(KD=M K) = f1; 3=2g. Thus from (4.23), it follows that ±(E=K) = 1. Next we consider the case (NAD2). By (4.16), we have D = Q2 (Wr [3]) with r = §2. We compute the conductor of the quadratic twists WrK : r

¡ 2

2

K

Q1

Q2

Q1

Q2

±(WrK =Q2 )

2

1

1

2

If ±(WrK =Q2 ) = 1, Proposition 4.11 and Lemma 3.1 (vi) imply that U(Q2 (WrK [3])=Dx ) = f3g. Similarly as above, we have U(D=Dx) = f21g. Thus D and Q2 (WrK [3]) are arithmetically disjoint and we obtain U(KD=KDx ) = f3g by Lemma 3.5. By a similar computation as in the case (NAD1), we have U(KD=M K) = f1; 3=2g and ±(E=K) = 1. The case ±(WrK =Q2 ) = 2 can be dealt similarly. We have U(Q2 (WrK [3])=Dx) = f9g and U(KD=KDx ) = f9g and U(KD=M K) = f1; 3g. We consequently have ±(E=K) = 2. (B) The case G(3) ¹= SD16 . First note that there are three biquadratic ¯elds over Q2 p containing Q2 ( ¡ 3). They are p p p p p p B1 = Q2 ( ¡ 1; ¡ 3); B2 = Q2 ( 2; ¡ 3); B3 = Q2 ( ¡ 2; ¡ 3):

(4.24)

p It is easy to see that a quadratic ¯eld K is contained in Bi if and only if K = Q2 ( ¡ 3) or K 2 Qi for each i = 1; 2; 3. It is also an easy task to compute the unique positive upper rami¯cation breaks in Bi =Q2 ; (i = 1; 2; 3). In fact, we have u(B1 =Q2 ) = 1; u(B2 =Q2 ) = u(B3 =Q2 ) = 2:

(4.25)

538


As we saw in the part (B) in Proposition 4.11, D contains a biquadratic ¯eld k4 corresponding to H4 . Applying Herbrand’s theorem (3.5) to the biquadratic sub¯eld, we always have u(k4 =Q2 ) = 1. Therefore we conclude that k4 = B1 is a sub¯eld contained in D. Thus our quadratic ¯eld K is contained in D if and only if K 2 Q1 . Moreover if this is the case, then the extension DK=M K agrees with D=B1 and the rami¯cation breaks can be computed easily by Proposition 4.11 and Lemma 3.1 (vi). Indeed, we have 8 > > > :5

if ±(E=Q2 ) = 4

if ±(E=Q2 ) = 6:

Now the result follows from ±(E=K) = 2u(D=B1 ) in view of (4.23). We now assume that K 2 Q2 or Q3 , namely, U(K=Q2 ) = f2g. If ±(K=Q2 ) = 3, then it is easy to see that that D and M K are arithmetically disjoint over by Proposition 4.11. By virtue of Lemma 3.5, we readily have U(DK=M K) = f1; 3=2g. Thus we obtain ±(E=K) = 3 by (4.23). We next consider the case ±(E=Q2 ) = 4. Let us write Dx instead of k2 . From Proposition 4.11 and Lemma 3.1 (iv), we have u(D=Dx) = 5. By Herbrand’s theorem we see U(Dx =M ) = f1g. On the other hand, from Lemma 4.2, U(KM=M ) = f2g follows. Thus by Lemma 3.5, we obtain u(KDx =Dx) = 5. Let D 0 be an intermediate ¯eld of DK=D x di®erent from D and KDx . By Lemma 3.6, we have u(D 0 =Dx ) µ 5. Since it is easy to verify that D 0 =Dx is rami¯ed, we have u(D 0 =Dx ) > 0. Further, it can be seen that the Galois group of D 0 =Q2 is also isomorphic to SD16 . It follows from Lemma 4.18 that D 0 is also obtained as a division ¯eld of a certain twisted elliptic curve. Therefore by Proposition 4.11, the possible values of u(D 0 =Dx ) are 3 and 5. If u(D 0 =Dx ) = 5, then the exponent of the discriminant of D 0 =Q2 is the same as that of D=Q2 , which is 36 by (4.19). But it is shown in [2, Table 9] that there are no two SD16 -extensions with same discriminant and same D8 sub¯eld. Hence we conclude u(D 0 =Dx) = 3. Since D 0 and D are arithmetically disjoint over Dx , we get u(DK=KDx) = 3. Now it is easy to see that L(DK=M K) = f1; 3g. Therefore we obtain U(DK=M K) = f1; 3=2g and ±(E=K) = 3. Next we assume ±(E=Q2 ) = 6. From Lemma 3.1 (vi), it follows U(D=B1 ) = f3; 7g. In particular, we have u(Dx=B1 ) = 3. On the other hand, the arithmetically disjointness of M K and B1 over M implies that u(M B1 =B1 ) is also 3. Let F be the intermediate ¯eld of KDx =B1 di®erent from both KB1 and Dx . Then it is an easy matter to check that F=Q2 is a Galois extension with Galois group isomorphic to D8 . By Lemma 3.6 (i), we have u(F =B1 ) µ 3. It is plain to show that F =B1 is rami¯ed. Here we shall prove u(F =B1 ) = 1. Indeed, if u(F=B1 ) = 2, then F and KB1 are arithmetically disjoint over B1 and we obtain u(KDx =KB1 ) = 2 and U(KDx =M K) = f1; 3=2g. This contradicts the Hasse-Arf theorem since the Galois group of KDx =M K is isomorphic to the abelian group E4 . Also if u(F =B1 ) = 3, then we obtain u(KDx =Dx) = u(KDx=KB1 ) = 3. This implies U(KDx =M K) = f1; 2g. This contradicts Lemma 4.12 (i) because e(M K=Q2 ) =


539

2. Thus we have u(F =B1 ) = 1 as claimed. Now it follows u(KDx =Dx ) = 1. Since u(D=Dx) = 7, we have u(DK=KDx ) = 13. As above, we compute L(DK=M K) = f1; 13g and thus U(DK=M K) = f1; 4g and conclude ±(E=K) = 8. (C) The case G(3) ¹= D12 . In this case, the division ¯eld D contains a biquadratic ¯eld which is one of Bi ’s in (4.24). As we have seen in (B), K is contained in D if and only if K 2 Qi and Bi » D. If this is the case, then it follows that DK = D and the extension DK=K is tamely rami¯ed. This implies ±(E=K) = 0. We here note that, by Herbrand’s theorem, it is readily seen that D ¼ B1 if and only if ±(E=Q2 ) = 2 (see Proposition 4.11). p We now assume K 6» D. By (4.25), the arithmetically disjointness of K( ¡ 3) and p Bi over Q2 ( ¡ 3) is veri¯ed for the following two cases: (AD1) ±(E=Q2 ) = 2 and u(K=Q2 ) = 2, (AD2) ±(E=Q2 ) = 4 and u(K=Q2 ) = 1. Applying Lemma 3.5 and Proposition 4.1 and (4.23), we compute p (AD1) u(B1 K=K( ¡ 3)) = 1 and u(DM=M K) = 3 and ±(E=K) = 2, p (AD2) u(Bi K=K( ¡ 3)) = 3 and u(DM=M K) = 9 and ±(E=K) = 6, where i = 2; 3. The remaining cases are where Bi » D and K 2 Qj (i; j = 2; 3 and i 6= j). Since the ¯eld KBi contains all quadratic ¯elds over Q2 , KBi = B1 Bi holds. Since B1 and p Bi are arithmetically disjoint over Q2 ( ¡ 3) by (4.25), we have u(KBi =Bi ) = 1 and ±(DK=M K) = 3 by Proposition 4.1. This yields ±(E=K) = 2. (D) The case G(3) ¹= D8 . By (4.20), D contains B3 . Thus if K 2 Q3 , then we have p K » D and u(D=K( ¡ 3)) = 4 by Lemma 3.1(vi) and conclude ±(E=K) = 8. If u(K=Q2 ) = 1, namely K 2 Q1 , then M K and D are arithmetically disjoint over p M . We have by Lemma 3.5, U(DK=M K) = f3; 5g, where M = Q2 ( ¡ 3). This yields ±(E=K) = 10 by (4.23). Now we assume K 2 Q2 . Then as in (C), KB3 contains all quadratic ¯elds over Q2 and thus we have u(KB3 =B3 ) = 1. Since u(D=B3 ) = 4, D and KB3 are arithmetically disjoint over B3 . It follows u(DK=KB3 ) = 7. On the other hand, it is easy to see u(KB3 =M K) = 1. As a result we have U(DK=M K) = f1; 4g and thus ±(E=K) = 8. (E) The case G(3) ¹= C8 . For the following three cases, we can show that K and D are arithmetically disjoint over Q2 by using Proposition 4.11. (AD1) ±(E=Q2 ) = 2 and u(K=Q2 ) = 2, (AD2) ±(E=Q2 ) = 4 and u(K=Q2 ) = 1, (AD3) ±(E=Q2 ) = 6 and u(K=Q2 ) = 1. For these three cases, we apply Lemma 3.5 and obtain (AD1) u(DK=M K) = 1 and ±(E=K) = 2, (AD2) u(DK=M K) = 3 and ±(E=K) = 6, (AD3) U(DK=M K) = f3; 5g and ±(E=K) = 10. Next we consider the non-arithmetically disjoint cases. We use the notation in the part (E) of the proof of Proposition 4.11. In particular, Kn is the n-th local cyclotomic ¯eld Q2 (³n ).

540


We ¯rst consider the case e(D=Q2 ) = 2. We have u = u(D=M ) = u(M K=M ) = u(K=Q2 ): Let D 0 be a quadratic extension of M that is di®erent from both M K and D. We have u(D 0 =M ) µ u by Lemma 3.6 (i). Moreover it is not di±cult to see that D 0 is a cyclic extension over Q2 . Therefore, as in the part (E) of Proposition 4.11, we can show that D 0 is contained in K(28 ¡1)24 . Since DD 0 = DK, it follows that DK is also contained in K(28 ¡1)24 . Assume further that ±(E=Q2 ) = 2. Then we have u = 1. By comparing the conp ductor, D is contained in K28 ¡1 ( ¡ 1). This implies that G = Gal(K(28 ¡1)24 =D) contains h¾i. Therefore it follows G = G3 , where G3 is de¯ned in (4.21). Thus if u(D 0 =M ) = 1, then we have D = D 0 , which contradicts our choice of D 0 . Therefore we have u(D 0 =M ) = 0. This means that D 0 =M is unrami¯ed. Then DK=M K is also unrami¯ed. Hence we have ±(E=K) = 0. Next we assume ±(E=Q2 ) = 4. We have u = 2. This implies that u(D 0 =M ) µ 2. If u(D 0 =M ) = 0, then, as above, we have ±(E=K) = 0. If u(D 0 =M ) = 1, then D and D 0 are arithmetically disjoint and we have u(DK=M K) = 1. From (4.23), it follows ±(E=K) = 2. Suppose now that u(D 0 =M ) = 2. Then DK = DD 0 is the compositum of the ¯elds corresponding to G2 and G3 . Therefore we have Gal(K(28 ¡1)24 =DK) = G2 \ G3 = h½4 ¾i ¼ h¾ 2 i, where the last group corresponds to K(28 ¡1)23 . From this we conclude DK » K(28 ¡1)23 . Since the discriminant of K(28 ¡1)23 over Q2 is 28 , the exponent of the di®erent DDK=Q2 is less than 3. This is impossible, because DK=M is totally and wildly rami¯ed. Thus we have u(D 0 =M ) 6= 2. p We next consider the case e(D=Q2 ) = 4. Then we see M = Q2 ( ¡ 3) and u(M K=M ) = u(Dx =M ) = u(K=Q2 ) = 2. Let F be a quadratic extension of M that is di®erent from Dx and M K. It follows from Lemma 3.6 (i) that u(F =M ) µ 2. If u(F =M ) = 0, then the extensions KDx =M K and KDx =Dx and DK=D are all unrami¯ed. By Lemma 3.1 (vi), we have u(D=Dx ) = 4. It now follows from Lemma 4.2 that u(DK=KDx ) = 4. Consequently we obtain ±(E=K) = 8. If u(F =M ) = 1, then F and Dx are arithmetically disjoint, since we have u(Dx =M ) = 2 by Herbrand’s theorem. As before, we have u(KDx =Dx ) = 1. Now KDx and D are arithmetically disjoint and we compute u(DK=KDx) = 7. Let U(DK=M K) = fu1 ; u2 g. Then it is easy to see that 7 = l(DK=M K) = 2(u2 ¡ u1 ) + u1 . Since u1 < u2 , we get u1 = 1 and u2 = 4. We now conclude ±(E=K) = 8. We shall show that u(F=M ) 6= 2. Suppose to the contrary that u(F =M ) = 2. Then there is no rami¯cation break in KDx =M and thus we have u(KDx =M K) = 2. This implies the ¯rst rami¯cation break in the cyclic extension KD=M K is 2. This contradicts Lemma 4.12 (ii) (a). (F) The case G(3) ¹= E4 . In this case, we have D = Bi , where Bi is a biquadratic ¯eld in (4.24). Therefore the discussion made in the part (C) above also applies to this case. This completes the proof of Theorem 4.16.


541

To complete the description of the variation of the conductor, we mention the results on the tame part. Though these results are proved in a more general context, we state them in a form relevant to our cases. The ¯rst part of the following proposition follows from [18, Theorem 3] and the second from [10, Corollary to Theorem 3]. Proposition 4.19. Let E be an elliptic curve de¯ned over Q2 . Let c4 ; c6 and ¢ be the usual c-invariants and the discriminant of a minimal model of E. (i) Assume that Type(E) = I¤n (n ¶ 1). If one of the following conditions is satis¯ed, then E has multiplicative reduction over a quadratic extension of Q2 and vice versa. (PM1) v2 (c4 ) = 4; v2 (c6 ) = 6; v2 (¢) = 8 + n; n ¶ 5, (PM2) v2 (c4 ) = 6; v2 (c6 ) = 9; v2 (¢) = 10 + n; n ¶ 9. (ii) If one of the following conditions is satis¯ed, then E has good reduction over a certain quadratic extension of Q2 . (PG1) v2 (c4 ) ¶ 6 and v2 (¢) = 6, (PG2) v2 (c4 ) = 4 and v2 (¢) = 12, (PG3) v2 (c4 ) = 6 and v2 (¢) = 18, (PG4) v2 (c4 ) ¶ 8 and v2 (¢) = 12. Looking up Table 2 in [18], the above conditions are satis¯ed for the following cases:

(P M 1) Type(E) = I¤n with n ¶ 5; f (E=Q2 ) = 4; (P M 2) Type(E) = I¤n with n ¶ 9; f (E=Q2 ) = 6; (P G1) Type(E) = II;

f (E=Q2 ) = 6;

(P G2) Type(E) = I¤4 ;

f (E=Q2 ) = 4;

(P G3) Type(E) = I¤8 ;

f (E=Q2 ) = 6;

(P G4) Type(E) = II¤ ;

f (E=Q2 ) = 4:

In the potential multiplicative cases, we have Type(EK ) = In0 with some n0 ¶ 1. This integer n0 can be calculated from the relation n0 = ¡ vK (j) = ¡ 2(v2 (c34 =¢)): Therefore, we have n0 = 2n ¡ 8 in (PM1) and n0 = 2n ¡ 16 in (PM2). If E attains good reduction (resp. multiplicative reduction) over a quadratic ¯eld K, then we have "(E=K) = 0 (resp. 1). And this exactly happens when ±(E=K) = 0. This completes the proof of the assertion on the conductor in Theorem 2.5 and we are able to ¯ll out f (E=K) in Table 3.

542

5


Variation of the reduction type

In this section we compute the variation of the reduction type and complete the proofs of our theorems using the results in the preceding section. Let K=k be an extension of local ¯elds with residue characteristic p and E an elliptic curve de¯ned over k. We write Ogg’s formula (2.4) for E and the base change EK : "(E=k) + ±(E=k) = vk (¢k ) + 1 ¡ "(E=K) + ±(E=K) = vK (¢K ) + 1 ¡

mk ; mK ;

(5.1) (5.2)

where ¢k and ¢K denote the minimal discriminants of E and EK respectively. The integers mk and mK are determined by the reduction type according to (2.5). Furthermore, by admissible variable change ( x = u2 x0 + r; [u; r; s; t] : (5.3) y = u3 y 0 + su2 x0 + t; the discriminant of a model changes by u12 (see [24, Table 1.2]). Namely ¢K and ¢k relate by ¢k = u12 ¢K (5.4) for some u 2 K £ . Taking vK = e(k=k)vk into account, we have vK (¢K ) = e(K=k)vk (¢k ) ¡

12vK (u):

(5.5)

Since we use Tate’s algorithm many times in proving our theorems, we recall the structure of the algorithm brie°y. We use [25, IV.9] as a standard reference and use the notation there. In particular, we write ai;r = ¼ ¡r ai , where ¼ is a prime element of the ¯eld of de¯nition. We are interested only in curves with bad reduction and the singular point on the reduced model may be assumed to be located at (0; 0). When such a model is given as an input of the algorithm, the algorithm checks whether it has In ; II; III or IV without changing the model. This is done till Step 5. In Step 6, we ¯nd ® and ¯ in the ring of integers in K satisfying ( Y 2 + a1 Y + a2 ² (Y ¡ ®)2 (mod ¼); (5.6) Y 2 + a3;1 Y ¡ a6;2 ² (Y ¡ ¯)2 (mod ¼); and apply [1; 0; ®; ¼¯] (see (5.3)) to obtain a model satisfying ¼ja1 and a2 , ¼ 2 ja3 and a4 , and ¼ 3 ja6 . Let ¹ be the number of distinct roots of P (T ) = T 3 + a2;1 T 2 + a4;2 T + a6;3

(5.7)

in an algebraic closure of the residue ¯eld. If ¹ = 3, then Type(E) = I¤0 and if ¹ = 2, then Type(E) = I¤n for n ¶ 1. Finally if ¹ = 1, namely P (T ) has a triple root, then Type(E) = IV ¤ ; III¤ ; II¤ or the model is not minimal. If the model is not minimal, we apply [¼; 0; 0; 0] and return to the ¯rst step. In most cases in what follows, our applications of Tate’s algorithm simply involve only computing the above ¹, since our results in Section 4 give a good restriction on possible reduction types.


543

5.1 Tamely rami¯ed extension (Proof of Theorem 2.1) We shall prove Theorem 2.1. If p ¶ 5, the theorem can be proved entirely by means of Tate’s algorithm. We here instead give a simpler and shorter uni¯ed proof for all p using the results we obtained in the preceding sections. Let e = e(K=k) be the rami¯cation index of the tame extension K=k. If "(E=k) = 0 or 1, then "(E=K) = "(E=k) by the stable reduction theorem. If "(E=k) = 0, there is nothing to prove. If "(E=k) = 1 and Type(E) = In , then vk (jE ) = ¡ n (see [25, p.365, Table 4.1]). It yields vK (jE ) = ¡ ne. Thus we have Type(EK ) = Ine . Now we assume "(E=k) = 2. If k(E` )=k is at most tamely rami¯ed, then we have ±(E=k) = ±(E=K) = 0 by De¯nition 3.7. Also if K=k is tamely rami¯ed, then we have ±(E=K) = e±(E=k) by Proposition 4.1. Combining (5.1), (5.2) and (5.5), we obtain mK = e(mk + 1) ¡

"(E=K) + 1 ¡

12vK (u)

(5.8)

(mod 12):

(5.9)

in either case. In particular, it follows mK ² e(mk + 1) ¡

"(E=K) + 1

The following lemma settles the case "(E=K) = 0. Lemma 5.1. Suppose that E has additive and potential good reduction and that k(E[`])=k is tamely rami¯ed. Let K=k be a ¯eld extension with e = e(K=k). Then we have Type(EK ) = I0 if and only if e(mk + 1) ² 0 (mod 12). Proof. If Type(EK ) = I0 , then from (5.9) it immediately follows e(mk +1) ² 0 (mod 12). Conversely, if e(mk + 1) ² 0 (mod 12), then by (5.9) we obtain mK ² ¡ "(E=K) + 1 (mod 12). Since E has potential good reduction, we have "(E=K) = 0 or 2. If "(E=K) = 2, then mK ² 1 (mod 12). If mK = 1, then we have Type(E) = II by (2.5) and it follows from Proposition 2.2 that K(E[`])=K is wildly rami¯ed for all p. This is a contradiction. If mK ¶ 13, then we have Type(E) = I¤n with some n. When p ¶ 3, then this is also impossible since E has potential good reduction. When p = 2, K(E` )=K is wildly rami¯ed by Proposition 2.2. Thus it is impossible, too. Hence we obtain "(E=K) = 0. This means that E has good reduction. If "(E=K) = 1, in other words, Type(EK ) = In0 for some n0 , then it is necessary to have Type(E) = I¤n for some n. In fact, the following lemma is known (see [25, Table 4.1]). Lemma 5.2. An elliptic curve with additive reduction has potential multiplicative reduction, then Type(E) = I¤n with some n. If the residue characteristic of the ¯eld of de¯nition is greater than 2, then the converse is also true. Therefore, if p ¶ 3 and Type(EK ) = In0 , then the theory of Tate curve shows that E attains multiplicative reduction provided 2 divides e. The relation of n and n0 can be computed from the j-invariant and, in fact, we obtain n0 = ne.

544


If "(E=K) = 2, then (5.9) implies mK + 1 ² e(mk + 1) (mod 12). By using this congruence, mK (mod 12) can be computed from mk . If mK 6= 7; 8; 9, then the reduction type is determined immediately by (2.5). Even if mK = 7; 8; 9, we can distinguish the reduction type provided p ¶ 3. Actually, by Lemma 5.2, we have Type(EK ) = I¤n with some n if and only if Type(E) = I¤n0 with some n0 . Otherwise we have Type(E) = IV¤ ; III¤ or II¤ . The remaining case is where p = 2. In this case, Type(E) = I¤n does not guarantee that E has potential multiplicative reduction (see [18, Theorem 3] and Proposition 4.19). Therefore we need another criterion to distinguish them. If Type(E) is either IV or IV¤ , then k(E[`])=k is tamely rami¯ed by Proposition 2.2 and so is K(E[`])=K. If Type(EK ) = I¤n with n ¶ 1, then K(E[`])=K is wildly rami¯ed and we have a contradiction. Thus if Type(E) = IV or IV ¤ and mK = 7, then we have Type(EK ) = IV ¤ . It remains to consider the following four cases: (T1) Type(E) = II and e ² 5 (mod 6), thus mK ² 9 (mod 12); (T2) Type(E) = III and e ² 3 (mod 4), thus mK ² 8 (mod 12); (T3) Type(E) = III¤ and e ² 1 (mod 4), thus mK ² 8 (mod 12); (T4) Type(E) = II¤ and e ² 1 (mod 6), thus mK ² 9 (mod 12). For these cases, we need the following lemma. Lemma 5.3. Let E be an elliptic curve de¯ned over a local ¯eld K with residue characteristic 2. Assume that mK = 8 or 9. If we can take a minimal Weierstrass model (2.2) of E over K satisfying vK (a1 ) ¶ 1; vK (a2 ) ¶ 2; vK (a3 ) ¶ 2; vK (a4 ) ¶ 3; vK (a6 ) ¶ 4; then Type(E) 6= I¤n with any n ¶ 1. Proof. Suppose that we can take a minimal model satisfying the above condition. We apply Tate’s algorithm to the minimal model. Let ¼K denote a prime element of K. By our hypothesis, the model is unchanged until Step 6 of the algorithm, since we can take ® = ¯ = 0 in (5.6). Since the model is minimal, the reduction type is determined the number of distinct roots of (5.7) in the algebraic closure of the residue ¯eld of K. Our assumption implies that P (T ) ² T 3 (mod ¼K ). Thus P (T ) has a triple root. This means that Type(E) 6= I¤n . Using this lemma, we shall prove the remaining cases. We begin with (T4). Applying Tate’s algorithm over k, we can take a minimal Weierstrass model (2.2) of E over k satisfying vk (a1 ) ¶ 1; vk (a2 ) ¶ 2; vk (a3 ) ¶ 3; vk (a4 ) ¶ 4; vk (a6 ) = 5: By writing e = 6t + 1 with a positive integer t, a new integral model obtained by the 5t transformation [¼K ; 0; 0; 0] satis¯es vK (a01 ) ¶ t + 1; vK (a02 ) ¶ 2t + 2; vK (a03 ) ¶ 3t + 3; vK (a04 ) ¶ 4t + 4; vK (a06 ) = 5:


545

This model is a minimal model. In fact, by the de¯nition of minimal discriminant, we have 5t vK (u) ¶ vK (¼K ) = 5t (5.10) where u is an element of K satisfying (5.5). Thus from (5.8) we have mK µ e(9 + 1) ¡

2+1¡

12 ¢ 5t = 9

(recall that t = (e¡ 1)=6). Therefore we get mK = 9 and the equality holds in (5.10). This shows that our model is minimal. Hence the model satis¯es the conditions in Lemma 5.3. We have Type(E) = II¤ . The proof for (T3) is similar to that of (T4), since we can take a model of E over k satisfying vk (a1 ) ¶ 1; vk (a2 ) ¶ 2; vk (a3 ) ¶ 3; vk (a4 ) = 3; vk (a6 ) ¶ 5: 3(e¡1)=4

We take [¼K ; 0; 0; 0] for the corresponding transformation. Next we consider (T1). By Tate’s algorithm, we can take a model over k satisfying vk (b2 ) ¶ 1; vk (a3 ) ¶ 1; vk (a4 ) ¶ 1; vk (a6 ) = 1; from which we derive vk (a1 ) ¶ 1. Since b2 = a21 + 4a2 ² 0 (mod ¼K ), the congruence Y 2 + a1 Y ¡ a2 ² 0 (mod ¼K ) has a double root ® 2 k. Applying [1; 0; ®; 0], we have a01 = a1 + 2®; a02 = a2 ¡

®a1 ¡

®2 ; a03 = a3 ; a04 = a4 ¡

®a3 ; a06 = a6

by the transformation formula [24, Table 1.2]. These yield vk (a01 ) ¶ 1; vk (a02 ) ¶ 1; vk (a03 ) ¶ 1; vk (a04 ) ¶ 1; vk (a06 ) ¶ 2: t Writing e = 6t + 5 and applying [¼K ; 0; 0; 0], we obtain a model with

vK (a001 ) ¶ 5t + 5; vK (a002 ) ¶ 4t + 5 vK (a003 ) ¶ 3t + 5; vK (a004 ) ¶ 2t + 5; vK (a006 ) = 5: We can show that this model is minimal by a similar manner as in (T4). Applying Lemma 5.3, we obtain the result. The proof for (T2) is similar to that of (T1). We can take a model over k satisfying vk (a1 ) ¶ 1; vk (a3 ) ¶ 1; vk (a4 ) ¶ 1; vk (a6 ) ¶ 2: (e¡3)=4

After applying [1; 0; ®; 0], we use [¼K ; 0; 0; 0] to obtain a minimal model satisfying the conditions in Lemma 5.3. This completes the proof of Theorem 2.1.

5.2 Cubic base change over Q3 (Proof of Theorem 2.4) We shall prove Theorem 2.4. Let E be an elliptic curve de¯ned over Q3 and K=Q3 a wildly rami¯ed cubic extension. Having settled the case where E has good or multiplicative reduction, we assume that E has additive reduction, namely "(E=Q3 ) = 2.

546


From Proposition 4.9 and remarks there, we know when "(E=K) = 0 or 1 occurs. Thus we consider the remaining case "(E=K) = 2. In (5.1), (5.2) and (5.5), we let k = Q3 and e(K=k) = 3 and then we obtain mK ² 3(±(E=Q3 ) + mQ3 ) ¡

±(E=K) + 2

(mod 12):

(5.11)

For a given reduction type over Q3 , mQ3 is given by (2.5) and the possible values of ±(E=Q3 ) are known from Papadopoulos’ computation [18, Tableau II]. Also Theorem 4.6 tells us the value of ±(E=K). Thus we can compute mK (mod 12). If p = 3, then Lemma 5.2 tells us when E has potential multiplicative reduction. Therefore mK (mod 12) is enough to determine Type(EK ) by means of (2.5). This completes the proof of Theorem 2.4.

5.3 Quadratic base change over Q2 (Proof of Theorem 2.5) We shall prove Theorem 2.5. As we go along, we will make various assumptions. These assumptions are cumulative and will be underlined for clarity. Let E be an elliptic curve de¯ned over Q2 and K=Q2 a rami¯ed quadratic extension. As before, we may assume "(E=Q2 ) = 2. By Proposition 4.19, we know when "(E=K) = 0 or 1 occurs. Therefore we also assume "(E=K) = 2. In this case p = 2, there exist elliptic curves having potential good reduction with Type(E) = I¤n for some n ¶ 1. Therefore mK (mod 12) is not enough to determine the reduction type. We have to determine the exact value of mK . For that purpose, we need a criterion to determine whether an initial minimal model over Q2 is still minimal over K. By making the following de¯nition, it will become easier to state our results. De¯nition 5.4. We de¯ne a partial order < on the set of Kodaira symbols by I0 < In < II < III < IV < I¤0 ; IV < I¤n ; IV < IV¤ < III¤ < II¤ : We also de¯ne, if n < n0 , then In < In0 and I¤n < I¤n0 . For a model E of an elliptic curve E and a Kodaira type T , we say Type(E) > T if either E is minimal and Type(E) > T or E is not minimal. If it is clear from the context what model we are discussing, then we write Type(E) > T instead of Type(E) > T . This partial order re°ects how Tate’s algorithm proceeds. For the criterion for the minimality, we need the following two lemmas. Lemma 5.5. Let E be any model of an elliptic curve E=K and ¢ the discriminant of E. Then we have f (E=K) µ vK (¢).


547

Proof. Let ¢K be the minimal discriminant of E. Then by Ogg’s formula (5.2) we obtain f (E=K) = vK (¢K ) + 1 ¡ mK µ vK (¢K ) µ vK (¢). The second lemma gives a criterion for (5.7) to have a triple root. Lemma 5.6. Let E be an integral Weierstrass model (2.2) of an elliptic curve de¯ned over K. Suppose that Type(E) > IV. Then E can be taken to satisfy vK (a1 ) ¶ 1; vK (a3 ) ¶ 2; vK (a4 ) ¶ 2; vK (a6 ) ¶ 2:

(5.12)

Suppose further that we take a model E satisfying (5.12) and that the coe±cients of E belong to Z2 . Then the equation (5.7) has a triple root in an algebraic closure of the residue ¯eld of K if and only if one of the following conditions holds. (i) v2 (a2 ) ¶ 1 and v2 (a6 ) ¶ 2 and v2 (a4 ) ¶ 2. (ii) v2 (a2 ) = 0 and v2 (a6 ) ¶ 2 and v2 (a3 ¡ a4 ) ¶ 2. (iii) u(K=Q2 ) = 2 and v2 (a6 ) = 1 and v2 (a3 ) ¶ 2 and v2 (a4 ) ¶ 2. (iv) u(K=Q2 ) = 1 and v2 (a6 ) = 1 and v2 (a2 ) ¶ 1 and v2 (a4 ) ¶ 2. (v) u(K=Q2 ) = 1 and v2 (a6 ) = 1 and v2 (a2 ) = 0 and v2 (a3 ¡ a4 ) ¶ 2. Proof. The ¯rst assertion (5.12) on the model follows directly from Step 5 of Tate’s algorithm. t Let ¼K be a prime element of K. We write ai;t = ai =¼K as before. Since K=Q2 is rami¯ed, we can take ® and ¯ in (5.6) to be 0 or 1. After taking a new model with these ® and ¯, it is easy to verify that the equation (5.7) has a triple root ° if and only if a02;1 ² a04;2 ² a06;3 ² ° (mod ¼K ): Here the new coe±cient a02 is given by a2 ¡ ®a1 ¡ ®2 and is divisible by ¼K . Since ® is taken to be 0 or 1, this implies a02 2 Z2 and thus vK (a02 ) ¶ 2. Therefore, if P (T ) has a triple root °, then we have ° ² 0 (mod ¼K ). Then the above condition becomes a04;2 ² a06;3 ² 0

(mod ¼K ):

Conversely if this condition is satis¯ed, it is readily seen that P (T ) has a triple root. (i) If v2 (a2 ) ¶ 1 and v2 (a6 ) ¶ 2, then we have ® = ¯ = 0. The model is unchanged. Therefore the condition is equivalent to vK (a4 ) ¶ 3, namely v2 (a4 ) ¶ 2. (ii) If v2 (a2 ) = 0 and v2 (a6 ) ¶ 2, then we have ® = 1 and ¯ = 0. The coe±cients of the new equation are a01 = a1 + 2; a02 = a2 ¡

a1 ¡

1; a03 = a3 ; a04 = a4 ¡

a3 ; a06 = a6 :

Hence the condition for this case is vK (a04 ) = vK (a3 ¡ a4 ) ¶ 3, namely v2 (a3 ¡ a4 ) ¶ 2. For the remaining cases, we show that, if a 2 Z2 with v2 (a) = 1, then ( 2 3 4 ¼K + ¼K (mod ¼K ) if u(K=Q2 ) = 1; a² (5.13) 2 4 ¼K (mod ¼K ) if u(K=Q2 ) = 2:

548


holds for any choice of ¼K . Suppose ¯rst that u(K=Q2 ) = 2. Writing a = 2 + 22 b with b 2 Z2 , we ¯nd 0 2 4 that it is enough to prove that 2 ² ¼K (mod ¼K ) for any choice of ¼K . Let ¼K 0 be another prime element of K. When we write ¼K = ¼K u with a unit u, we can take u such that vK (u ¡ 1) ¶ 1 holds, since K=Q2 is a rami¯ed quadratic 0 0 2 4 ² ¼ 0 2K (1 + ¼K extension. Writing u = 1 + ¼K v, we have ¼K v)2 ² ¼ 0 2K (mod ¼K ). 03 3 4 ² Similarly we have ¼K ¼ K (mod ¼K ). Thus we have only to show the claim p for a particular choice of ¼K . If we write K = Q2 ( m) with m = 2; 6; ¡ 2 or 10, p the claim is readily veri¯ed by choosing ¼K = m. 2 When u(K=Q2 ) = 1, it is similarly shown that we have only to prove 2 ² ¼K + p 3 4 ¼K (mod ¼K ) for a particular ¼K . If we write K = Q2 ( m) with m = ¡ 1 or 3, p then the claim is veri¯ed for ¼K = 1 + m. Thus we have shown (5.13). Now we can show the remaining cases of Lemma 5.6. (iii) We assume that u(K=Q2 ) = 2 and v2 (a6 ) = 1. Then we have ¯ = 1. The new 2 2 coe±cient a06 is a6 ¡ ¼K a3 ¡ ¼K . It follows from (5.13) that a6 ¡ ¼K is always 0 4 divisible by ¼K . Hence we see vK (a6 ) ¶ 4 if and only if v2 (a3 ) ¶ 2. Suppose that this condition is satis¯ed. Since we have a04 = a4 ¡ ¼K a1 or a4 ¡ a3 ¡ ¼K a1 ¡ 2¼K according to ® = 0 or 1, where we have v2 (a1 ) ¶ 1, the condition a04;2 ² 0 (mod ¼K ) is equivalent to v2 (a4 ) ¶ 2 in either case. 2 3 ² 2¼K (iv) and (v) If u(K=Q2 ) = 1 and v2 (a6 ) = 1, we have a6 ¡ ¼K a3 ¡ ¼K 4 (mod ¼K ). Thus the condition for a06 always holds. The conditions for a04 is obtained in a similar manner. This completes the proof of Lemma 5.6. We can determine whether a given model over Q2 is minimal over K by the following algorithm. The input of the algorithm is f (E=K), vK (¢Q2 ) and a minimal model of E over Q2 . The conductor f (E=K) is derived from Theorem 4.16 and vK (¢Q2 ) is computed from f (E=Q2 ) and mQ2 by Ogg’s formula. The algorithm also returns the reduction type Type(EK ) if the given model is minimal over K. Algorithm 5.7. Input f (E=K)(¶ 2); vK (¢Q2 ) and a minimal model E over Q2 . Output Type(EK ) or \E is not minimal". Step 1 Let v := vK (¢Q2 ). If f (E=K) > v ¡ 12, then let mK := v + 1 ¡ f , otherwise goto Step 3. Step 2 If mK 6= 7; 8 or 9, then return Type(EK ) according to (2.5). If mK = 7, then return Type(EK ) = IV¤ or I¤2 according as f (E=K) = 2 or not. Step 3 If E does not satisfy (5.12) in Lemma 5.6, then make a variable change to satisfy the condition and apply the criterion in Lemma 5.6. If mK has not determined yet, go to Step 4, otherwise go to Step 5. Step 4 If P (T ) does not have a triple root, then return Type(EK ) = I¤n with n = v ¡ f (E=K)¡


549

4, else return \E is not minimal". Step 5 If P (T ) has a triple root, then return Type(E) = II¤ or III¤ according to mK . If it does not have a triple root, then return Type(EK ) = I¤3 or I¤4 according to mK . Proof. If the condition in Step 1 is satis¯ed, then E is minimal over K by Lemma 5.5. In Step 2, when mK = 7, we have Type(E) = IV¤ if and only if f (E=K) = 2 by Proposition 2.2. After Step 2, we have either f (E=K) µ v ¡ 12 or mK = 8 or 9. Suppose that we have f (E=K) µ v¡ 12. If E is minimal, then mK = v¡ f (E=K)+1 ¶ 13 and we have Type(E) = I¤n with n ¶ 1. Thus it follows Type(E) > IV. Therefore we can apply Lemma 5.6. If P (T ) has a triple root, then E is not minimal, since mK ¶ 13. If P (T ) does not have a triple root, then E is minimal and, as we noted, Type(E) = I¤n where n = mK ¡ 5 = v ¡ f (E=K) ¡ 4. If mK = 8 or 9, then we have Type(E) > IV by (2.5). Hence we can apply Lemma 5.6 in Step 5. Since E is minimal in this case, the reduction type is determined by the number of the distinct roots of P (T ) as claimed. The following example illustrates how to apply this algorithm. Example 5.8. We consider an elliptic curve E=Q2 having Type(E) = II and f (E=Q2 ) = 6. By Tate’s algorithm (over Q2 ), it is easy to deduce v2 (c4 ) ¶ 4. By Ogg’s formula, we have v2 (¢Q2 ) = 6. Since the discriminant is cube, the Galois group G(3) cannot be GL2 (Z=3Z). From Theorem 4.16, it follows that ±(E=K) is one of 0; 2; 3; 6. When ±(E=K) = 0, it has good reduction by Proposition 4.19. Thus we have to consider the cases where f (E=K) = 2 + ±(E=K) = 4; 5 or 8. For these cases, we have vK (¢Q2 ) = 12 by the ¯rst step in Algorithm 5.7. If f (E=K) = 8, then mK = 5. Therefore we have Type(EK ) = I¤0 . For the remaining cases, we have mK = 8 or 9 and we go to Step 5. If f (E=K) = 4, then while EK has bad reduction, E has good reduction over a certain quadratic ¯eld by Theorem 4.16. By Proposition 4.19 (PG1), we have v2 (c4 ) ¶ 6. Since 1728¢Q2 = c34 ¡ c26 and v2 (¢Q2 ) = 6, we ¯nd v2 (c6 ) = 6. On the other hand, it is readily seen from (5.12) that v2 (b2 ) ¶ 2; v2 (b4 ) ¶ 2 and v2 (b6 ) ¶ 2. Since v2 (c6 ) = 6, we obtain v2 (b6 ) ¶ 3. This yields v2 (a23 ) ¶ 3 and thus v2 (a3 ) ¶ 2. Now if v2 (a4 ) = 1, then we have v2 (b4 ) = 1. This implies that v2 (c4 ) = 4, which is a contradiction. We conclude v2 (a4 ) ¶ 2. Further if v2 (a6 ) ¶ 2, then we have v2 (b6 ) ¶ 4 and thus v2 (c6 ) ¶ 7. We again have a contradiction. Thus we have v2 (a6 ) = 1. Consequently, we can take a model satisfying Lemma 5.6 (iii). Therefore P (T ) de¯ned by (5.7) has a triple root and we have Type(EK ) = II¤ . If f (E=K) = 5, then v2 (c4 ) µ 5. In fact, if v2 (c4 ) ¶ 6, then E attains good reduction over a certain quadratic extension by Proposition 4.19 (PG1). This is not our case because of Theorem 4.16. Now by a similar manner as above, we can take a model satisfying v2 (a6 ) = 1; v2 (a3 ) ¶ 2 and v2 (a4 ) ¶ 1. Thus P (T ) has a triple root if and only if v2 (a4 ) ¶ 2 by Lemma 5.6 (ii). If this condition is satis¯ed, then we haveType(EK ) = III¤ ,

550


otherwise we have Type(EK ) = I¤3 . This ¯nal part also proves Theorem 2.5 (iii). We now have to consider the case where the minimal model over Q2 is no longer minimal over K. From (5.1) and (5.2), we deduce 2mQ2 = f (E=K) ¡

2f (E=Q2 ) + mK + 1 + 12vK (u);

where u is de¯ned by (5.4). From Theorem 4.16, it follows that f (E=K) ¡ ¡ 11. Thus we obtain 2mQ2 ¶ ¡ 10 + mK + 12vK (u):

(5.14) 2f (E=Q2 ) ¶ (5.15)

If vK (u) = 1, then mK = 2v2 (¢Q2 ) + 1 ¡ f (E=K). If mK 6= 8; 9, then the reduction type is determined as in the second step in Algorithm 5.7. Thus we assume mK = 8 or 9. Then by (5.15), we have mQ2 ¶ 5. Thus by (2.5) we have Type(E) > IV. If vK (u) ¶ 2, then, we have vK (¢Q2 ) ¶ vK (¢K )+24 ¶ 25. This implies v2 (¢Q2 ) ¶ 13. From [18, Tableau IV], it follows Type(E) > IV. Therefore we assume Type(E) > IV. If Type(E) = IV¤ , then Proposition 2.2 (iii) implies ±(E=K) = 0 and we have nothing to do. Also if Type(E) = I¤0 , then vK (u) = 1 by (5.15) and mK µ 7 by calculation using (5.14). Therefore we have also done in this case. Therefore what remains is divided into two cases (see De¯nition 5.4): (i) Type(E) ¶ I¤1 , (ii) Type(E) ¶ III¤ . The second case is easier to deal with. Lemma 5.9. Suppose that Type(E) ¶ III¤ . Then we can take a model of E over Q2 satisfying v2 (a1 ) ¶ 1; v2 (a2 ) ¶ 2; v2 (a3 ) ¶ 3; v2 (a4 ) ¶ 3; v2 (a6 ) ¶ 5: If this model is not minimal over K, then, after applying [¼K ; 0; 0; 0], the polynomial P (T ) has a triple root if and only if Type(E) = II¤ . Equivalently, P (T ) does not have a triple root if and only if Type(E) = III¤ . Proof. By Tate’s algorithm, we can take a desired model. If it is not minimal, we apply [¼K ; 0; 0; 0] and obtain an integral model over K satisfying vK (a01 ) ¶ 1; vK (a02 ) ¶ 2; vK (a03 ) ¶ 3; vK (a04 ) ¶ 2; vK (a06 ) ¶ 4: In (5.6), we can take ® = ¯ = 0. A similar observation as in Lemma 5.6 implies that the triple root of P (T ) must be congruent to 0 modulo ¼K . Thus the condition for triple root is vK (a04 ) ¶ 3, which is equivalent to vK (a4 ) ¶ 7, namely v2 (a4 ) ¶ 4. Step 10 of Tate’s algorithm tells us that this is equivalent to Type(E) = II¤ . We next consider the ¯rst case Type(E) ¶ I¤1 . If Type(E) = I¤1 , then we have f (E=Q2 ) = 3 and v2 (¢Q2 ) = 4 by [18, Tableau IV]. Therefore this model is also minimal over K. Thus we have done.


551

If Type(E) ¶ I¤2 , then we make use of the following lemma. Lemma 5.10. Assume that Type(E) = I¤n with n ¶ 2. (i) We can take a model of E over Q2 satisfying v2 (a1 ) ¶ 1; v2 (a2 ) = 1; v2 (a3 ) ¶ º; v2 (a4 ) ¶ º + 1; v2 (a6 ) ¶ 2º; if n = 2º ¡

3 (º ¶ 3) is odd and

v2 (a1 ) ¶ 1; v2 (a2 ) = 1; v2 (a3 ) ¶ º + 1; v2 (a4 ) ¶ º + 1; v2 (a6 ) ¶ 2º + 1; if n = 2º ¡ 2 (º ¶ 2) is even. (ii) Let c4 be the invariant of the model taken in (i). If v2 (c4 ) = 6, then we can take a model in (i) with a1 = 0. (iii) If the model in (i) is not minimal over K, then, after applying [¼K ; 0; 0; 0], the 2 polynomial P (T ) has a triple root if and only if vK (a2 + ¼K a1 ¡ ¼K ) ¶ 4 and v2 (a4 ) ¶ 4. (iv) If further n ¶ 3, then the condition in (iii) is equivalent to v2 (c4 ) = 6 when u(K=Q2 ) = 2 and to v2 (c4 ) = 4 when u(K=Q2 ) = 1. Proof. (i) This follows from [18, Part III, Proposition 1]. (ii) If v2 (a1 ) = 1, then we have v2 (b2 ) = 2 and v2 (b4 ) ¶ º +1. These yield v2 (c4 ) = 4. Thus, if v2 (c4 ) = 6, then v2 (a1 ) ¶ 2. Applying [1; 0; ¡ a1 =2; 0], we have a new integral model with a01

= 0;

a02

a21 0 a1 = a2 + ; a3 = a3 ; a04 = a4 + a3 ; a06 = a6 : 4 2

It is easy to verify that this new model satis¯es our requirement. (iii) Applying [¼K ; 0; 0; 0], we have new models satisfying vK (a01 ) ¶ 1; vK (a02 ) = 0; vK (a03 ) ¶ 2º ¡

3; vK (a04 ) ¶ 2º ¡

2; vK (a06 ) ¶ 4º ¡

6

1; vK (a04 ) ¶ 2º ¡

2; vK (a06 ) ¶ 4º ¡

4

for odd n and vK (a01 ) ¶ 1; vK (a02 ) = 0; vK (a03 ) ¶ 2º ¡

for even n. In both cases, it is readily seen that the model is unchanged until Step 6 of Tate’s algorithm. We can take ® = 1 and ¯ = 0 in (5.6) and the new coe±cients are a1 00 = a01 + 2; a2 00 = a02 ¡

a01 ¡

1; a003 = a03 ; a4 00 = a04 ¡

a03 ; a6 00 = a06 :

By a similar observation in the proof of Lemma 5.6, the triple root of P (T ), if exists, is congruent to 0 (mod ¼K ). Thus the conditions for the triple root are

552


vK (a2 00 ) ¶ 2 and vK (a4 00 ) ¶ 3, which are equivalent to vK (a2 + ¼K a1 ¡ ¼ 2 ) ¶ 4 and v2 (a4 ) ¶ 4 as claimed. (iv) We ¯rst note that we have v2 (c4 ) = 4 or 6 by [18, Tableau IV] if Type(E) = I¤n . If u(K=Q2 ) = 2, then, by (5.13), v2 (a2 ) = 1 implies vK (a2 ¡ ¼ 2 ) ¶ 4. Hence the condition vK (a2 + ¼K a1 ¡ ¼ 2 ) ¶ 4 is equivalent to v2 (a1 ) ¶ 2. If this holds, then we have v2 (b2 ) = 3 and v2 (b4 ) ¶ º + 2. Consequently, we obtain v2 (c4 ) = 6. On the other hand, if v2 (a1 ) = 1, then, as we observed in the proof of (ii), we have v2 (c4 ) = 4. The proof for the case u(K=Q2 ) = 1 is similar. This completes the proof of Lemma 5.10. We illustrate how to apply these lemmas by the following example. Example 5.11. Let us consider the case Type(E) = I¤n with n ¶ 9. Two possible values for f (E=Q2 ) are 4 and 6 by [18, Tableau IV]. We consider the latter case here. The former case can be treated similarly. If f (E=Q2 ) = 6, then by (2.4) we see v2 (¢Q2 ) = 10 + n. By Proposition 4.19 (PM2), this curve has potential multiplicative reduction and we have v2 (c4 ) = 6. A model of E over Q2 satisfying the condition in Lemma 5.10 (i) ful¯lls Lemma 5.6 (i). Therefore P (T ) has a triple root. Thus Step 4 of Algorithm 5.7 implies that the model is not minimal. Applying [¼K ; 0; 0; 0], we have a model with discriminant ¢ satisfying vK (¢) = 2(10 + n) ¡ 12 = 8 + 2n. We ¯rst consider a quadratic base change K=Q2 with u(K=Q2 ) = 1. Then by Theorem 4.16 we have f (E=K) = 8. By Lemma 5.10 (iv), the polynomial P (T ) corresponding to the new model does not have a triple root because v2 (c4 ) = 6. In particular, this model is minimal. Therefore we have mK = vK (¢) + 1 ¡ f (E=K) = 2n + 1 ¶ 10 and we conclude Type(EK ) = I¤2n¡4 . Next we consider the case u(K=Q2 ) = 2. Since vK (c4 ) = 12, we have vK (u) µ 3 in (5.14). If vK (u) = 3, then for any minimal model, we have vK (c04 ) = vK (c06 ) = 0. This implies vK (b02 ) = 0. We conclude that EK has multiplicative reduction and f (E=K) = 1. Therefore, if f (E=K) ¶ 2, then we have vK (u) µ 2. By Theorem 4.16, the possible f (E=K)’s are now 3; 4 or 5. It follows from Lemma 5.10 (iv), P (T ) does have a triple root. Since vK (¢) + 1 ¡ f (E=K) ¶ 4 + 2n ¶ 10, we ¯nd that this model is not minimal again. Applying [¼K ; ¤; ¤; ¤], we have a new model with vK (¢) = 2n ¡ 4. This model must be minimal as we have seen. Hence we conclude that mK = 2n ¡ 3 ¡ f (E=K) ¶ 10 and Type(EK ) = I¤2n¡8¡f(E=K ). If vK (u) ¶ 3, then from (5.15), it follows mQ2 ¶ 14 and therefore Type(E) = I¤n with n ¶ 9. We must have v2 (c4 ) ¶ 6. Hence this happens only in the case of Example 5.11. In all other cases, we have v2 (u) µ 2. If v2 (u) = 1, then the model becomes minimal in Lemma 5.10 (iii) and we can determine Type(EK ). Hence we now assume vK (u) = 2. Since the reduction type has already been determined provided mK 6= 8; 9, we assume mK = 8 or 9. Then from (5.15), it follows mQ2 ¶ 11. Thus we ¯nd Type(E) = I¤n with n ¶ 6. We recall that f (E=Q2 ) = 4 or 6.


553

If f (E=Q2 ) = 4, then by Proposition 4.19 (PM1), we have v2 (c4 ) = 4 and v2 (v6 ) = 6. Then any minimal model over K satis¯es vK (c4 ) = vK (c6 ) = 0. As we have seen in Example 5.11, EK has multiplicative reduction. Therefore we assume f (E=Q2 ) = 6. Since we have dealt with the case n ¶ 9 in the above example, we also assume 6 µ n µ 8. An explicit computation shows that the following two cases are all we have to consider. (i) Type(E) = I¤7 ; u(K=Q2 ) = 2. (ii) Type(E) = I¤8 ; u(K=Q2 ) = 2. We prove (i) only. The second one can be proved similarly. In (i), we have v2 (¢Q2 ) = 17. Therefore G(3) is not isomorphic to SD16 . From Theorem 4.16, it follows f (E=K) = 3 or 4 and thus mK = 8 or 7 accordingly. Thus we have only to consider the case f (E=K) = 3. By Lemma 5.10 (i) we can take a model satisfying v2 (a1 ) ¶ 1; v2 (a2 ) = 1; v2 (a3 ) ¶ 5; v2 (a4 ) ¶ 6; v2 (a6 ) ¶ 10: Since P (T ) associated to this model has a triple root, we have v2 (a1 ) ¶ 2 and v2 (c4 ) = 6 by the proof of the lemma. As in the proof of Lemma 5.10 (iv), we apply [¼K ; 0; 0; 0] and then [1; 0; 1; 0] as Tate’s algorithm for this model proceeds. This model is still not minimal. Thus we again apply [¼K ; 0; 0; 0]. It is readily observed that the new model remains unchanged until Step 6. Up to this step, we have the variable change [¼; 0; 0; 0] ¯ 2 [1; 0; 1; 0] ¯ [¼; 0; 0; 0] = [¼K ; 0; ¼K ; 0] and we obtain a new model satisfying vK (a01 ) = 1; vK (a02 ) ¶ 0; vK (a03 ) ¶ 4; vK (a04 ) ¶ 3; vK (a06 ) ¶ 8:

(5.16)

4 0 2 Here we take a1 = 0 by Lemma 5.10 (ii) and we have ¼K a2 = a2 ¡ ¼K . This model remains unchanged again until Step 6 of Tate’s algorithm. If vK (a02 ) = 0, we have ® = 1 and ¯ = 0 in (5.6). The total variable change is given 2 2 2 by [1; 0; 1; 0] ¯ [¼K ; 0; ¼K ; 0] = [¼K ; 0; ¼K + ¼K ; 0]. The condition for P (T ) to have a triple root is then µ ¶ 1 00 2 3 4 vK (a2 ) = vK (a2 ¡ ¼K ¡ 2¼K ¡ ¼K ) ¶ 2: 4 ¼K 2n+2 2n By a similar manner in proving (5.13), we can show 2n ² ¼K (mod ¼K ) if u(K=Q2 ) = 2. Thus the ¼K -adic expansion of a2 consists only of even-power terms. This implies 2 4 ¡ ¼K vK (a2 ¡ ¼K ) ¶ 6. Thus we have vK (a002 ) = 1 and we conclude that P (T ) has a double root and a single root. Thus we ¯nd Type(EK ) = I¤n with some n ¶ 1. We compute this n by Tate’s algorithm. It is easy to verify that our current model satis¯es

vK (a001 ) = 1; vK (a002 ) = 1; vK (a003 ) ¶ 4; vK (a004 ) ¶ 3; vK (a006 ) ¶ 8: Since Y 2 +a003;2 Y ¡ a006;4 ² Y 2 (mod ¼K ), this has a double root. Thus we have Type(EK ) > I¤1 . Suppose that a002;1 X 2 + a004;3 X + a006;5 has a double root. Then the root is 0. Since Y 2 + a003;3 Y ¡ a006;6 has a double root, we then have Type(EK ) > I¤3 and mK > 8. This is impossible. Therefore a002;1 X 2 +a004;3 X +a006;5 has distinct roots and we conclude Type(EK ) = I¤2 .

554


If vK (a02 ) ¶ 1, then ® = ¯ = 0 in (5.6). Thus the model is unchanged and satis¯es 4 0 2 2 (5.16). As before, vK (¼K a2 ) = vK (a2 ¡ ¼K ) ¶ 1 implies vK (a2 ¡ ¼K ) ¶ 2. Therefore 00 P (T ) has a triple root. If v2 (a4 ) ¶ 4, then the model becomes non-minimal. Thus we have v2 (a004 ) = 3. We can easily deduce Type(EK ) = III¤ by Tate’s algorithm. Having covered all possible cases, the proof of Theorem 2.5 is now complete. Note that, if we have not determined the value of mK , we would have to give sharper estimates for the coe±cients of models coming out from Tate’s algorithm and the proof would be more complicated. We conclude this paper by a comment on the general case. Our assumptions on the base ¯eld and on the degree of base change enable us to limit the number of ¯elds with given rami¯cation property and thus to determine the possible higher rami¯cation groups. Hence if we ease the assumptions, we might need more data to determine the conductor and the reduction type. Therefore we might not be able to expect the `simple’ description like Theorem 4.16.

Acknowledgements The author would like to thank Professor Syuji Yamagata for his various advice and useful discussion during the preparation of this paper.

References [1] Clemens Adelmann: The decomposition of primes in torsion point ¯elds, SpringerVerlag, Berlin, 2001. [2] Pilar Bayer and Anna Rio: \Dyadic exercises for octahedral extensions", J. Reine Angew. Math., Vol. 517, (1999), pp. 1{17. [3] J.E. Cremona: Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, second edition, 1997. [4] Marcus du Sautoy and Ivan Fesenko: \Where the wild things are: rami¯cation groups and the Nottingham group", In: New horizons in pro-p groups, BirkhÄauser Boston, Boston, MA, 2000, pp. 287{328. [5] Jean-Marc Fontaine: \Groupes de rami¯cation et repr¶esentations d’Artin", Ann. ¶ Sci. Ecole Norm. Sup. (4), Vol. 4, (1971), pp. 337{392. [6] Helmut Hasse: \Arithmetische Theorie der kubischen ZahlkÄorper auf klassenkÄorpertheoretischer Grundlage", Math. Z., Vol. 31, (1930), pp. 565{582. [7] Luise-Charlotte Kappe and Bette Warren: \An elementary test for the Galois group of a quartic polynomial", Amer. Math. Monthly, Vol. 96, (1989), pp. 133{137. [8] Masanari Kida: \Rami¯cation in the division ¯elds of an elliptic curve", To appear in Abh. Math. Sem. Univ. Hamburg. [9] Masanari Kida; \Computing elliptic curves using KASH", In: Arjeh M. Cohen, Xiao-Shan Gao, Nobuki Takayama, (Eds): Mathematical Software, World Scienti¯c, 2002, pp. 250{260.


555

[10] Alain Kraus: \Sur le d¶efaut de semi-stabilit¶e des courbes elliptiques µa r¶eduction additive", Manuscripta Math., Vol. 69, (1990), pp. 353{385. [11] Pascual Llorente and Enric Nart: \E®ective determination of the decomposition of the rational primes in a cubic ¯eld", Proc. Amer. Math. Soc., Vol. 87, 1983, pp. 579{585. [12] Paul Lockhart, Michael Rosen, Joseph H. Silverman: \An upper bound for the conductor of an abelian variety", J. Algebraic Geom., Vol. 2(1993), pp. 569{601. [13] E. Maus: \Arithmetisch disjunkte KÄorper", J. Reine Angew. Math., Vol. 226, (1967), pp. 184{203. [14] Hirotada Naito: \Dihedral extensions of degree 8 over the rational p-adic ¯elds", Proc. Japan Acad. Ser. A Math. Sci., Vol. 71(1995), pp. 17{18. [15] Hirotada Naito: \Local ¯elds generated by trisection points of elliptic curves", S¹ urikaisekikenky¹usho K¹oky¹uroku, Vol. 971, (1996), pp. 153{159, Algebraic number theory and Fermat’s problem (Japanese), Kyoto, 1995. [16] Hirotada Naito: \Local ¯elds generated by 3-division points of elliptic curves", Proc. Japan Acad. Ser. A Math. Sci., Vol. 78, (2002), pp. 173{178. [17] JÄ urgen Neukirch: Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992. [18] Ioannis Papadopoulos: \Sur la classi¯cation de N¶eron des courbes elliptiques en caract¶eristique r¶esiduelle 2 et 3", J. Number Theory, Vol. 44, (1993), pp. 119{152. [19] Sebastian Pauli: \Factoring polynomials over local ¯elds", J. Symbolic Comput., Vol. 32, (2001), pp. 533{547. [20] Sebastian Pauli and Xavier-Fran»cois Roblot: \On the computation of all extensions of a p-adic ¯eld of a given degree", Math. Comp., Vol. 70, (2001), pp. 1641{1659 (electronic). [21] Jean-Pierre Serre: Corps locaux. Hermann, Paris, 1968. Publications de l’Universit¶e de Nancago, No. VIII.

Deuxiµeme ¶edition,

[22] Jean-Pierre Serre: \Propri¶et¶es galoisiennes des points d’ordre ¯ni des courbes elliptiques", Invent. Math., Vol. 15, (1972), pp. 259{331. [23] Jean-Pierre Serre and John Tate: \Good reduction of abelian varieties", Ann. of Math. (2), Vol. 88, (1968), pp. 492{517. [24] Joseph H. Silverman: The arithmetic of elliptic curves, Springer-Verlag, New York, 1986. [25] Joseph H. Silverman: Advanced topics in the arithmetic of elliptic curves, SpringerVerlag, New York, 1994. [26] Andr¶e Weil: \Exercices dyadiques", Invent. Math., Vol. 27, (1974), pp. 1{22. [27] Masakazu Yamagishi: \On the number of Galois p-extensions of a local ¯eld", Proc. Amer. Math. Soc., Vol. 123, (1995), pp. 2373{2380. [28] Sunao Yamamoto: \On a property of the Hasse’s function in the rami¯cation theory", Mem. Fac. Sci. Kyushu Univ. Ser. A, Vol. 22, (1968), pp. 96{109.

556


Type(E)

Type(EK )

I0

I0

In

Ine

II

I0

e ² 0 (mod 6)

II

e ² 1 (mod 6)

IV

e ² 2 (mod 6)

p 6= 2

I¤0

e ² 3 (mod 6)

p 6= 3

IV¤

e ² 4 (mod 6)

p 6= 2

II¤

e ² 5 (mod 6)

I0

e ² 0 (mod 4)

III

e ² 1 (mod 4)

I¤0

e ² 2 (mod 4)

III¤

e ² 3 (mod 4)

I0

e ² 0 (mod 3)

IV

e ² 1 (mod 3)

IV¤

e ² 2 (mod 3)

III

IV

e = e(K=k)

condition on p

p 6= 2; 3

p 6= 2

p 6= 2

p 6= 3

Table 1 Variation of reduction type (tame case)


Type(E)

Type(EK )

e = e(K=k)

I¤0

I0

e ² 0 (mod 2)

I¤0

e ² 1 (mod 2)

Ine

e ² 0 (mod 2)

I¤ne

e ² 1 (mod 2)

I0

e ² 0 (mod 3)

IV¤

e ² 1 (mod 3)

IV

e ² 2 (mod 3)

I0

e ² 0 (mod 4)

III¤

e ² 1 (mod 4)

I¤0

e ² 2 (mod 4)

III

e ² 3 (mod 4)

I0

e ² 0 (mod 6)

II¤

e ² 1 (mod 6)

IV¤

e ² 2 (mod 6)

p 6= 2

I¤0

e ² 3 (mod 6)

p 6= 3

IV

e ² 4 (mod 6)

p 6= 2

II

e ² 5 (mod 6)

I¤n IV¤

III¤

II¤

condition on p p 6= 2

p 6= 2

p 6= 3

p 6= 2

p 6= 2

p 6= 2; 3

Table 1 (Cont.) Variation of reduction type (tame case)

557

Table 2 Variation of reduction type at 3

3

4

5

11

12

13

II

¤

11

2

5

10

9

4

9

IV¤

III¤

3

6 6+n

I¤0 I¤n

2 2

9

6

n2 2 3

9

6

2 2 n 2 3

IV¤

IV¤

III III¤ IV¤

II

I¤0 I¤3n III II II

7

3 n 0 4

2

7

3 n 2 4

2 2

IV I0 II¤ II¤

¤

III

IV

II I¤0 IV

I¤0 I¤3n

4 n 2 3

3

4 n 2 3 2

3

2 2

II¤ III II

IV

¤

IV III¤ IV¤ III

II

I¤0 I¤3n

Q3 C1 C2 C3 Type(E) v3 (¢Q3 ) f (E=Q3 ) f (E=K) Type(EK ) f (E=K) Type(EK ) f (E=K) Type(EK ) I0 0 0 0 I0 0 I0 0 I0 In n 1 1 I3n 1 I3n 1 I3n n 2 III¤ II 3 3 3 IV¤ 3 IV ¤ 3 IV¤ n I0 0 ¤ 4 4 6 IV 4 II¤ 4 II¤ n 2 III 5 5 9 IV¤ 7 II¤ 3 II ¤ ¤ ¤ III 3 2 2 III 2 III 2 III n 2 III IV 5 3 3 II 3 II 3 II n ¤ 2 I0 6 4 6 II 4 IV 4 IV n 2 III¤ 7 5 9 II 7 IV 3 IV¤ y 2 = x3 + 1 y 2 + y = x3 ¡ y 2 + xy = x3 ¡

6x + 8

63

7 x2 ¡

27

y 2 + y = x3 ¡

42x ¡

61

x2 + 25x + 1

270x ¡ 1708 y 2 + xy + y = x3 ¡

y 2 + y = x3 ¡

y 2 = x3 ¡

y 2 + y = x3 + 20

y 2 + xy = x3 ¡

y 2 + y = x3 ¡

100

x2 ¡ 5x + 5

y 2 + y = x3 ¡ 3x ¡ 5 y 2 + xy = x3 ¡ x2 ¡ 5

y 2 + y = x3 + 2

30x ¡ x2 ¡

1

y 2 + y = x3 ¡

y 2 + xy + y = x3 ¡

y 2 + y = x3

Example y 2 + y = x3 ¡ x2 ¡ 10x ¡ 20 y 2 + xy + y = x3 ¡ x

558 M. Kida / Central European Journal of Mathematics 4 (2003) 510{560

Table 3 Variation of reduction type at 2

I¤4

I¤3

I¤1 I¤2

IV I¤0

III

6

14

6

10

3 4 6 7 4 5 4

5

9

8 10 12 13 11 12 12

7 3 5 7 8 2 4

6

7 4 6 8 9 4 8

6

GL2

GL2 SD16

8

otherwise

8

3 3 8 10 3 6 0

8

10 3 6 10 12 2 3 2 6

8

fL 0 1 3 2 8

GL2

otherwise

GL2

otherwise

SD16

otherwise

Q2 Type(E) v2 (¢Q2 ) f (E=Q2 ) condition I0 0 0 In n 1 II 4 4 GL2 Q1 Q2 Q3 Type(EK ) f (E=K) Type(EK ) f (E=K) Type(EK ) I0 0 I0 0 I0 I2n 1 I2n 1 I2n I¤1 4 I¤0 4 I¤0 ¤ ¤ IV 4 I0 4 I¤0 ¤ ¤ ¤ ¤ I0 III or I3 5 III or I¤3 n5 ¤ I0 4 0 II I¤0 0 4 I0 II¤ ¤ ¤ I0 8 I2 8 I¤2 ¤ ¤ I1 3 I1 3 I¤1 I¤2 5 I¤3 5 I¤3 ¤ ¤ I2 8 I4 8 I¤4 ¤ ¤ I2 10 I4 10 I¤4 ¤ ¤ IV 2 IV 2 IV¤ III 4 II 4 II IV 4 II 4 II II 5 III 5 III n 4 I¤0 3 I¤1 II ¤ 3 4 I1 I¤0 n I¤0 4 2 IV¤ II ¤ 2 4 IV I¤0 III 3 III 3 III ¤ ¤ I1 4 I0 4 I¤0 I¤0 5 I¤3 5 I¤3 ¤ ¤ I0 8 I2 8 I¤2 ¤ ¤ III 4 I2 4 I¤2 ¤ ¤ I2 5 III 5 III¤ ¤ I0 I4 4 I¤4 n4 3 III 4 II I¤4 4 3 II III Example y 2 ¡ y = x3 +x2 ¡ 10x¡ 20 y 2 +xy +y = x3 ¡ 11x+12 y 2 = x3 + x2 + x y 2 = x3 ¡ x2 ¡ x y 2 = x3 + x y 2 = x3 ¡ x2 ¡ x ¡ 1 y 2 = x3 + x2 ¡ x + 1 y 2 = x3 + x2 ¡ 2x ¡ 2 y 2 = x3 ¡ x2 + x y 2 = x3 ¡ x y 2 = x3 + x2 + x + 1 y 2 = x3 + x2 ¡ 3x + 1 y 2 = x3 ¡ 27 y 2 = x3 + x2 ¡ 4x ¡ 4 y 2 = x3 ¡ x2 + 4x ¡ 4 y 2 = x3 ¡ 11x ¡ 14 y 2 = x3 ¡ x2 + 3x ¡ 3 y 2 = x3 + x2 + 3x + 3 y 2 = x3 + x2 ¡ 5x ¡ 5 y 2 = x3 ¡ x2 ¡ 5x + 5 y 2 = x3 ¡ x2 ¡ 4x + 4 y 2 = x3 + x2 ¡ 64x ¡ 220 y 2 = x3 ¡ 4x y 2 = x3 + x2 ¡ 9x + 7 y 2 = x3 + x2 + 16x + 180 y 2 = x3 + 4x y 2 = x3 + x2 ¡ 12 y 2 = x3 + x2 ¡ 17x + 15 y 2 = x3 ¡ x2 ¡ 17x ¡ 15 ?

?

?

?

??

note

M. Kida / Central European Journal of Mathematics 4 (2003) 510{560 559

II¤

IV III¤

6

6

14

10 + n

I¤n (n ¶ 9)

6

2 3 5 7 8 3 4

18

I¤8

6

6

8 10 12 14 15 11 12

17

I¤7

¤

16

8 8

otherwise

2 3 6 10 12 3 0

8

8

8

8

fL 1 8

GL2

Q2 v2 (¢Q2 ) f (E=Q2 ) condition 8+n 4 15 6

I¤6

Type(E) I¤n (n ¶ 5) I¤5

Q1 Q2 Q3 Type(EK ) f (E=K) Type(EK ) f (E=K) Type(EK ) I2n¡8 4 I¤2n¡4 4 I¤2n¡4 ¤ I6 III 5 III n5 ¤ 3 I 4 I¤0 1 I¤8 ¤ 4 3 I0 I¤1 n ¤ ¤ I2 3 4 III I¤10 4 3 I¤2 III¤ n 0 I0 4 I¤4 I¤12 ¤ 4 0 I4 I0 n ¤ I I 4 1 2n¡16 2n¡12 I¤2n¡4 1 4 I¤2n¡12 I2n¡16 IV 2 IV 2 IV I¤1 3 I¤1 3 I¤1 ¤ ¤ I2 5 I3 5 I¤3 ¤ ¤ I2 8 I4 8 I¤4 ¤ ¤ I2 10 I4 10 I¤4 ¤ ¤ III 3 III 3 III¤ ¤ I0 II 4 II¤ n4 4 II 3 III II¤ 3 4 III II n IV 4 II 2 II¤ 2 4 IV II¤

Example note 2 3 2 y = x ¡ x + 8x ¡ 16 y 2 = x3 ¡ 44x ¡ 112 y 2 = x3 + x2 ¡ 97x ¡ 385 ? y 2 = x3 ¡ x2 ¡ 97x + 385 y 2 = x3 + x2 + 63x ¡ 1377 ? y 2 = x3 ¡ x2 + 63x + 1377 y 2 = x3 + x2 ¡ x + 95 ? y 2 = x3 ¡ x2 ¡ x ¡ 95 y 2 = x3 + x2 ¡ 673x ¡ 6945 ? y 2 = x3 ¡ x2 ¡ 673x + 6945 2 3 2 y = x + x + 4x + 4 y 2 = x3 + x2 ¡ 24x ¡ 36 y 2 = x3 + x2 + 3x + 11 y 2 = x3 + x2 + 3x ¡ 5 y 2 = x3 + x2 ¡ 13x ¡ 21 y 2 = x3 ¡ x2 + 16x ¡ 180 y 2 = x3 + x2 ¡ 5x ¡ 13 y 2 = x3 ¡ 16x ¡ 32 ? y 2 = x3 ¡ 16x + 32 y 2 = x3 + x2 + 11x + 19 ? y 2 = x3 ¡ x2 + 11x ¡ 19

560 M. Kida / Central European Journal of Mathematics 4 (2003) 510{560

Table 3 (cont.) Variation of reduction type at 2

CEJM 4 (2003) 561{572

Representation of Hilbert algebras and Implicative Semilattices Sergio A. Celani ¤ Departamento de Matem¶atica, Facultad de Ciencias Exactas, Universidad Nacional del Centro and CONICET, Pinto 399, 7000 Tandil, Argentina

Received 13 March 2003; accepted 26 June 2003 Abstract: In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring. c Central European Science Journals. All rights reserved. ® Keywords: Hilbert algebras, implicative semilattices, topological representation, duality, Boolean ring. MSC (2000): 03G25, 06A12, 06E15

1

Introduction

¤

In this paper we shall continue the line of research initiated in [2] and [3] on the representation of Hilbert algebras and distributive meet-semilattices. In [2] it was mentioned that some results on the representation for Hilbert algebras by means of ordered sets can be used to give a representation for implicative semilattices. On the other hand, in [3] a topological duality for distributive meet-semilattices in terms of ordered topological spaces was developed. As shown in [3], this approach is particularly useful to give a duality for homomorphism of distributive meet-semilattices. Since implicative semilattices are also distributive meet-semilattices, we can apply the mentioned duality to give a duality of implicative semilattices, this being the main objective of this paper. As we will show, many of the results can also be applied to give a topological representation for Hilbert algebras. We prove that every Hilbert algebra is isomorphic to a Hilbert subalE-mail: [email protected]

562

S.A. Celani / Central European Journal of Mathematics 4 (2003) 561{572

gebra of an implicative semilattice. These results are a re¯nement of the representation given by A. Diego in [4], which assert that every Hilbert algebra is isomorphic to a Hilbert subalgebra of a Heyting algebra given as the open subsets of a T0 topological space. We will show that our representation for Hilbert algebras is a duality just when they are implicative semilattices. Thus, we do not give a full duality for Hilbert algebras using this topological representation. The existence of a duality for Hilbert algebras remains open to futher study. In Section 2 we introduce the de¯nitions and necessary concepts to develop this paper. In Section 3 we present the topological representation for Hilbert algebra and implicative semilattices. Applying the results given in [3] on the duality between homomorphism of distributive meet-semilattices and meet-relations, we prove that there exists a duality between homomorphism of implicative semilattices and the so called functional relations. To conclude, we give a topological representation for Boolean implicative semilattices, also called Boolean rings.

2

Preliminaries

Let us recall that a meet-semilattice with last element 1 is an algebra hA; ^; 1i of type (2; 0) such that the operation ^ is idempotent, commutative, associative, and a ^ 1 = a, for all a 2 A: As usual, the binary relation µ de¯ned by a µ b if and only if a ^ b = a is an partial order. In what follows we will say semilattice instead of meet-semilattice. A ¯lter of a semilattice A is a subset F ³ A such that 1 2 F; if a µ b and a 2 F , then b 2 F; and if a; b 2 F; then a ^ b 2 F . The ¯lter generated by a subset H ³ A we will denoted by F (H ) : If H = fag; then we will write F (a) : We will denote by F i (A) the set of all ¯lters of A: A semilattice A is called distributive semilattice, or DS-algebra, if for all a; b; c 2 A such that a ^ b µ c there exist a1 , b1 2 A such that a µ a1 ; b µ b1 and c = a1 ^ b1 . Consequently, the set F i (A) is a lattice. A semilattice A is distributive semilattice if and only if the lattice of the ¯lters F i (A) is a distributive lattice (see [3]). Let X be a set. The set of all subsets of X is denoted by P (X ). A subset K ³ P (X ) is called dually directed if for any U; V 2 K there exists W 2 K such that W ³ U \ V: Let us consider a poset hX; µi. A subset U ³ X is said to be increasing (decreasing) if for all x; y 2 X such that x 2 U (y 2 U ) and x µ y, we have y 2 U (x 2 U ). The set of all increasing subsets of X is denoted by Pi (X ). We note that if hX; µi is a poset then hPi (X) ; \; X i is a distributive semilattice. For each Y ³ X; the increasing (decreasing) set generated by Y is [Y ) = fx 2 X : 9y 2 Y : y µ xg ((Y ] = fx 2 X : 9y 2 Y : x µ yg). If Y = fyg; then we will write [y) and (y] instead of [fyg) and (fyg], respectively. The set complement of a subset Y ³ X will be denoted by Y c or X ¡ Y: De¯nition 2.1. A Hilbert algebra, or H -algebra, is an algebra hA; !; 1i where A is a non-empty set, ! is a binary operation on A; 1 is an element of A such that the following conditions are satis¯ed for every a; b; c 2 A :


563

H1 a ! (b ! a) = 1: H2 (a ! (b ! c)) ! ((a ! b) ! (a ! c)) = 1: H3 If a ! b = 1 and b ! a = 1; then a = b: Let A be a Hilbert algebra. It is known that the relation µ de¯ned by a µ b i® a ! b = 1 is a partial order on A where 1 is the largest element of A with respect to this ordering (see [1, 5, 4], or [7]). De¯nition 2.2. [6, 8, 9] An implicative semilattice, or IS-algebra, is an algebra hA; ^; !; 1i where A is a non-empty set, ^ and ! are binary operations on A; and 1 is an element of A such that the following conditions are satis¯ed for every a; b; c 2 A : IS1 a ! a = 1 IS2 (a ! b) ^ b = b IS3 a ^ (a ! b) = a ^ b IS4 a ! (b ^ c) = (a ! c) ^ (a ! b) : If hA; ^; !; 1i is an IS-algebra, then hA; !; 1i is an H -algebra and hA; ^; 1i is a DSalgebra. On the other hand, if hA; !; 1i is a Hilbert algebra such that for any a; b 2 A the set A (a; b) = fx 2 A : a µ b ! xg has a least element x; then A is an IS-algebra (see [6]). A deductive system of an H-algebra or an IS-algebra A is a subset D of A such that 1 2 D; and if a; a ! b 2 D; then b 2 D: The set of all deductive systems of A is denoted by Ds (A) : It is known that if A is an IS-algebra, then Ds (A) = F i (A) (see [6]). Let X ³ A: The deductive system generated by X; in symbols hXi ; is the set (see [7]) hXi = fa 2 A : 9 fx0 ; :::; xn g ³ X : x0 ! (::: (xn ! a) :::) = 1g : If A is an IS-algebra, then hXi = fa 2 A : 9 fx0 ; :::; xn g ³ X : x0 ^ ::: ^ xn µ ag : Let A be an H -algebra or an IS-algebra. Let D be a deductive system of A: We shall say that D is irreducible if for any D1 ; D2 2 Ds (A) such that D = D1 \ D2 ; then D = D1 or D = D2 : A similar notion can be de¯ned for ¯lters of DS-algebras (see [3]). The set of all irreducible deductive systems of A will be denoted by X (A). Theorem 2.3. [2, 3] Let A be an H-algebra or an IS-algebra. Then (1) D 2 X (A) if for every a; b 2 = D there exists c 2 = D such that a µ c and b µ c: (2) For all a; b 2 A if a £ b there exists D 2 X (A) such that a 2 D and b 2 = D: Let hA; µi be a poset. A subset I of A is called an order-ideal of A if I is decreasing and for all a; b 2 I there exists an element c 2 I such that a µ c and b µ c: The following result was proved for H-algebras in [2] and for DS-algebras in [3]. Theorem 2.4. Let A be an H-algebra or an IS-algebra. Let D 2 Ds (A) and let I be an order-ideal of A such that F \ I = ;: Then there exists P 2 X (A) such that D ³ P and P \ I = ;:

564

3


Topological representation

De¯nition 3.1. Let us consider the structure F = hX; µ; Di ; where hX; µi is a poset and D ³ Pi (X ) such that X 2 D: We shall say that F is a general H-frame if for every U; V 2 D; U ) V = fx 2 X : [x) \ U ³ V g 2 D: We will say that F is a general IS-frame if it is an H -frame such that for every U; V 2 D; U \ V 2 D. We note that if F = hX; µ; Di is a general H -frame then hD; ); X i is an H -algebra, and if F is a general IS-frame then hD; \; ); X i is an IS-algebra. Theorem 3.2. Let A be an H -algebra or an IS-algebra. Then: (1) A is isomorphic to a subalgebra of Pi (X (A)) by means of the mapping ¯A : A ! Pi (X (A)) de¯ned by ¯A (a) = fP 2 X (A) : a 2 P g : (2) The family ¯A (A)c = f¯A (a)c : a 2 Ag is a basis for a topology on X (A) : Proof. (1) It is easy (see [2] and [3]). S (2) Since the irreducible deductive systems are proper, we have that X (A) = ¯A (a)c : a2A

Moreover, by Theorem 2.3 we deduce that for any a; b 2 A and for any P 2 X (A) ; if P 2 ¯A (a)c \ ¯A (b)c ; then there exists c 2 A such that P 2 ¯A (c)c ³ ¯A (a)c \ ¥ ¯A (b)c : Thus, ¯A (A)c = f¯A (a)c : a 2 Ag is a basis for a topology on X (A) :

If A is an H-algebra or an IS-algebra, then the structure F (A) = hX (A) ; ³; ¯A (A)i is a general H-frame or a general IS-frame, respectively. On F (A) we can de¯ne a topology using the set ¯A (A)c as a base. This topological space will be called the topological space associated with A: Let us consider the following subset of Pi (X (A)) : D (X (A)) = fU ³ X (A) : U c is compact-open in the topological space F (A)g : Theorem 3.3. Let A be an H -algebra. Then hD (X (A)) ; \; ); Xi is an IS-algebra. Proof. Let U 2 D (X (A)) : Since ¯A (A)c is a basis of the space F (A), Uc =

[

f¯A (a)c : a 2 B ³ Ag ;

and as U c is compact, there exists fa1 ; :::; an g ³ B such that U c = ¯A (a1 )c [ ::: [ ¯A (an )c = (¯A (a1 ) \ ::: \ ¯A (an ))c : So, every U 2 D (X (A)) is a ¯nite intersection of elements of the base ¯A (A)c : Thus, U \ V 2 D (X (A)) ; for every U; V 2 D (X (A)).


565

Let U; V 2 D (X (A)) : We prove that U ) V 2 D (X (A)) : As there exist a1 ; :::; an ; b1 ; :::; bk 2 A such that U = ¯A (a1 ) \ ::: \ ¯A (an ) and V = ¯A (b1 ) \ ::: \ ¯A (bk ) ; then U ) V = U ) (¯A (b1 ) \ ::: \ ¯A (bk )) = (U ) ¯A (b1 )) \ ::: \ (U ) ¯A (bk )) = (¯A (a1 ) ) (::: (¯A (an ) ) ¯A (b1 )) :::)) \ :::: \ (¯A (a1 ) ) (::: (¯A (an ) ) ¯A (bk )) :::)) : By assumption (¯A (a1 ) ) (::: (¯A (an ) ) ¯A (bj )) :::)) 2 D (X (A)) for any 1 µ j µ k: ¥ Then, U ) V 2 D (X (A)), and thus D (X (A)) is an IS-algebra. Theorem 3.4. Let A be an H -algebra. Then the mapping ¯A : A ! D (X (A)) is an onto homomorphism of Hilbert algebras if and only if A is an IS-algebra. Proof. If A is an IS-algebra, then is also a DS-algebra. Thus, by the results of [3], we have that the mapping ¯A is onto. Suppose that ¯A is onto. Let a; b 2 A: As ¯A (a) ; ¯A (b) 2 D (X (A)) ; ¯A (a) \ ¯A (b) 2 D (X (A)) : Hence ¯A is onto, there exists c 2 A such that ¯A (c) = ¯A (a) \ ¯A (b) : We prove that c = a^b: As, ¯A (c) ³ ¯A (a) ; ¯A (c) ³ ¯A (b) and ¯A is an order-isomorphisms, then c µ a and c µ b: Let z 2 A such that z µ a and z µ b: Then, ¯A (z) ³ ¯A (a) and ¯A (z) ³ ¯A (b) implies that ¯A (z) ³ ¯A (a) \ ¯A (b) = ¯A (c). So, z µ c and therefore c = a ^ b: With this we have hA; î is a ^-semilattice. We prove the axiom IS4 of De¯nition 2.2. The other axioms are straight forward and left to the reader. Let a; b; c 2 A: Since, ¯A (a ! (b ^ c)) = ¯A (a) ) ¯A (b ^ c) = ¯A (a) ) (¯A (b) \ ¯A (c)) = (¯A (a) ) ¯A (b)) \ (¯A (a) ) ¯A (c)) = (¯A (a ! b)) \ (¯A (a ! c)) = ¯A ((a ! b) ^ (a ! c)) ; and taking into account that ¯A is an order-isomorphisms, we get a ! (b ^ c) = (a ! b)^ ¥ (a ! c) : Thus, hA; !; ^; 1i is an IS-algebra. Remark. From Theorem 3.2 and Theorem 3.3 we have that a Hilbert algebra A is isomorphic to a subalgebra of the implicative semilattice D (X (A)). This result extends the topological representation given by A. Diego in [4]. On other hand, by Theorem 3.4 we deduce that this representation is not a full duality for H -algebras, because A is isomorphic to D (X (A)) if and only if D (X (A)) is an IS-algebra.

4

Duality

Let F = hX; µ; Di be a general IS-frame. Let us consider the mapping HX : X ! X (D) given by HX (x) = fU 2 D : x 2 U g :

566


Let hX; T i be a topological space. The set of all compact and open subsets of X will be denoted by K O (X ), and the set fU : U c 2 K O (X)g will be denoted by D (X ) or D: An ordered topological space is a triple hX; µ; T i where hX; T i is a topological space and hX; µi is a poset. An ordered topological space hX; µ; T i where K O (X) is a basis for the topology T will be denoted by hX; µ; Di. Let us recall that a DS-space is an ordered topological space hX; µ; Di where K O (X ) forms a basis for the topology, all closed subsets are increasing and the mapping HX : X ! X (D) is onto and an order-isomorphism (see [3]). De¯nition 4.1. An IS-space is a structure hX; µ; Di such that it is a DS-space and a general IS-frame. Applying the results given in [3] for DS-algebras and DS-spaces we have the next two theorems. Theorem 4.2. Let A be an IS-algebra. Then the structure F (A) is an IS-space such that the mapping ¯A : A ! D (X (A)) is an isomorphism of IS-algebras. Theorem 4.3. Let F = hX; µ; Di be an IS-space. Then the mapping HX : X ! X (D) is an order-isomorphism and a homeomorphism. Thus, we have that there exists a duality between IS-spaces and IS-algebras. Now we will give other characterization of IS-spaces. Lemma 4.4. Let F = hX; µ; Di be an H-frame or an IS-frame. Then for each x 2 X; the set HX (x) = fU 2 D : x 2 U g belongs to X (D) if and only if the set D c = fU c : U 2 Dg is a basis for a topology de¯ned on X: Proof. Suppose that HX (x) 2 X (D) for every x 2 X: Since HX (x) is proper, then S there exists U 2 D such that x 2 = U . Thus, X = fU c : U 2 Dg : Let U; V 2 D and let x 2 U c \ V c . So, U; V 2 = HX (x) ; and since HX (x) is irreducible, then there exits W 2 D such that W 2 = HX (x) and U ³ W and V ³ W . Thus, x 2 W c ³ U c \ V c : We conclude that D c is a basis for a topology on X: ¥ If D c is a basis for a topology on X; then it is easy to see that HX (x) 2 X (D) : De¯nition 4.5. Let F = hX; µ; Di be a general IS-frame such that HX (x) 2 X (D) for every x 2 X. We shall say that: (1) F is saturated if for each x 2 X and for pair (¡; §) ³ D £ D; with §c =fU 2 D : U 2 = §g dually directed, such that for any fO1 ; :::; On g ³ ¡ and for any fV1 ; :::; Vk g ³ §; [x) \ O1 \ ::: \ On \ V1c \ ::: \ Vkc 6= ;;


then it holds [x) \

\

¡\

\

567

§c 6= ;:

(2) F is replete if for each x; y 2 X such that HX (x) ³ HX (y) ; there exists z 2 X with x µ z and HX (z) = HX (y). Theorem 4.6. Let F = hX; µ; Di be a general IS-frame such that HX (x) 2 X (D) for every x 2 X . If F is saturated, then it is replete. Proof. Let us suppose that F is saturated. Let x; y 2 X such that HX (x) ³ HX (y) : It is clear that HX (y)c = fU 2 D : x 2 = U g is a subset of D dually directed. Moreover, for every fO1 ; :::; On g ³ HX (y) and for every fV1 ; :::; Vk g ³ HX (y) ; we have that [x) \ O1 \ ::: \ On \ V1c \ ::: \ Vkc 6= ;; c c because in opposite case, as HX (y) is an order-ideal, then there exists W 2 = HX (y) such that W c ³ V1c \ ::: \ Vkc : So, [x) \ O1 \ ::: \ On \ W c 6= ; implies that x 2 O1 ) (O2 ) : : : (On ) W ) : : :) 2 HX (x) ³ HX (y), and as HX (y) is a deductive system of D, T T W 2 HX (y) ; which is a contradiction. Thus, [x) \ HX (x) \ HX (y)c 6= ;, i.e., there ¥ exists z 2 X such that x µ z and HX (z) = HX (y) :

Theorem 4.7. Let F = hX; µ; Di be an IS-frame such that the mapping HX is onto and HX (x) 2 X (D) for every x 2 X. Then F is saturated if and only if F is replete. Proof. The direction ) follows by Theorem 4.6. Suppose that F is replete. Let us consider subsets ¡ and § of D such that §c is dually directed. Suppose that [x) \ O1 \ ::: \ On \ V1c \ ::: \ Vkc 6= ;; for any fO1 ; :::; On g ³ ¡ and for any fV1 ; :::; Vk g ³ §: Let us consider the deductive system h¡ [ HX (x)i and the order-ideal I (§) = fU 2 D : 9V 2 § : U ³ V g : Since § is dually directed, I (§) is an ideal. Then, h¡ [ HX (x)i \ I (§) = ;; because in opposite case there exist U1 ; :::; Un 2 ¡ and V 2 § such that U1 ) (U2 ) : : : (Un ) V ) : : :) 2 HX (x) : So, [x) \ U1 \ ::: \ Un \ V c = ;; which is a contradiction. Thus, there exists P 2 X (D) such that ¡ ³ P , HX (x) ³ P and § \ P = ;: Since the mapping HX is onto, there exists y 2 X such that P = HX (y) : Thus, HX (x) ³ HX (y) ; and as F is replete, there exists z 2 X such that x µ z and T T ¥ HX (z) = HX (y) : Therefore, z 2 [x) \ ¡ \ §c and F is saturated.

Theorem 4.8. Let F = hX; µ; Di be a general IS-frame such that D c is a basis of open and compact for a topology de¯ned on X: Suppose that the mapping HX : X ! X (D) is injective. Then the following conditions are equivalent:

568


(1) F is an IS-space. (2) The mapping HX is onto and F is saturated. (3) The mapping HX is onto and F is replete. Proof. 1 ) 2: Since F is an IS-space, then the mapping HX is onto. Let x 2 X and let us consider subsets ¡ and § of D such that §c is dually directed. Suppose that [x) \ O1 \ ::: \ On \ V1c \ ::: \ Vkc 6= ;; for any fO1 ; :::; On g ³ ¡ and for any fV1 ; :::; Vk g ³ §: Let us consider the ¯lter h¡ [ HX (x)i and the set I (§) = fU 2 D : 9V 2 § : U ³ V g : Since § is dually directed, I (§) is an ideal. It is easy to see that h¡ [ HX (x)i \ I (§) = ;:As F is onto, there exists T T y 2 X such that ¡ [ HX (x) ³ HX (y) and § \ HX (y) = ;, i.e. y 2 [x) \ ¡ \ §c : Thus; F is saturated. The equivalence between 2 and 3 follows by Theorem 4.7. We prove 3 ) 1: It is clear that D c ³ K O (X) : Let A 2 K O (X ) : Since A is open, S A = fUic : Ui 2 K ³ Dg ; and since A is compact, A = U1c [ ::: [ Unc ; for some ¯nite family fU1 ; :::; Un g ³ K ³ D. As D is an IS-algebra, Ac 2 D: Thus, D c = K O (X) : We now prove that all closed subsets are increasing. Let F be a closed subset. Then F c S T is open. Thus, F c = fUic : Ui 2 Z ³ Dg ; i.e., F = fUi : Ui 2 Z ³ Dg. So, F is an intersection of increasing subsets and consequently is an increasing subset. We prove that HX is an order-isomorphism. Let x; y 2 X such that HX (x) ³ HX (y) : Since F is replete, there exists z 2 X such that x µ z and HX (y) = HX (z) ; and as HX is injective, then y = z: So, x µ y; and thus HX is an order-isomorphism. We conclude ¥ that F is an IS-space: Now, we shall study the duality for homomorphisms of IS-algebras. Let A and B be two IS-algebras A function h : A ! B is a semilattice homomorphism if h (1) = 1 and h (a ^ b) = h (a) ^ h (b), for every a; b 2 A. We note that a semilattice homomorphism h : A ! B satis¯es the property h (a ! b) µ h (a) ! h (b) for every a; b 2 A. An IS-homomorphism h is a semilattice homomorphism such that h (a) ! h (b) µ h (a ! b) for all a; b 2 A: De¯nition 4.9. Let F1 ; F2 be two IS-spaces. Let R ³ X1 £ X2 . We shall say that R is a meet-relation if: (1) For every U 2 D (X2 ) ; hR (U ) = fx 2 X1 : R (x) ³ U g 2 D (X1 ) ; T (2) R (x) = fU 2 D (X2 ) : R (x) ³ U g ; for every x 2 X1 : A meet-relation R ³ X1 £ X2 is a functional relation if for all x 2 domR and for all y 2 R (x) there exists z 2 X1 such that x µ z and R (z) = [y) : Let A1 and A2 be two IS-algebras and let h : A1 ! A2 be a semilattice homomorphism. Then the relation Rh : X (A2 ) £ X(A1 ) given by (P; Q) 2 Rh if and only if h¡1 (P ) ³ Q;


569

for each P 2 X(A2 ) is a meet-relation [3]. Let X, Y be two IS-spaces. If R : X1 £ X2 is a meet-relation, then the mapping hR : D (X2 ) ! D (X1 ) given by hR (U ) = fx 2 X1 : R (x) ³ U is a semilattice homomorphism. Thus, by the results given for distributive semilattice (see [3]) there exists a duality between semilattice homomorphisms and meet-relations. Now, we shall prove that the class of functional relations are equivalent to certain classes of (partial) mapping between two IS-spaces. De¯nition 4.10. Let F1 and F2 be two IS-spaces. A partial function f : X1 ! X2 is a IS-morphism if: (1) For all x; y 2 domf , if x µ y then f (x) µ f (y). (2) If x 2 domf and f (x) µ y; then there exists z 2 domf such that x µ z and f (z) = y: c (3) For all U 2 D2 ; (f ¡1 (U )c ] 2 D1 : Theorem 4.11. There exists a bijective correspondence between functional relations and IS-morphisms de¯ned between two IS-spaces. Proof. Let F1 and F2 be two IS-spaces. Let f : X1 ! X2 be an IS-morphism. De¯ne the binary relation Rf ³ X1 £ X2 by: (x; y) 2 Rf , (9z 2 domf ) x µ z and f (z) = y: We prove that Rf is a functional relation. Let x 2 domf and let y 2 Rf (x) : Then there exists z 2 domf such that x µ z and f (z) = y: We prove that Rf (z) = [y) : Let a 2 Rf (z) : So, there exists b 2 domf such that z µ b and f (b) = a: Since f is increasing, f (z) = y µ f (b) = a: Thus, y µ a: Suppose that y µ a: As f (z) = y µ a and f is an IS-morphism, then there exists k 2 domf such that that z µ k and f (k) = a: So, a 2 Rf (z) : c We prove that hRf (U ) = (f ¡1 (U )c ] ; for all U 2 D1 : Suppose that x 2 (f ¡1 (U )c ] : Then there exists f (z) 2 = U such that x µ z: Let y = f (z) : Then, (x; y) 2 Rf and y 2 = U: Thus, Rf (x) Ã U; i.e., x 2 = hRf (U ) : c c ¡1 Let x 2 (f (U ) ] and suppose that x 2 = hRf (U ) : So, there exists y 2 X2 such that (x; y) 2 Rf and y 2 = U: By the de¯nition of Rf , we get an element z 2 X1 such that x µ z and f (z) = y: Thus, x µ z and z 2 f ¡1 (U )c , and this implies that x 2 (f ¡1 (U )c ] ; which is a contradiction. We conclude that Rf is a functional relation. Let us consider a functional relation R ³ X1 £ X2 : Let us consider the set XR = fz 2 X1 : (9x 2 X1 ) (9y 2 X2 ) x µ z and R (z) = [y)g : De¯ne a partial mapping fR : X1 ! X2 with domfR = XR as follows: For z 2 XR ; let fR (z) = y such that R (z) = [y) : We prove that fR is an IS-morphism.

570


Let x; y 2 XR such that x µ y: Then, there exists x0 ; y 0 2 X2 such that R (x) = [x0 ) and R (y) = [y 0 ) : Since x µ y; then R (x) = [x0 ) ³ R (y) = [y 0 ) ; which implies that x0 µ y 0 ; i.e., fR (x) µ fR (y) : Thus, fR is increasing. Let y = fR (z) µ k: Then, R (z) = [y) : Since (z; y) 2 R, y µ k and R is a functional relation, there exists d 2 X1 such that z µ d and R (d) = [k) : So, fR (d) = k: ¡ ¤ c c Finally, it is easy to prove that hR (U ) = fR¡1 (U ) ; for all U 2 D1 : Thus, fR is an ¥ IS-morphism. Now we prove that the functional relation, or IS-morphisms, are the dual of the IS-homomorphisms. Theorem 4.12. Let F1 , and F2 be two IS-spaces. A meet-relation R ³ X1 £ X2 is a functional relation if and only if hR (U ) ) hR (V ) ³ hR (U ) V ) for any U; V 2 D (X2 ) : Proof. )) First we prove that if R is a meet-.relation, then (R¯ µ) ³ R, where ¯ denotes the composition of relations. Suppose the contrary. Then, there exists x 2 X1 and y; k 2 X2 T such that (x; y) 2 R, y µ k and (x; k) 2 = R: Since, R (x) = fU 2 D (X2 ) : R (x) ³ U g ; then k 2 = U; for some U 2 D (X2 ) such that R (x) ³ U: So, y 2 U , but since U is increasing and y µ k, we get y 2 = U , which is a contradiction. Thus, (R¯ µ) ³ R: Suppose that there exist U; V 2 D (X2 ) such that hR (U ) ) hR (V ) * hR (U ) V ) : Then there exist x 2 X1 and y; k 2 X2 such that [x) \ hR (U ) ³ hR (V ), (x; y) 2 R, [y) \ U * V , y µ k; k 2 U; and k 2 = V: So, (x; k) 2 R, and as R is a functional relation, there exists z 2 X1 such that x µ z and R (z) = [k) : Thus, z 2 [x) \ hR (U ) ³ hR (V ). So, R (z) = [k) ³ V , which is a contradiction. Thus, hR (U ) ) hR (V ) ³ hR (U ) V ) for any U; V 2 D (X2 ) : () Let (x; y) 2 X1 £ X2 such that (x; y) 2 R: Let us consider ¯nite subsets fU1 ; :::; Un g ³ HX2 (y) and fV1 ; :::; Vk g ³ HX2 (y)c : We prove that [x) \ hR (U1 ) \ ::: \ hR (Un ) \ hR (V1 )c \ ::: \ hR (Vk )c 6= ;: c c Suppose that contrary. Since HX2 (y) is an order-ideal, then there exists W 2 HX2 (y) such that Vi ³ W; for 1 µ i µ k: So,

[x) \ hR (U1 ) \ ::: \ hR (Un ) \ hR (W )c = [x) \ hR (U1 \ ::: \ Un ) \ hR (W )c = ;: Thus, x 2 hR (U1 \ ::: \ Un ) ) hR (W ) ³ hR ((U1 \ ::: \ Un ) ) W ) ; i.e. R (x) ³ (U1 \ ::: \ Un ) ) W: Since (x; y) 2 R and fU1 ; :::; Un g ³ HX2 (y), we have that W 2 HX2 (y) ; which is a contradiction. As F1 is saturated, we get \ \ fhR (U ) : y 2 U g \ fhR (V )c : y 2 [x) \ = V g 6= ;: So, \

there exists z 2 X1 such that x µ z, z 2

\

fhR (U ) : y 2 U g, and z 2

fhR (V )c : y 2 = V g : Now, it is easy to see that z 2 [x) and R (z) = [y) :

¥


571

By the above results we have that there exists a duality between the category of ISalgebras with IS-homomorphisms and the category of IS-spaces with functional relations (or IS-morphisms).

5

Boolean implicative semilattices

An implication algebra (or Tarski algebra, in the terminology of A. Monteiro [7]) is a Hilbert algebra A verifying the additional condition: P ((a ! b) ! a) ! a = 1: Let A be an implication algebra. It is well known that A is a join-semilattice where the supremum of two elements a; b 2 A is de¯ned by a _ b = (a ! b) ! b (see [7]). A Boolean implicative semilatticeis an implicative semilattice satisfying property P. The class of Boolean implicative semilattices form a variety and will be denoted by BIS. It is easy to see that the variety BIS is equivalent to the variety of Boolean rings, i.e. commutative ring in which every element is idempotent. Let us recall that the variety of unitary Boolean ring is equivalent to the variety of Boolean algebras. Example 5.1. Let A 2 IS: Let us consider the ¯lter \ fP » A : P is a maximal ¯lter of Ag : F =

It is clear that F is a proper ¯lter. Moreover, for each a; b 2 F , ((a ! b) ! a) µ a; because if ((a ! b) ! a) £ a for some a; b 2 F; then we can determinate a maximal ¯lter P such that (a ! b) ! a 2 P and a 2 = P , which is a contradiction. Since every ¯lter is closed under ^ and !, then hF; ^; !; 1i is an implicative semilattice satisfying the identity P, i.e., is a Boolean implicative semilattice. An implication algebra can be characterized in terms of irreducible deductive system. More precisely, a Hilbert algebra A is an implication algebra if and only if every irreducible deductive system is a maximal deductive system (see [7]). Taking into account that the implicative semilattices are also Hilbert algebras, we have the following result: Theorem 5.2. Let A 2 IS. Then A 2 BIS if and only if every irreducible ¯lter is a maximal ¯lter. Let A be an implication algebra. From Theorem 3.3 we can deduce that D (X (A)) is a Boolean implicative semilattice, and from Theorem 3.4 we have that the homomorphism of Hilbert algebras ¯A is onto if and only if A is a Boolean implicative semilattice. Taking these considerations into account, we say that an IS-space hX; µ; Di is a Boolean ISspace if the partial order µ is the equality. In this case we shall write hX; Di : Theorem 5.3. Let A 2 BIS. Then the associated IS-space of A is Boolean IS-space. Theorem 5.4. Let hX; Di be a Boolean IS-space. Then hD; ); \; Xi 2 BIS :

572


Proof. Since the partial order is the equality, then for every U; V 2 D we get that U ) V = U c [ V . So, it is clear that ((U ) V ) ) U ) ) U = X, for every U; V 2 D: Thus, hD; ); \; Xi 2 BIS . ¥ As conclusion of the previous results we can say that there exists a duality between Boolean implicative semilattices and Boolean IS-spaces.

References [1] D. Busneag: \A note on deductive systems of a Hilbert algebra", Kobe Journal of Mathematics, Vol. 2, (1985), pp. 29{35. [2] S.A. Celani: \A note on Homomorphisms of Hilbert Algebras", International Journal of Mathematical and Mathematics Science, Vol. 29, (2002), pp. 55{61. [3] S.A. Celani: \Topological Representation of Distributive Semilattices", Scientiae Mathematicae Japonicae online, Vol. 8, (2003), pp.41{51. [4] A. Diego: \Sur les alg¶ebras de Hilbert", Coll¶ection de Logique Math., Serie A, No. 21, Gauthiers-Villars, Paris, (1966). [5] D. Gluschankof and M. Tilli: \Maximal deductive systems and injective objects in the category of Hilbert algebras", Zeitschr. f. math. Logik und Grundlagen d. math., Vol. 34, (1988), pp. 213{220. [6] J. Meng, Y.B. Jun, S.M. Hong: \Implicative semilattices are equivalent to positive implicative BCK-algebras with condition (S)", Math. Japonica, Vol. 48, (1998), pp. 251{255. [7] A. Monteiro: \Sur les algµebras de Heyting symm¶etriques.", Portugaliae Mathematica, Vol. 39, (1980), pp. 1{239. [8] P. KÄohler: \Brouwerian semilattices", Trans. Amer. Math. Soc., Vol. 268, (1981), pp. 103{126. [9] W.C. Nemitz: \Implicative semi-lattices", Trans. Amer. Math. Soc., Vol. 117, (1965), pp. 128{142.

CEJM 4 (2003) 573{643

The Closure Diagram for Nilpotent Orbits of the Split Real Form of E8 · {D okovi¶c¤ Dragomir Z. Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Received 15 April 2003; accepted 8 July 2003 Abstract: Let O1 and O2 be adjoint nilpotent orbits in a real semisimple Lie algebra. Write O1 ¶ O2 if O2 is contained in the closure of O1 : This de nes a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra E8 : The proof is based on the fact that the Kostant{Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreducible components of the boundaries @O1i and of the intersections O1i \ O1j . c Central European Science Journals. All rights reserved. ® Keywords: Exceptional Lie groups, adjoint action, closures of nilpotent orbits, normal triples, Kostant{Sekiguchi bijection, prehomogeneous vector spaces. AMS: 17B25, 17B45

1

Introduction

¤

The closure diagrams for adjoint nilpotent orbits in noncompact real forms of the exceptional simple complex Lie algebras were determined in [10, 11, 12, 13, 14], except for the split real form of E8 (also known as the form of type E VIII, in Cartan’s notation). In this paper we handle this last case. The paper can be viewed as a continuation of our paper [14], and we shall often refer to it. By A we denote a simple complex Lie algebra of type E8 , by A 0 a split real form of A , and by G (respectively G0 ) the adjoint group of A (respectively A 0 ). As usual, let A 0 = k 0 © p0 be a Cartan decomposition of A 0 ; A = k © p its complexi¯cation, and µ E-mail: [email protected]

574

µ {D okovi´ c / Central European Journal of Mathematics 4 (2003) 573{643 D.Z.

the Cartan involution of A (and also of G). We recall that k is of type D8 and has the dimension 120. Let K ¹= Spin16 =Z2 (the half-spin group) be the connected subgroup of G with the Lie algebra k . (By Zk we denote a cyclic group of order k.) Since G is simply connected, we also have K = fx 2 G : µ(x) = xg. The space p is the irreducible (half-spin) k -module of dimension 27 = 128, and is faithful as a K-module. Denote by N the nilpotent variety of A and set N

R

=N

\ A 0;

N

1

=N

\ p:

It is known that the orbit spaces N R =G0 and N 1 =K, equipped with the quotient topologies, are homeomorphic and that the Kostant{Sekiguchi bijection is a homeomorphism N R =G0 ! N 1 =K (see [6, 1]). Hence, instead of N R =G0 we may (and we shall) study the quotient N 1 =K. In Table 1 we give a comprehensive list of representatives of the nonzero nilpotent K-orbits in p, which should prove useful for further study of these orbits. In many cases several representatives of di®erent types are given. (See Appendix B for the de¯nition of types.) Each of the representatives is written, in abbreviated form, as a sum of root vectors such that the associated roots are linearly independent. Our main result, Theorem 4.1, determines the topology of N 1 =K. This is accomplished via the closure diagram § which is described in Section 4. The diagram itself is not exhibited due to its complicated structure, but all the information needed for its construction is given in Table 2. The construction of § and the proof of its validity are based on extensive and nontrivial computations for which we used heavily Maple [5] and LiE [19] (in addition to our own programs). We shall now describe the contents of the paper in more details. In Section 2 we recall the result of K. Mizuno [15] (for the case of the complex ¯eld, C, only) describing the closures of the G-orbits in N . His closure diagram, with some additional information, is exhibited in Figures 2 and 3. We make use of this diagram later in our proofs and to describe the closures of the K-orbits in N 1 . In Section 3 we recall our classi¯cation (see [7] or [6]) of the K-orbits in N 1 , and for each of these orbits O1i we list one or more simple representatives. Each representative is the sum of certain root vectors and the corresponding roots determine a Dynkin-type diagram to which we refer as the type of the representative. The list of orbits and their representatives is given in Table 1. Given a nilpotent element X 2 N 1 , it is a nontrivial task to decide to which nilpotent K-orbit it belongs. Assuming that a suitable software is available, one can easily compute the dimension of the orbit K ¢ X. In general, this information is not su±cient since there are many pairs of orbits having the same dimension. The list given in Table 1 can often be used to identify quickly the orbit K ¢ X . In Section 4 we state our main result. If O1i is a K-orbit in N 1 , then its boundary, @O1i , is an a±ne variety whose irreducible components are of type O1j (the bar denotes the Zariski closure). For each orbit O1i , we list the irreducible components O1j of @O1i in Table 2. By using this table, it is a simple matter to construct the closure diagram for

µ {D oković / Central European Journal of Mathematics 4 (2003) 573{643 D.Z.

575

the K-orbits in N 1 . However, since the diagram is too complicated (it has 116 nodes and 219 lines), we do not show it explicitly. The auxilliary Table 3 can be used, together with the Figures 2 and 3, to read o® the closure of any particular K-orbit in N 1 . Given a K-orbit O1i » N 1 , one can choose a normal triple (E; H; F ) with E; F 2 N 1 , H in the ¯xed Cartan subalgebra h of k , and E 2 O1i . This triple induces a Z-gradation of the Z2 -graded Lie algebra A = k © p, i.e., A becomes a Z2 £ Z-graded Lie algebra: A = ©A H (i; j), i 2 f0; 1g, j 2 Z. The sum of the A H (i; j) over j 2 Z is equal to k if i = 0 and to p if i = 1. The centralizer, KH , of H in K plays an important role. There are two important prehomogeneous vector spaces (PV) associated to (E; H; F ). The ¯rst one is (KH ; A H (1; 2)) whose generic orbit is precisely the intersection of A H (1; 2) with O1i . This P.V. is regular, and its singular set is the union of several irreducible hypersurfaces S1 ; : : : ; Sr , where r is the number of the basic relative invariants of this P.V. The second important P.V. that we attach to (E; H; F ) is (QH ; p2 (H)), where QH P A H (0; j), j ¶ 0, and p2 (H) = is the parabolic subgroup of K with Lie algebra qH = P j j A H (1; j), j ¶ 2. This P.V. has the important property that O1 » O1i i® O1 \ p2 (H ) 6= ;. The theory of prehomogeneous vector spaces is not well developed in the nonregular case. Hence it is much harder to study the orbits in this second P.V. The singular set of (QH ; p2 (H)) is the union of the irreducible hypersurfaces S^1 ; : : : ; S^r , where S^k = Sk +p3 (H ) P A H (1; j), j ¶ 3. and p3 (H) = While each Sk is a quasi-homogeneous KH -variety, the hypersurfaces S^k of p2 (H), in general, are not quasi-homogeneous QH -varieties. However there is a unique K-orbit O1jk » N 1 such that S^k \ O1jk is a dense open subset of S^k . Our main tool is Proposition 4.3 which asserts that if O1j is an irreducible component of @O1i , then j 2 fj1 ; : : : ; jr g. In Section 5 we prove that if i ! j, which means that the pair (i; j) is one of the pairs listed in Table 2, then O1j » O1i . For each of these pairs, we give in Table 4 an element E which belongs to O1j \ p2 (H i ), where (E i ; H i; F i ) is a normal triple with E i 2 O1i . In Section 6 we complete the proof of our main result. For certain pairs (i; j), with dim(O1i ) > dim(O1j ), one has to show that O1j 6» O1i . We ¯rst determine the minimal collection of such pairs needed for the proof of the theorem and refer to these pairs as the critical pairs. Several of the critical pairs can be eliminated by using the Mizuno’s diagram and several other by using [10, Theorem 4.1]. Still 37 critical pairs remain to be dealt with by using the theory of prehomogeneous vector spaces. The proof presented in Sections 5 and 6 is an alternative quicker proof than the one that we actually used to construct Table 2. In the original proof we examined each of the nodes in our diagram, i.e., an orbit O1i , computed the integers j1 ; : : : ; jr , and then determined the irreducible components O1j of @O1i . The ¯rst appendix, A, contains the important Table 7. Let (E i ; H i; F i ) be a normal triple associated with the K-orbit O1i » N 1 . Then we list all the roots ® such that A ® » p2 (H i ). First we list those ®’s for which A ® » A H i (1; 2) and separate them by a semi-colon from the other ®’s. For the sake of brevity, we actually list only the indices j of the roots ® = ®j . This table is used for instance to ¯nd the representatives E i listed


576

in Table 1, as well as to verify the entries of Table 4. In our computations, we actually decomposed each A H i (1; j), for j ¶ 2, into simple KH i -modules, but this information is not included in Table 7. In the Appendix B we give the de¯nition of types for representatives of nilpotent K-orbits in p. In many cases, these types are not standard, i.e., they are not Dynkin diagrams. We have compiled a list of connected nonstandard types which was used often to identify quickly the K-orbit to which a particular nilpotent element X 2 p belongs. This list is not exhaustive in any way. (An exhaustive list would be much longer.) The Appendix C contains the Table 8 in which we have collected some data obtained by applying the M l2 -theory to the adjoint module A = k © p, viewed as the Z2 -graded module over the Z2 -graded Lie algebra A . This table was used in Section 6 (to construct Table 6) to simplify the proof of the main result. In the Appendix D we give in Table 9 the list of the irreducible components O1k of the intersection O1i \ O1j for the pairs (i; j) such that i < j and O1i 6» O1j . We point out that our enumeration of the K-orbits in N 1 is such that i < j implies dim(O1i ) µ dim(O1j ). A list of representatives of the nilpotent G0 -orbits in A 0 is given in [9, Table 4]. These representatives are not the simplest possible because they were embedded in the so called real Cayley triples. We use this opportunity to point out that the representative of orbit 113 given in that table is not valid. It should be replaced by the following element: p p p p p E = 2 15X1 + 74X7 + 38X8 + 34X12 + 6 3X13 ´ ´ p p p 1 ³ ³p +p 2 429X2 + 385X10 + 910X11 ¡ 11 6X3 : 37 There is an obvious error in [8, Table 4], not reported so far. The representative E p p for the orbit 5 should read 6X1 + 10X2 . Finally, let us also mention two misprints in our previous paper [14]. First, the nodes 24 and 25 in the closure diagram in Figure 3 should be connected by a dotted horizontal line to indicate that these two K-orbits are contained in the same G-orbit. Second, the entry for k = 54 in Table 1 should have the label E8 (b6 ) instead of E8 (b2 ).

2

Preliminaries

The closure diagram for adjoint nilpotent orbits in A was determined by Mizuno [15] and veri¯ed later by Beynon and Spaltenstein [2]. We will use the version of Mizuno’s diagram as given in [14, Figures 1 and 2]. For reader’s convenience, we reproduce this diagram in Figures 2 and 3 with some additional information. Each node represents a G-orbit in N and is labeled by the corresponding Bala{Carter symbol (see [6]). The orbits having the same dimension are positioned at the same level. Because of its length, the diagram is split into three pieces. The bottom and the top portions of the diagram are shown in Figure 2, while the middle part is shown separately in Figure 3. The dimensions of the orbits are indicated on the left of the diagrams. There are 70 nilpotent G-orbits in A (including the trivial orbit O 0 = f0g). The nonzero ones are listed in [14, Table 1]. The k-th orbit, i.e., the one that appears as the


577

k-th entry of that table, will be denoted by O k . The second column of this table contains the Bala{Carter symbol of O k , and the third one gives the weights of the weighted Dynkin diagram of O k . The complex dimension of O k is recorded in the last column. The nonzero G0 -orbits in N R ; or equivalently the nonzero K-orbits in N 1 ; were classi¯ed in [7] (see also [6]). We shall keep the same numbering for these orbits as in these two references. The i-th nontrivial G0 -orbit in N R will be denoted by O0i ; and we denote by O1i the nontrivial K-orbit in N 1 that corresponds to O0i under the Kostant{Sekiguchi bijection. The fourth column of [14, Table 1] gives the superscripts i of the orbits O0i (or, equivalently, O1i ) which are contained in O k . These superscripts are also exhibited in Figures 2 and 3 near the node representing the orbit O k . For instance, if k = 10 (the orbit O 10 has Bala{Carter label 2A2 ) then: O 10 \ A

0

= O014 [ O015 [ O016 ;

O 10 \ p = O114 [ O115 [ O116 :

Let ¾ be the complex conjugation of A with respect to A 0 , and let h be a ¾-stable Cartan subalgebra of k : Since A 0 is of inner type, h is also a Cartan subalgebra of A . We denote the root system of (A ; h) by R; choose a system of positive roots R + » R and a base B = f®i : 1 µ i µ 8g » R + of R: The simple roots ®i 2 B are indexed as in Figure 1. We extend the enumeration of simple roots ®i ; 1 µ i µ 8, to the enumeration ®i ; 1 µ i µ 120, of R + (see [14, Table 6]). A negative root ¡ ®i will be also written as ®¡i . The coroot of ®i is denoted by Hi 2 h. The set I = f§1; §2; : : : ; §120g will be used to index the roots, the coroots, etc. For ® 2 R we denote by A ® the root space of ®. A nonzero element X® 2 A ® is called a root vector of ®: We assume that a root vector Xi has been chosen for each root ®i , i 2 I. One can easily distinguish the roots ® for which A ® » k from those for which A ® » p. Indeed, if ® = k1 ®1 + k2 ®2 + ¢ ¢ ¢ + k8 ®8 , then in the former case k1 is even (k1 = 0; §2) and in the latter it is odd (k1 = §1). By adjoining the negative of the highest root, ®0 = ¡ ®120 = ®¡120 , to B we obtain the extended base Be = B [ f®0 g (see Figure 1). Let R0 be the root system of (k ; h) where + we view R0 as a subsystem of R: We set R+ 0 = R0 \ R and denote by B0 the unique base of R0 contained in R0+ . It turns out that B0 » Be . Explicitly we have: B0 = f®0 ; ®2 ; ®3 ; ®4 ; ®5 ; ®6 ; ®7 ; ®8 g = f¯i : 1 µ i µ 8g; where, as in [7, Table IV], ¯1 = ®0 ; ¯2 = ®8 ; ¯3 = ®7 ; ¯4 = ®6 ; ¯5 = ®5 ; ¯6 = ®4 ; ¯7 = ®3 ; ¯8 = ®2 : We say that a triple (E; H; F ) in A is a standard triple if [H; E] = 2E, [H; F ] = ¡ 2F , [F; E] = H and E; H; F are nonzero. Such a triple is normal if also H 2 k and E; F 2 p. We recall some notations from [14] concerning a normal triple (E; H; F ). For any integer j we de¯ne the subspaces: A H (0; j) = fX 2 k : [H; X ] = jXg; A H (1; j) = fX 2 p : [H; X] = jX g;

578


and for integers i ¶ 1 we set pi (H ) =

X

A H (1; j):

j¸i

By QH we denote the parabolic subgroup of K with Lie algebra X qH = A H (0; j): j¸0

3

Representatives of the nonzero nilpotent K-orbits in

p

There are exactly 116 nilpotent K-orbits in p. They are denoted by O1i , 0 µ i µ 115, where O10 = f0g is the trivial orbit. Thus N

1

=

115 [

O1i :

i=0

In this section we provide a comprehensive list of representatives for these orbits (see Table 1). For each i 2 f1; 2; : : : ; 115g we choose a normal triple (E i ; H i; F i ) with E i 2 O1i , H i 2 h, and such that ¯j (H i ) ¶ 0, 1 µ j µ 8. Such H i is uniquely determined by the orbit O1i (and vice versa). The centralizer KH i = ZK (H i ) of H i in K is a connected reductive subgroup of K which can be easily determined (up to local isomorphism) from the integers ¯j (H i ) given in Table 1. Furthermore KH i is a Levi factor of the parabolic subgroup QH i (de¯ned in Section 2). The pair (KH i ; A H i (1; 2)) is a prehomogeneous vector space (PV) which means that KH i , under the restriction of the adjoint action, has an open dense orbit in A H i (1; 2). It is known that this generic orbit, i , is just the intersection of O1i with A H i (1; 2) (see [10, Theorem 3.1, (a)], but note that there is a misprint in that assertion: A H (0; 2) should be replaced by A H (1; 2)). A generic element of A H i (1; 2) is just an element of i . The theory of prehomogeneous vector spaces plays an important role in our proofs, and we refer the reader to [17, 16] for the de¯nitions and facts that we use from this theory. For any prehomogeneous vector space, one de¯nes its singular set as the complement of the generic orbit. Since the centralizer MH i = ZKH i (E i ) of E i in KH i is a reductive (not necessarily connected) subgroup, the PV (KH i ; A H i (1; 2)) is regular [17]. Consequently, the singular set of this PV is a hypersurface (not necessarily irreducible). The irreducible components of this hypersurface will play an important role later on. P For any J » I we de¯ne XJ = j2J Xj and refer to J as the support of XJ . Let us also introduce the subset Ii of I de¯ned by Ii = fj 2 I : ®j (H i ) = 2g = fj 2 I : A

®j

»A

H i (1; 2)g:

The sets Ii are listed in Table 7 (Appendix A). We say that J » Ii is independent if the roots ®j , j 2 J, are linearly independent, and that J is generic (in Ii ) if XJ 2 i . We now give a description of Table 1. For each i 2 f1; 2; : : : ; 115g we record in the ¯rst column the integer k such that O1i » O k (see also [14, Table 1]). We construct normal


579

triples (E; H; F ) with H = H i and E 2 i . The third column lists the integers ¯j (H i ). The last two columns give the representatives E = E i 2 O1i and their types (see below for more details about types). Each representative E i is of the form E i = XJ where J is a minimal independent generic subset of Ii . In the fourth column we record the support J of E i and in the last column its type. In many cases we give several representatives of di®erent types, but we did not attempt to classify all minimal independent generic subsets of Ii . If the type of E i is written as a sum of two or more symbols, then we enclose in parentheses the contribution (to the support of E i ) of each of these symbols. For instance the support of the ¯rst representative of O125 , of type D4 (a1 ) + A1 , is given in Table 1 as (99; ¡ 53; ¡ 67; ¡ 72) + (¡ 63): This means that E 25 = (X99 + X¡53 + X¡67 + X¡72 ) + (X¡63 ): The contribution of D4 (a1 ) (resp. A1 ) is X99 + X¡53 + X¡67 + X¡72 (resp. X¡63 ). The roots ®99 , ®¡53 , ®¡67 , ®¡72 , ®¡63 are linearly independent. By de¯nition, the type of E i is the diagram shown in Figure 4. The dotted line in this diagram joins the two roots whose angle is ¼=3 while the solid lines (as usual) join two roots whose angle is 2¼=3. In Figures 5-9 (Appendix B) we exhibit the diagrams ¡J for the connected nonstandard types that occur in Table 1. It is not hard to verify Table 1. Indeed one just has to check that each E i belongs to A H i (1; 2) and that KH i ¢ E i = i . For the former condition it su±ces to verify that the support of E i is contained in Ii . The latter condition can be veri¯ed, for instance, by computing the dimension of KH i ¢ E i (or K ¢ E i ). This computation can be often avoided by verifying that none of the basic relative invariants of the PV (KH i ; A H i (1; 2)) vanishes on E i .

4

Construction of the closure diagram

The boundary of the orbit O1i is the a±ne variety @O1i = O1i n O1i : Our goal is to determine its irreducible components. Each of these components is the closure of another nilpotent orbit of smaller dimension. If O1j is an irreducible component of @O1i , then we say that O1j (resp. j) is a child of O1i (resp. i). Thus we have [ j O1 ; @O1i = j

where the union is taken over all children j of i. It is clear that O1i ¶ O1j holds i® there exists a sequence i = i0 ; i1 ; ¢ ¢ ¢ ; im¡1 ; im = j (m ¶ 0)

580


such that ij is a child of ij¡1 for 1 µ j µ m. Our main result is given in Table 2 where, for each i, we list the children of i. Theorem 4.1. For each i 2 f0; 1; : : : ; 115g, Table 2 gives the list of all children j of i or, equivalently, the list of the irreducible components O1j of @O1i . For example, the node 42 has only one child, namely 37, which in its turn has two children: 29 and 35. As @O10 is empty, the node 0 has no children. We shall write i ! j if the pair (i; j) is listed in Table 2. There are 219 such pairs. Let us de¯ne the diagram § as follows: Position the nodes 0-115 at various levels according to the dimensions of the corresponding orbits (see Figures 2 and 3) and join the nodes i and j by a line if i ! j. (It is assumed that the nodes are positioned so that the line joining nodes i and j contains no other nodes.) Our main result can now be given an equivalent formulation. Theorem 4.2. The diagram § is the closure diagram of the nilpotent K-orbits in p. Since § is rather complicated, we do not provide its pictorial representation. Consequently, it is not straightforward to list the orbits contained in the closure of one of them. To make this task easier we provide Table 3. If O1i » O k , then we list in this table the superscripts j for which dim(O1j ) < dim(O1i ) and O1j » O k but O1j 6» O1i . By using Figures 2 and 3 and Table 3, it is now easy to write down the list of all K-orbits contained in O1i . For instance, let us determine the closure of the orbit O142 . In this case the ambient G-orbit has the Bala{Carter label A4 + 2A1 (see Figure 3), i.e., k = 26 (see Table 1). The orbits O1j such that O1j » O 26 are clearly visible from Figures 2 and 3. From these we have to remove the orbits 43 and 44 as they have the same dimension as the orbit 42, and also the orbits 39, 38, 36, 34, 31 and 30 listed in Table 3. Hence O142 is the union of the orbits O1j for j = 42; 37; 35; 32 and those with j µ 29 except O121 . The proof of Theorem 4.1 will be given in subsequent sections. However we shall now describe the lengthy and tedious procedure that was actually used to construct Table 2. Let us consider a ¯xed node i corresponding to the orbit O1i . Let S be the singular set of the PV (KH i ; A H i (1; 2)), a hypersurface in A H i (1; 2). We note that the representative E i of O1i is a generic element of this PV. The hypersurface S is a union of irreducible conical hypersurfaces Sk de¯ned by equations fk = 0, (k = 1; 2; : : : ; r), where the fk are the basic relative invariants of this PV. Recall that these invariants are algebraically independent irreducible homogeneous polynomials on A H i (1; 2). One knows that r µ l, where l is the 0 0 length of A H i (1; 2) as a KH i -module. More precisely, r = dim(KH i =KH i MH i ) where K H i is the derived subgroup of KH i (see [16, 18]). The pair (QH i ; p2 (H i )) is also a PV and its singular set S^ is the union of the irreducible hypersurfaces S^k = Sk + p3 (H i ). For each k extend the polynomial function fk : A H i (1; 2) ! C to the polynomial function f^k : p2 (H i ) ! C by setting f^k (x+y) = fk (x) for


581

x 2 A H i (1; 2) and y 2 p3 (H i). These f^k are the basic relative invariants of (QH i ; p2 (H i )) and S^k is de¯ned by the equation f^k = 0. The following proposition proved in [13] enables us to ¯nd the children of i. Proposition 4.3. Use the above notations. For each k 2 f1; 2; : : : ; rg there exists a unique node jk in § such that S^k \ O1jk is a dense open subset of S^k . Every child of i belongs to the set fj1 ; j2 ; : : : ; jr g. To simplify notation, we set Pk = S^k \ O1jk for 1 µ k µ r. We now make several remarks concerning this proposition. First, it is clear from the de¯nitions that O1i > O1jk for 1 µ k µ r. Second, the nodes j1 ; j2 ; : : : ; jr are not necessarily distinct. Third, there exist examples such that O1jk > O1js for some k and s. Then js is not a child of i. In most cases, the hypersurface S^k contains a dense open QH i -orbit which is necessarily contained in Pk . In such cases it is not di±cult to determine jk . Once all the jk are computed, it is easy to identify all the children of the node i. Let us give an example. We shall determine the children of the node 26, assuming that we already know the children of the nodes i < 26. The group KH 26 is locally isomorphic to the direct product SL4 £ SL2 £ T4 where T4 is a 4-dimensional complex torus. The KH 26 module A H 26 (1; 2) has length l = 4: It is the direct sum of three 1-dimensional modules V1 = hX¡69 i, V2 = hX¡79 i, V3 = hX112 i and a 12-dimensional module V4 spanned by the root vectors with indices ¡ 87, ¡ 82, ¡ 83, ¡ 76, ¡ 77, ¡ 78, ¡ 70, ¡ 71, ¡ 72, ¡ 64, ¡ 65, ¡ 58. These modules are pairwise nonisomorphic. As a module for SL4 £ SL2 , V4 is isomorphic to ^2 C4 « C2 . De¯ne xk = XJk 2 p2 (H 26 ), 1 µ k µ 4, where the supports Jk and the types of xk are: k = 1 : (¡ 58; ¡ 87) + (112) + (¡ 30) + (¡ 45) + (¡ 79); k = 2 : (112; ¡ 69) + (¡ 58; ¡ 87) + (¡ 54) + (¡ 59); k = 3 : (¡ 69; ¡ 79) + (¡ 58; ¡ 87);

A2 + 4A1

2A2 + 2A1

2A2

k = 4 : (112; ¡ 69; ¡ 79) + (¡ 64) + (¡ 65) + (¡ 76);

A3 + 3A1

It is easy to show that the elements x1 ; x2 ; x3 ; x4 belong to the K-orbits 15; 22; 14; 24, respectively. In fact x3 is the representative E 14 of type 2A2 and x4 is the representative E 24 (see Table 1). A computation shows that each of the orbits QH 26 ¢ xk has codimension 1 in p2 (H 26 ). Hence the closures of these orbits are the four irreducible hypersurfaces S^k that constitute the singular set of the PV (QH 26 ; p2 (H 26 )). We conclude that j1 = 15, j2 = 22, j3 = 14, and j4 = 24. As we already know that the orbits 22 and 15 (but not 14) are contained in the closure of the orbit 24, we can conclude that the children of the node 26 are 24 and 14. In the remaining sections of the paper we shall present an alternative way of proving our result, which is essentially just a veri¯cation that the diagram § is correct, i.e., that it is the closure diagram for the nilpotent K-orbits in p.


582

5

Justi¯cation for the lines in the diagram

We write O1i ¶ O1j if O1i ¼ O1j , and O1i > O1j if also O1i 6= O1j . In this section we prove that if i ! j, i.e., the pair (i; j) is listed in Table 2, then O1i > O1j . Clearly, we may assume that j 6= 0, which explains why Table 4 has only 218 entries i ! j. Since p2 (H i ) » O1i , in order to prove that O1i > O1j it su±ces to exhibit an element E 2 p2 (H i ) \ O1j . Such elements are given in Table 4. The fact that E 2 p2 (H i ) is easy to verify by using Table 7 (Appendix A). In most cases we have chosen E to be a representative of O1j . Failing that, we have tried to choose E in A H j (1; 2). In that case the type of E ensures that E 2 O1j . It is more complicated to verify that E 2 O1j in the remaining cases which are marked by an asterisk at the end of the E-column. In these cases one embeds E in a normal triple (E; H; F ) with H 2 h and shows that H and H j are in the same orbit of the Weyl group of (k ; h). Several examples of such computations are given in [14, Section 4] and [13, Section 5].

6

Critical pairs

We say that a pair of nodes (i; j) of § is a critical pair if the following four conditions are satis¯ed: (i) dim(O1i ) > dim(O1j ): (ii) There is no downward path in § from i to j. (iii) If l ! i, then there is a downward path in § from l to j. (iv) If j ! k, then there is a downward path in § from i to k. We say that a critical pair (i; j) is good if O1i 6> O1j . In order to complete the proof of the main theorem, it su±ces to show that all critical pairs are good. It is very tedious to ¯nd the critical pairs by inspection of Table 2, and so we used a short program to ¯nd all such pairs. They are listed in Table 5. From the Mizuno’s diagram (see Figures 2 and 3), it is obvious that the six critical pairs in Table 5 that are marked by an asterisk are good. Let us consider the adjoint module V of A as a Z2 -graded module V = V0 © V1 with V0 = k and V1 = p. The integers di (j; k) de¯ned in the Appendix C are recorded in Table 8. We now invoke [10, Theorem 4.1] to obtain a list of pairs (i; j) such that dim(O1i ) > dim(O1j ) and O1i 6> O1j . Some of these pairs (those that are needed for our proof) are listed in Table 6. By inspecting this table, we see that the 11 critical pairs in Table 5 that are marked by a dagger are good. We are now faced with the most tedious part of the proof: Show that the 36 unadorned critical pairs in Table 5 are good. For that purpose we use essentially a variant of the method outlined in Section 4, with some shortcuts provided by Table 6. Since the arguments are of similar nature and too long to be included in the paper, we shall give the proofs only for ¯ve critical pairs. The pair (11; 9) As a KH 11 -module, A H 11 (1; 2) is the direct sum of two simple modules, i.e., l = 2. Let X be the representative E 10 of type A2 + 2A1 of the orbit O111 given in


583

Table 1, and Y = XJ 2 p2 (H 11 ) where J = f102; ¡ 9; ¡ 37; ¡ 79g. The space p2 (H 11 ) has the dimension 30, and a computation shows that each of the QH 11 -orbits through X or Y has dimension 29. The element Y is of type (4A1 )00 and J » I5 . It follows that Y 2 O15 . Hence the closures of the two QH 11 -orbits, mentioned above, are two distinct irreducible hypersurfaces in p2 (H 11 ). Since 10 ! 8 ! 5, we conclude that the node 11 has only one child, namely, 10. It follows that O111 6> O19 , i.e., the critical pair (11; 9) is good. The pair (21; 12) Since 21 ! 19; 20 (see Table 2), a glance at Figure 2 shows that 19 and 20 are children of the node 21. As the KH 21 -module A H 21 (1; 2) has length 2, the node 21 has no other children. Since 19; 20 ! 18, it follows from Figure 2 that 18 is a common child of the nodes 19 and 20. Since the KH 20 -module (A H 20 (1; 2) is simple, the node 20 has no other children. The KH 19 -module (A H 19 (1; 2) has the length 2. Consequently, the node 19 has at most two children. Let X be the representative E 18 of type A3 + A1 of the orbit O118 given in Table 1, and Y = XJ where J = f105; ¡ 80; ¡ 57; ¡ 79; ¡ 70g » I19 . The space p2 (H 19 ) has the dimension 43, and a computation shows that each of the QH 19 orbits through X and Y has dimension 42. The element Y is of type 2A2 + A1 and so Y 2 O117 . Hence the closures of the two QH 19 -orbits, mentioned above, are two distinct irreducible hypersurfaces in p2 (H 19 ). Since O118 > O117 , we conclude that the orbit 19 has only one child, namely, 18. Since 18 ! 11; 17 (see Table 2), a glance at Figure 2 shows that 11 and 17 are children of the node 18. As the KH 18 -module A H 18 (1; 2) has length 2, the node 18 has no other children. Since O111 6> O112 (see Figure 2), and O117 6> O112 (see Table 6), we can conclude that O121 6> O112 , i.e., the critical pair (21; 12) is good. The pair (34; 14) Since 34 ! 31 (see Table 2), a glance at Figure 2 shows that 31 is a child of the node 34. As the KH 34 -module A H 34 (1; 2) is simple, the node 21 has no other children. Since 31 ! 27, it follows from Figure 2 that 27 is a child of the node 31. Since the KH 31 -module (A H 31 (1; 2) is simple, the node 31 has no other children. Since O127 6> O114 (see Table 6), we have shown that O134 6> O114 , i.e., the critical pair (34; 14) is good. The pair (41; 14) This case is more complicated than the ones we handled so far. We have to determine the children for several nodes. Since 41 ! 33; 38 (see Table 2), a glance at Figure 2 shows that 33 and 38 are children of the node 41. The KH 41 -module A H 41 (1; 2) has the length l = 4, and so the node 41 has at most 4 children. Let X = XJ1 2 p2 (H 41 ); 41

J1 = f¡ 63; 93; ¡ 75; 103; ¡ 66; ¡ 65g » I30 ;

Y = XJ2 2 p2 (H );

J2 = f¡ 63; 93; ¡ 71; ¡ 65; ¡ 59; 103g:

6 O139 (see Figure 3), X must The element X has the type (2A3 )0 or (2A3 )00 . Since O141 > have the type (2A3 )00 . We conclude that X belongs to one of the orbits 29 or 30 (see Table 1). Since J1 » I30 , it follows that X 2 O130 . The element Y has the type A3 + A2 + A1 and so Y belongs either to the orbit 31 or 32. By using the method described in the previous section, one can show that Y 2 O131 . The space p2 (H 41 ) has the dimension 43, and a computation shows that each of the QH 41 -orbits through X, Y , and the representatives E 33 and E 38 (from Table 1) has dimension 42. Hence the closures of these four QH 41 -

584


orbits are the four distinct irreducible hypersurfaces in p2 (H 41 ). Since 41 ! 38 ! 30 and 41 ! 38 ! 36 ! 31, we conclude that the orbit 41 has only two children, namely, 33 and 38. Since 33 ! 21; 32 (see Table 2), a glance at Figure 2 shows that 21 and 32 are children of the node 33. The KH 33 -module A H 33 (1; 2) has the length l = 3, and so the node 33 has at most 3 children. Let X be the element E from the row with i = 33 and j = 32 of Table 4, and let Y = XJ 2 p2 (H 33 ) where J = f¡ 69; 93; ¡ 71; ¡ 59; ¡ 65g. The element Y has the type A3 + A2 and by computing its characteristic we ¯nd that Y 2 O127 . The space p2 (H 33 ) has the dimension 42, and a computation shows that each of the QH 33 orbits through X, Y , and the representative E 21 (from Table 1) has dimension 41. Hence the closures of these three QH 33 -orbits are the three distinct irreducible hypersurfaces in p2 (H 33 ). Since 32 ! 27, we conclude that the orbit 33 has only two children, namely, 21 and 32. Since 14 ! 12 and the pair (21; 12) is good, we infer that O121 6> O114 . Since 32 ! 27; 28 (see Table 2), a glance at Figure 2 shows that 27 and 28 are children of the node 32. The KH 32 -module A H 32 (1; 2) has the length l = 3, and so the node 32 has at most 3 children. Let X = XJ 2 p2 (H 32 ) where J = f108; ¡ 71; ¡ 77; ¡ 59; ¡ 69; ¡ 73g. This element has the type A3 + 3A1 and by computing its characteristic we ¯nd that X 2 O124 . The space p2 (H 32 ) has the dimension 40, and a computation shows that each of the QH 33 -orbits through X and the representatives E 27 of type A3 + A2 and E 28 of type D4 (a1 ) + 2A1 (from Table 1) has dimension 39. Hence the closures of these three QH 32 -orbits are the three distinct irreducible hypersurfaces in p2 (H 32 ). Since 28 ! 24, we conclude that the orbit 32 has only two children, namely, 27 and 28. Since O127 6> O114 and O128 6> O114 (see Table 6), we infer that O132 6> O114 , and also O133 6> O114 . Since 38 ! 30; 36 (see Table 2), a glance at Figure 2 shows that 30 and 36 are children of the node 38. The KH 38 -module A H 38 (1; 2) has the length l = 4, and so the node 38 has at most 4 children. Let X = XJ1 2 p2 (H 38 );

J1 = f103; ¡ 73; ¡ 75; 98; ¡ 63; ¡ 71g;

Y = XJ2 2 p2 (H 38 );

J2 = f103; ¡ 73; ¡ 75; 98; ¡ 65; ¡ 71g:

Both X and Y have the type A3 + A2 + A1 , but by computing the characteristics we ¯nd that X 2 O131 and Y 2 O132 . The space p2 (H 38 ) has the dimension 42, and a computation shows that each of the QH 38 -orbits through X, Y , and the representatives E 30 of type (2A3 )00 and E 36 of type A3 + A2 + 2A1 (from Table 1) has dimension 41. Hence the closures of these four QH 38 -orbits are four distinct irreducible hypersurfaces in p2 (H 38 ). Since 38 ! 36 ! 31; 32, we conclude that the orbit 38 has only two children, namely, 30 and 36. Since 36 ! 31; 32 (see Table 2), a glance at Figure 2 shows that 31 and 32 are children of the node 36. The KH 36 -module A H 36 (1; 2) has the length l = 2, and so the node 36 has no more children. Since we have already shown that O131 6> O114 and O132 6> O114 , we infer that O136 6> O114 . Since 30 ! 27; 28 (see Table 2), a glance at Figure 2 shows that 27 and 28 are children of the node 30. The KH 30 -module A H 30 (1; 2) has the length l = 2, and so the node 30 has


585

6 O114 and O128 6> O114 (see Table 6), we infer that O130 6> O114 . no more children. Since O127 > It follows that O141 6> O114 , i.e., the critical pair (41; 14) is good. The pair (45; 34) As A H 45 (1; 2) is a simple KH 45 -module, the node 45 has only one child. Since 45 ! 43, and so O145 > O143 , this child must be 43 (see Figure 3). Similarly, A H 43 (1; 2) as a KH 43 -module has length 2, and so the node 43 has at most two children. Since 43 ! 37 and 43 ! 39, by inspection of Figure 3 we conclude that 37 and 39 are the children of the node 43. >From Table 6 we see that O139 6> O134 and O142 6> O134 . As 42 ! 37, we infer that O137 6> O134 . Hence we have shown that O145 6> O134 , i.e., the critical pair (45; 34) is good.

7

Appendix A: Root spaces in

p2 (H i )

For each i 2 f1; 2; : : : ; 115g we list in Table 7 the indices j for which A ®j » p2 (H i ). We list ¯rst those j that belong to the set Ii , de¯ned in Section 3, and separate them by a semicolon from the other indices.

8

Appendix B: The list of connected nonstandard types

Consider one of the representatives E i from Table 1. It is of the form E i = XJ for some J » Ii . Let RJ = f®j : j 2 J g and recall that RJ is a linearly independent set. We assign to E i the diagram ¡J whose set of nodes is RJ . If ® and ¯ are distinct roots in RJ such that ®(H¯ ) = ¡ 1 (resp. ®(H¯ ) = 1), then we join the corresponding nodes by a solid (resp. dotted) line. Note that ®(H¯ ) = ¯(H® ) for all roots ® and ¯. The diagram ¡J is the type of E i . If this diagram is not a Dynkin diagram of a semisimple Lie algebra, then we say that the type is nonstandard. In this appendix we list the connected nonstandard types that arise from Table 1. We did not search systematically for all possible such diagrams. We merely recorded those that we encountered in our computations and considered them useful. If we replace the dotted lines in ¡J by solid ones, we obtain a new diagram which we denote by ¢J . The diagrams ¢J with the additional condition that they contain no circuits of odd length, called admissible diagrams, were classi¯ed by Carter [4]. Most of the connected nonstandard admissible diagrams on 6, 7 or 8 nodes in E8 , listed in loc. cit., appear in our list (after replacing dotted lines by solid ones). There is an example, see the diagrams E6 (a3 ) and E6 (a3 )0 , of two nonisomorphic diagrams ¡J1 and ¡J2 such that ¢J1 and ¢J2 are isomorphic. Di®erent representatives of a nilpotent K-orbit O1i given in Table 1 always have different types. If X is a nonstandard Bala{Carter label for the ambient G-orbit O k ¼ O1i , then we use X; X 0 ; X 00 ; : : : to denote the types of di®erent representatives E of O1i when these types are connected and nonstandard. This notation is also used for the connected components of disconnected types. When X is one of the standard types 4A1 , A3 + 2A1 , 2A3 , A5 + A1 or A7 the meaning of X 0 and X 00 is di®erent (see [14, Section 3] or [9]). We remark that di®erent nilpotent K-orbits in p may have representatives of the same


586

type. This can happen only for the K-orbits contained in the same G-orbit. For instance each of the orbits O17 and O18 has a representative of type 5A1 .

9

Appendix C: Z2 -gradation of the adjoint module

The direct decomposition A = k + p makes A into a Z2 -graded Lie algebra and we can view the adjoint module V = A as a Z2 -graded A -module V = V0 © V1 with V0 = k and V1 = p. Note that dim V0 = 120 and dim V1 = 128, and that the highest weight vector belongs to V0 . We introduce the integers di (j; k) = dim(Vi \ ker ½(E k )j ) where i = 0; 1; j ¶ 1; 1 µ k µ 115, and ½ is the representation of A on V . (E k is a representative of O1k from Table 1.) These integers are easy to compute using the M l2 theory (see [10]) and are displayed in Table 8 for 1 µ k µ 94. (The other values of k are not needed.)

10

Appendix D: Intersections of the closures of two nilpotent orbits

In this appendix we give in Table 9 the list of the irreducible components O1k of the intersection O1i \ O1j . It su±ces to do that only for the pairs (i; j) such that i < j and O1i 6» O1j . The enumeration of the K-orbits in N 1 is such that i < j implies dim(O1i ) µ dim(O1j ). It is often the case that di®erent pairs (i; j) produce the same intersection O1i \ O1j . All such pairs are listed together in the table and are separated by semi-colons. For instance for each of the pairs (29; 35), (29; 39) and (29; 44), this intersection is the union of O126 and O128 .

Acknowledgments The author was supported in part by the NSERC Grant A-5285.

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587

[5] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, S.M. Watt: Maple V Language reference Manual, Springer{Verlag, New York, 1991, xv+267 pp. [6] D.H. Collingwood and W.M. McGovern: Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993. · {D okovi¶c: \Classi¯cation of nilpotent elements in simple exceptional real Lie [7] D.Z. algebras of inner type and description of their centralizers", J. Algebra, Vol. 112, (1988), pp. 503{524. · {D okovi¶c: \Explicit Cayley triples in real forms of G2 , F4 , and E6 ", Paci¯c J. [8] D.Z. Math., Vol. 184, (1998), pp. 231{255. · {D okovi¶c: \Explicit Cayley triples in real forms of E8 ;", Paci¯c J. Math., Vol. [9] D.Z. 194, (2000), pp. 57{82. · {D okovi¶c: \The closure diagrams for nilpotent orbits of real forms of F4 and G2 ", [10] D.Z. J. Lie Theory, Vol. 10, (2000), pp. 491{510. · {D okovi¶c: \The closure diagrams for nilpotent orbits of real forms of E6 ", J. Lie [11] D.Z. Theory, 11, (2001), pp. 381{413. · {D okovi¶c: \The closure diagrams for nilpotent orbits of the real forms E VI and [12] D.Z. E VII of E7 ", Represent. Theory, Vol. 5, (2001), pp. 17{42. · {D okovi¶c: \The closure diagram for nilpotent orbits of the split real form of E7 ", [13] D.Z. Represent. Theory, Vol. 5, (2001), pp. 284{316. · {D okovi¶c: \The closure diagram for nilpotent orbits of the real form E IX of E8 ", [14] D.Z. Asian J. Math., Vol. 5, (2001), pp. 561{584. [15] K. Mizuno: \The conjugate classes of unipotent elements of the Chevalley groups E7 and E8 ;", Tokyo J. Math., Vol. 3, (1980), pp. 391{461. [16] A. Mortajine: Classi¯cation des espaces pr¶ehomogµenes de type parabolique r¶eguliers et de leurs invariants relatifs, Hermann, Paris, 1991. [17] M. Sato and T. Kimura: \A classi¯cation of irreducible prehomogeneous vector spaces and their relative invariants", Nagoya Math. J., Vol. 65, (1977), pp. 1{155. [18] M. Sato, T. Shintani, M. Muro: \Theory of prehomogeneous vector spaces (algebraic part)", Nagoya Math. J., Vol. 120, (1990), pp. 1{34. [19] M.A.A. van Leeuwen, A.M. Cohen, B. Lisser: LiE, version 2.1, a software package for Lie group theoretic computations, Computer Algebra Group of CWI, Amsterdam, The Netherlands.


588

k

i

j (H i )

Support of E i

Type of E i

1

1

00000010

(¡ 1)

A1

2

2

00010000

(¡ 23) + (¡ 24)

2A1

3

3

01000010

(¡ 24) + (¡ 38) + (¡ 53)

3A1

4

4

02000000

(¡ 23; ¡ 82)

A2

(¡ 1) + (¡ 44) + (¡ 71) + (¡ 89)

(4A1 )00

4

5

00000020

(105) + (¡ 31) + (¡ 46) + (¡ 60)

(4A1 )00

5

6

10001000

(¡ 38) + (¡ 44) + (¡ 47) + (¡ 53)

(4A1 )0

6

7

11000001

(¡ 53; ¡ 57) + (¡ 47)

A2 + A1

(¡ 23) + (¡ 47) + (¡ 52) + (¡ 64) + (¡ 76)

5A1

6

8

00010010

(110) + (¡ 44) + (¡ 45) + (¡ 47) + (¡ 59)

5A1

7

9

20010000

(¡ 53; ¡ 72) + (¡ 38) + (¡ 63)

A2 + 2A1

(¡ 31) + (¡ 46) + (¡ 60) + (¡ 69) + (¡ 70)

6A1

+(¡ 72) 7

10

01000100

(¡ 44; ¡ 66) + (110) + (¡ 47)

A2 + 2A1

(112) + (¡ 45) + (¡ 54) + (¡ 57) + (¡ 58)

6A1

+(¡ 59) 8

11

00010020

(102; ¡ 44; ¡ 79)

A3

9

12

30000001

(¡ 23; ¡ 96) + (¡ 52) + (¡ 64) + (¡ 77)

A2 + 3A1

(¡ 23) + (¡ 52) + (¡ 64) + (¡ 76) + (¡ 77)

7A1

+(¡ 78) + (¡ 79) 9

13

10010001

(¡ 53; ¡ 72) + (112) + (¡ 38) + (¡ 63)

A2 + 3A1

(112) + (¡ 38) + (¡ 53) + (¡ 63) + (¡ 64)

7A1

+(¡ 65) + (¡ 67) 10

14

40000000

(¡ 58; ¡ 87) + (¡ 69; ¡ 79)

2A2

(¡ 83; ¡ 95) + (¡ 1) + (¡ 44) + (¡ 71)

A2 + 4A1

+(¡ 89) (¡ 1) + (¡ 44) + (¡ 71) + (¡ 83) + (¡ 89)

8A1

+(¡ 90) + (¡ 91) + (¡ 92) Table 1 The representatives E i of the nonzero nilpotent orbits O1i in p (AA

0

= E VIII)


k

i

j (H i )

10

15

20000002

Support of E i (¡ 64; ¡ 73) + (112) + (¡ 53) + (¡ 63)

589

Type of E i A2 + 4A1

+(¡ 65) (112) + (¡ 23) + (¡ 52) + (¡ 64) + (¡ 76)

8A1

+(¡ 77) + (¡ 78) + (¡ 79) 10

16

00020000

(105; ¡ 51) + (¡ 47; ¡ 75)

2A2

(¡ 53; ¡ 72) + (96) + (112) + (¡ 38)

A2 + 4A1

+(¡ 63) (96) + (112) + (¡ 38) + (¡ 53) + (¡ 63)

8A1

+(¡ 64) + (¡ 65) + (¡ 67) 11

17

01010010

(105; ¡ 51) + (¡ 60; ¡ 64) + (¡ 66)

2A2 + A1

12

18

01000110

(105; ¡ 51; ¡ 79) + (¡ 64)

A3 + A1

13

19

02000020

(108; ¡ 57; ¡ 79) + (¡ 70) + (¡ 80)

(A3 + 2A1 )00

13

20

00000200

(89; ¡ 37; ¡ 66; ¡ 71)

D4 (a1 )

(¡ 51; 93; ¡ 64) + (¡ 37) + (¡ 66)

(A3 + 2A1 )00

(99; ¡ 53) + (105; ¡ 65) + (¡ 64; ¡ 73)

3A2

14

21

02000040

(¡ 57; 93; ¡ 70; ¡ 80)

D4

15

22

10100100

(110; ¡ 63) + (¡ 64; ¡ 73) + (¡ 53) + (¡ 65)

2A2 + 2A1

16

23

10010011

(108; ¡ 64; ¡ 73) + (¡ 63) + (¡ 65)

(A3 + 2A1 )0

17

24

11001010

(112; ¡ 69; ¡ 79) + (¡ 64) + (¡ 65) + (¡ 76)

A3 + 3A1

17

25

00100101

(99; ¡ 53; ¡ 67; ¡ 72) + (¡ 63)

D4 (a1 ) + A1

(¡ 59; 106; ¡ 73) + (¡ 63) + (¡ 71) + (¡ 72)

A3 + 3A1

(99; ¡ 53) + (105; ¡ 65) + (¡ 64; ¡ 73)

3A2 + A1

+(¡ 63) 18

26

20100011

(112; ¡ 69; ¡ 79) + (¡ 64; ¡ 83)

A3 + A2

(112; ¡ 69; ¡ 79) + (¡ 58) + (¡ 72) + (¡ 82)

A3 + 4A1

+(¡ 83) 18

27

10001002

(¡ 59; 106; ¡ 73) + (¡ 71; ¡ 77)

A3 + A2

(106; ¡ 79; ¡ 52; ¡ 63) + (¡ 64) + (¡ 78)

D4 (a1 ) + 2A1

(¡ 52; 106; ¡ 79) + (¡ 64) + (¡ 76) + (¡ 77)

A3 + 4A1

+(¡ 78) Table 1 The representatives E i of the nonzero nilpotent orbits O1i in p (AA

0

= E VIII)


590

k

i

j (H i )

Support of E i

Type of E i

18

28

01010100

(106; ¡ 67; ¡ 66; ¡ 71) + (¡ 69) + (¡ 70)

D4 (a1 ) + 2A1

(103; ¡ 66; ¡ 65) + (108) + (¡ 63)

A3 + 4A1

+(¡ 67) + (¡ 75) 19

29

02020000

(¡ 71; 105; ¡ 65; ¡ 89)

A4

(96; ¡ 66; ¡ 77) + (112; ¡ 93; ¡ 54)

(2A3 )00

19

30

00020020

(¡ 63; 98; ¡ 65) + (103; ¡ 73; ¡ 75)

(2A3 )00

20

31

00100003

(¡ 46; 91; ¡ 60) + (¡ 70; ¡ 78) + (106)

A3 + A2 + A1

(91; ¡ 46; ¡ 60; ¡ 83) + (106) + (¡ 70)

D4 (a1 ) + 3A1

+(¡ 72) 20

32

10101001

(¡ 59; 106; ¡ 73) + (¡ 71; ¡ 77) + (¡ 69)

A3 + A2 + A1

(106; ¡ 59; ¡ 73; ¡ 83) + (¡ 69) + (¡ 71)

D4 (a1 ) + 3A1

+(¡ 72) 21

33

11001030

(¡ 69; 93; ¡ 70; ¡ 71) + (¡ 65)

D4 + A1

22

34

00000004

(¡ 23; 69; 91; ¡ 57) + (¡ 59; ¡ 83)

D4 (a1 ) + A2

(¡ 64; 102; ¡ 77) + (106; ¡ 79) + (¡ 76)

A3 + A2 + 2A1

+(¡ 78) (69; ¡ 45; ¡ 30; ¡ 72) + (91) + (106)

D4 (a1 ) + 4A1

+(¡ 70) + (¡ 71) (69; ¡ 30) + (91; ¡ 53) + (106; ¡ 73)

4A2

+(¡ 71; ¡ 77) 22

35

20002000

(108; ¡ 72; ¡ 71; ¡ 80) + (¡ 70; ¡ 73)

D4 (a1 ) + A2

(¡ 52; 106; ¡ 79) + (¡ 64; ¡ 86) + (¡ 76)

A3 + A2 + 2A1

+(¡ 77) (106; ¡ 79; ¡ 52; ¡ 69) + (¡ 64) + (¡ 76)

D4 (a1 ) + 4A1

+(¡ 77) + (¡ 78) 22

36

00200002

(103; ¡ 73; ¡ 70) + (108; ¡ 71) + (¡ 65)

A3 + A2 + 2A1

+(¡ 69) (91; ¡ 60; ¡ 46; ¡ 71) + (106) + (¡ 69)

D4 (a1 ) + 4A1

+(¡ 70) + (¡ 72) Table 1 The representatives E i of the nonzero nilpotent orbits O1i in p (AA

0

= E VIII)


Support of E i

591

k

i

j (H i )

Type of E i

23

37

11110010

(¡ 76; 110; ¡ 77; ¡ 85) + (¡ 73)

A4 + A1

23

38

01010110

(103; ¡ 73; ¡ 75) + (¡ 63; 98; ¡ 65) + (¡ 71)

2A3 + A1

24

39

10110100

(103; ¡ 66; ¡ 81) + (108; ¡ 78; ¡ 70)

(2A3 )0

25

40

20100031

(¡ 69; 93; ¡ 58; ¡ 82; ¡ 87)

D5 (a1 )

(¡ 69; 93; ¡ 58; ¡ 87; 98)

D5 (a1 )0

(¡ 64; 93; ¡ 69; ¡ 76) + (¡ 77) + (¡ 78)

D4 + 2A1

25

41

01010120

(¡ 69; 93; ¡ 70; ¡ 71) + (106) + (¡ 72)

D4 + 2A1

26

42

21010100

(¡ 76; 110; ¡ 77; ¡ 85) + (¡ 73) + (¡ 84)

A4 + 2A1

26

43

01200100

(¡ 70; 99; ¡ 60; ¡ 85) + (105) + (¡ 72)

A4 + 2A1

(99; ¡ 70; ¡ 60; ¡ 73) + (105; ¡ 80; ¡ 72)

D4 (a1 ) + A3

(102; ¡ 72; ¡ 70; ¡ 76) + (106; ¡ 79; ¡ 69)

D4 (a1 ) + A3

(¡ 64; 102; ¡ 77) + (106; ¡ 79; ¡ 69)

2A3 + 2A1

26

44

10101011

+(¡ 76) + (¡ 78) 27

45

00400000

(¡ 65; ¡ 82; 96; ¡ 73) + (112; ¡ 80)

A4 + A2

(¡ 54; 79; ¡ 82; ¡ 39)

2D4 (a1 )

+(102; ¡ 75; ¡ 77; ¡ 83) 27

46

02000200

(¡ 69; 98; ¡ 78; ¡ 70) + (106; ¡ 67)

A4 + A2

(94; ¡ 60; ¡ 63; ¡ 82)

2D4 (a1 )

+(102; ¡ 72; ¡ 75; ¡ 80) 28

47

01020110

(102; ¡ 81; ¡ 71; 100; ¡ 79)

A5

29

48

30001030

(¡ 80; 93; ¡ 57; ¡ 70; ¡ 88) + (¡ 81)

D5 (a1 ) + A1

(¡ 70; 93; ¡ 57; ¡ 91; 102) + (¡ 83)

D5 (a1 )0 + A1

(¡ 57; 93; ¡ 70; ¡ 80) + (¡ 81) + (¡ 83)

D4 + 3A1

+(¡ 84) 29

49

10101021

(¡ 69; 93; ¡ 70; ¡ 71; ¡ 78) + (106)

D5 (a1 ) + A1

(¡ 69; 93; ¡ 64; ¡ 83; 98) + (106)

D5 (a1 )0 + A1

(¡ 64; 93; ¡ 69; ¡ 76) + (106) + (¡ 77)

D4 + 3A1

+(¡ 78) 30

50

11010101

(¡ 69; ¡ 79; 106; ¡ 82) + (105; ¡ 78)

A4 + A2 + A1


0

= E VIII)


592

k

i

j (H i )

31

51

40000040

Support of E i

Type of E i

(¡ 78; 105; ¡ 80; ¡ 88) + (¡ 81; ¡ 82)

D 4 + A2

(¡ 44; 93; ¡ 71; ¡ 89; ¡ 99) + (¡ 83) + (¡ 90)

D5 (a1 ) + 2A1

(¡ 44; 93; ¡ 71; ¡ 99; 102) + (¡ 83) + (¡ 90)

D5 (a1 )0 + 2A1

(¡ 44; 93; ¡ 71; ¡ 89) + (¡ 83) + (¡ 90)

D4 + 4A1

+(¡ 91) + (¡ 92) 31

52

00200022

(¡ 69; 98; ¡ 77; ¡ 78) + (103; ¡ 76)

D 4 + A2

(¡ 69; 93; ¡ 64; ¡ 76; ¡ 83) + (91) + (106)

D5 (a1 ) + 2A1

(¡ 69; 93; ¡ 64; ¡ 83; 98) + (91) + (106)

D5 (a1 )0 + 2A1

(¡ 64; 93; ¡ 69; ¡ 76) + (91) + (106)

D4 + 4A1

+(¡ 77) + (¡ 78) 31

53

20002020

(¡ 80; 93; ¡ 70; ¡ 57; ¡ 78) + (106) + (¡ 81)

D5 (a1 ) + 2A1

(¡ 70; 93; ¡ 57; ¡ 91; 102) + (106) + (¡ 83)

D5 (a1 )0 + 2A1

(¡ 80; 105; ¡ 83; ¡ 84) + (106) + (¡ 57)

D4 + 4A1

+(¡ 70) + (¡ 81) 32

54

02020020

(102; ¡ 81; ¡ 76; 103; ¡ 79) + (¡ 85)

(A5 + A1 )00

32

55

00020200

(94; ¡ 69; ¡ 79; 100; ¡ 78) + (102)

(A5 + A1 )00

(98; ¡ 77; ¡ 71; 96; ¡ 73; 93)

E6 (a3 )

(96; ¡ 66; 89; ¡ 63; ¡ 77; ¡ 75)

E6 (a3 )0

33

56

02020040

(¡ 85; 93; ¡ 76; 103; ¡ 81)

D5

34

57

11101011

(¡ 77; 102; ¡ 75; ¡ 78) + (¡ 79; 106; ¡ 82)

A4 + A3

35

58

10111011

(102; ¡ 75; ¡ 78; 100; ¡ 79) + (¡ 76)

(A5 + A1 )0

36

59

11010111

(¡ 77; 102; ¡ 75; ¡ 84; ¡ 78) + (103; ¡ 76)

D5 (a1 ) + A2

(¡ 77; 98; ¡ 69; ¡ 84; 102) + (103; ¡ 76)

D5 (a1 )0 + A2

37

60

21011011

(¡ 80; 105; ¡ 83; ¡ 84) + (106; ¡ 82; ¡ 81)

D 4 + A3

37

61

10102100

(¡ 79; 99; ¡ 75; ¡ 83; 105; ¡ 77)

D6 (a2 )

(¡ 84; 105; ¡ 86; ¡ 70; 99; ¡ 79)

D6 (a2 )0

(¡ 70; 99; ¡ 79; ¡ 81) + (¡ 80; 105; ¡ 83)

D 4 + A3

(102; ¡ 77; ¡ 80; 103; ¡ 79) + (¡ 78)

A5 + 2A1

38

62

11110110


0

= E VIII)


593

k

i

j (H i )

Support of E i

38

63

01011101

(94; ¡ 69; ¡ 79; 100; ¡ 78) + (102) + (¡ 76)

A5 + 2A1

(¡ 75; ¡ 73; 94; ¡ 77; 100; ¡ 69) + (¡ 76)

E6 (a3 )0 + A1

(99; ¡ 67; 90; ¡ 69; ¡ 84; ¡ 78; 102)

E7 (a5 )

(¡ 72; 90; ¡ 69; ¡ 79; 99; ¡ 73) + (105)

D6 (a2 ) + A1

(¡ 72; 90; ¡ 69; ¡ 79; 99; ¡ 77) + (105)

D6 (a2 )0 + A1

(99; ¡ 79; ¡ 69; 90; ¡ 72) + (105; ¡ 83)

A5 + A2

(¡ 79; 99; ¡ 75; ¡ 83; 105; ¡ 77) + (¡ 82)

D6 (a2 ) + A1

(¡ 79; 99; ¡ 75; ¡ 83; 105; ¡ 84) + (¡ 82)

D6 (a2 )0 + A1

39

39

64

65

01003001

11101101

Type of E i

40

66

11110130

(¡ 80; 93; ¡ 82; 106; ¡ 81) + (¡ 78)

D5 + A1

41

67

20200200

(¡ 79; 99; ¡ 81; ¡ 82; 108; ¡ 80; ¡ 83; 105)

E8 (a7 )0

(105; ¡ 83; ¡ 85; 106; ¡ 79; ¡ 70; ¡ 81; 99)

E8 (a7 )00

(¡ 80; 105; ¡ 83; ¡ 84)

D4 + D4 (a1 )

+(¡ 82; 106; ¡ 81; ¡ 90) (¡ 79; 99; ¡ 81; ¡ 70; ¡ 82)

D5 (a1 ) + A3

+(¡ 80; 105; ¡ 83) (¡ 79; 99; ¡ 70; ¡ 90; 106)

D5 (a1 )0 + A3

+(¡ 80; 105; ¡ 83) (¡ 79; 99; ¡ 70; ¡ 86; 105; ¡ 81) + (¡ 85)

D6 (a2 ) + 2A1

+(¡ 87) (¡ 79; 106; ¡ 82; ¡ 81; 102; ¡ 84) + (¡ 80)

D6 (a2 )0 + 2A1

+(¡ 83) 41

68

00004000

(90; ¡ 52; 82; ¡ 64; ¡ 83; ¡ 76; 102; 98)

E8 (a7 )

(82; ¡ 52; 90; ¡ 78; ¡ 75; ¡ 77; ¡ 79; 99)

E8 (a7 )0

(94; ¡ 77; 102; ¡ 80) + (¡ 79; 96; ¡ 57; ¡ 78)

D4 + D4 (a1 )

(¡ 79; 99; ¡ 77; ¡ 75; ¡ 80) + (82; ¡ 52; 90)

D5 (a1 ) + A3

(¡ 64; 82; ¡ 52; 90; ¡ 86) + (¡ 77; 99; ¡ 79)

D5 (a1 )0 + A3

(90; ¡ 52; 82; ¡ 64; ¡ 83; ¡ 76; 98) + (99)

E7 (a5 ) + A1

(103; ¡ 80; 93; ¡ 70; ¡ 78; ¡ 77) + (95)

D6 (a2 ) + 2A1


0

= E VIII)


594

k

i

j (H i )

41

68

00004000

Support of E i (¡ 63; 82; ¡ 64; ¡ 73; 90; ¡ 69) + (99)

Type of E i D6 (a2 )0 + 2A1

+(105) (82; ¡ 63; ¡ 79; 99) + (90; ¡ 72; ¡ 80; 105)

2A4

(99; ¡ 79; ¡ 63; 82; ¡ 64) + (105; ¡ 80)

A5 + A2 + A1

+(90) 41

69

02002002

(¡ 84; 102; ¡ 77; ¡ 80; 103; 94; ¡ 79) + (¡ 82)

E7 (a5 ) + A1

(¡ 78; 100; ¡ 80; ¡ 67; 94; ¡ 79) + (102)

D6 (a2 ) + 2A1

+(¡ 82) (¡ 72; 90; ¡ 69; ¡ 79; 99; ¡ 77) + (105)

D6 (a2 )0 + 2A1

+(¡ 82) (99; ¡ 79; ¡ 69; 90; ¡ 72) + (105; ¡ 83)

A5 + A2 + A1

+(¡ 82) 42

70

40040000

(¡ 66; 96; ¡ 78; ¡ 99; 112; ¡ 93)

A6

(¡ 66; 96; ¡ 78; ¡ 88)

2D4

+(¡ 93; 112; ¡ 94; ¡ 95) 42

71

02020200

(¡ 80; 100; ¡ 79; 99; ¡ 81; ¡ 82)

A6

(94; ¡ 77; 102; ¡ 85) + (¡ 76; 96; ¡ 78; ¡ 79)

2D4

43

72

21011031

(¡ 80; 93; ¡ 82; 106; ¡ 81) + (¡ 83) + (¡ 84)

D5 + 2A1

43

73

01201031

(¡ 82; 93; ¡ 75; 99; ¡ 83; ¡ 77)

D6 (a1 )

(¡ 82; 93; ¡ 75; 95; ¡ 72; ¡ 80)

D6 (a1 )0

(¡ 82; 93; ¡ 75; 95; ¡ 78) + (103) + (¡ 77)

D5 + 2A1

44

74

11111101

(¡ 80; 100; ¡ 79; 99; ¡ 81; ¡ 82) + (¡ 83)

A6 + A1

45

75

11101121

(¡ 82; 93; ¡ 80; 103; ¡ 77; ¡ 83) + (¡ 84)

D6 (a1 ) + A1

45

76

10300130

(¡ 87; ¡ 80; 93; ¡ 70; 99; ¡ 81; 103)

E7 (a4 )

(¡ 81; 99; ¡ 70; 93; ¡ 80; ¡ 87; 98)

E7 (a4 )0

(¡ 87; ¡ 80; 93; ¡ 70; 99; ¡ 81; ¡ 90)

E7 (a4 )00

(¡ 80; 93; ¡ 70; 99; ¡ 90; ¡ 81) + (¡ 83)

D6 (a1 ) + A1

(¡ 82; 93; ¡ 75; 99; ¡ 77; ¡ 80) + (¡ 90)

D6 (a1 )0 + A1

(100; ¡ 79; 99; ¡ 81; ¡ 89; ¡ 93)

E6 (a1 )

(96; ¡ 73; 94; ¡ 77; ¡ 89; ¡ 78)

E6 (a1 )0

46

77

04020200

Table 1 The representatives E i of the nonzero nilpotent orbits O1i in p (AA

0

= E VIII)


Support of E i

595

k

i

j (H i )

Type of E i

46

77

04020200

(102; ¡ 77; 94; ¡ 89; ¡ 78; 96; ¡ 79)

(A7 )00

46

78

02020220

(99; ¡ 81; ¡ 82; 93; ¡ 80; 100; ¡ 84)

(A7 )00

47

79

02002022

(¡ 82; 93; ¡ 69; 90; ¡ 84) + (99; ¡ 77)

D5 + A2

(¡ 82; 93; ¡ 69; 90; ¡ 84; ¡ 72) + (99)

D6 (a1 ) + 2A1

+(¡ 83) (¡ 82; 93; ¡ 69; 90; ¡ 72; ¡ 80) + (99)

D6 (a1 )0 + 2A1

+(¡ 83) 47

80

00400040

(¡ 72; 84; ¡ 58; 93; ¡ 82; ¡ 86; 98) + (99)

E7 (a4 )0 + A1

(¡ 86; ¡ 82; 93; ¡ 58; 84; ¡ 72; 96) + (99)

E7 (a4 ) + A1

(¡ 83; 98; ¡ 81; ¡ 82; 99) + (100; ¡ 80)

D5 + A2

(100; ¡ 79; 99; ¡ 89; ¡ 81; ¡ 72) + (¡ 58)

D6 (a1 ) + 2A1

+(¡ 83) (¡ 82; 93; ¡ 58; 84; ¡ 72; ¡ 80) + (99)

D6 (a1 )0 + 2A1

+(¡ 83) 47

81

20200220

(¡ 83; 103; ¡ 80; 93; ¡ 82; ¡ 81; 98) + (¡ 84)

E7 (a4 )0 + A1

(¡ 87; ¡ 80; 93; ¡ 70; 99; ¡ 81; 103)

E7 (a4 ) + A1

+(¡ 84) (¡ 81; 98; ¡ 83; 103; ¡ 85; ¡ 80) + (¡ 70)

D6 (a1 ) + 2A1

+(¡ 84) (¡ 85; 93; ¡ 70; 99; ¡ 81; ¡ 76) + (¡ 84)

D6 (a1 )0 + 2A1

+(¡ 87) 48

82

04020240

(100; 93; ¡ 84; ¡ 89; 99; ¡ 81)

E6

49

83

21031031

(¡ 80; 93; ¡ 82; 96; ¡ 83; ¡ 84)

D6

50

84

31010211

(¡ 81; 98; ¡ 83; 103; ¡ 80; ¡ 92; ¡ 85)

D7 (a2 )

50

85

11111111

(¡ 83; 98; ¡ 81; 99; ¡ 82) + (¡ 80; 100; ¡ 84)

D5 + A3

51

86

12111111

(99; ¡ 81; 98; ¡ 83; ¡ 85; 100; ¡ 84)

(A7 )0

52

87

13111101

(100; ¡ 79; 99; ¡ 81; ¡ 89; ¡ 93) + (¡ 83)

E6 (a1 ) + A1

(100; ¡ 79; 99; ¡ 81; ¡ 89; ¡ 84) + (¡ 83)

E6 (a1 )0 + A1

(99; ¡ 81; ¡ 82; 93; ¡ 80; 100; ¡ 84) + (¡ 83)

A7 + A1

52

88

11111121


0

= E VIII)


596

k

i

j (H i )

53

89

11121121

(¡ 80; 93; ¡ 82; 96; ¡ 83; ¡ 84) + (99)

D6 + A1

53

90

30130130

(¡ 85; 93; ¡ 76; 96; ¡ 84; ¡ 87; ¡ 90)

E7 (a3 )

(¡ 84; 96; ¡ 76; ¡ 90; 98; 93; 100)

E7 (a3 )0

(¡ 85; 93; ¡ 76; 96; ¡ 84; ¡ 87) + (¡ 86)

D6 + A1

(¡ 88; 99; ¡ 81; ¡ 82; 93; ¡ 80; 100; 98)

D8 (a3 )

(¡ 84; 95; ¡ 75; 93; ¡ 82; ¡ 86; 103; ¡ 88)

D8 (a3 )0

(¡ 83; 98; ¡ 81; 99; ¡ 89; ¡ 84; 100; ¡ 82)

D8 (a3 )

(¡ 78; 87; ¡ 67; 95; ¡ 85; ¡ 88; 103; ¡ 86)

D8 (a3 )0

(95; ¡ 67; 87; ¡ 78; ¡ 82; ¡ 93) + (103; ¡ 88)

E6 (a1 ) + A2

(95; ¡ 67; 87; ¡ 78; ¡ 82; ¡ 81) + (103; ¡ 88)

E6 (a1 )0 + A2

(¡ 80; 93; ¡ 82; ¡ 81; 94; 95; ¡ 83)

D7 (a1 )

(¡ 80; 93; ¡ 82; ¡ 81; 90; ¡ 78; 96)

D7 (a1 )0

(¡ 80; 93; ¡ 82; 96; ¡ 83; ¡ 84) + (90) + (99)

D6 + 2A1

(¡ 87; 98; ¡ 86; 100; ¡ 84; ¡ 85; ¡ 89)

D7 (a1 )

(¡ 89; 93; ¡ 71; 96; ¡ 92; ¡ 86; 100)

D7 (a1 )0

(¡ 89; 93; ¡ 71; 96; ¡ 83; ¡ 92) + (¡ 90)

D6 + 2A1

54

54

55

55

91

92

93

94

20202022

04004000

02022022

40040040

Support of E i

Type of E i

+(¡ 91) 55

95

20220220

(¡ 87; 98; ¡ 86; 100; ¡ 84; ¡ 85) + (99)

D6 + 2A1

+(¡ 76) (¡ 76; 93; ¡ 85; 100; ¡ 84; ¡ 86; ¡ 83) + (99)

E7 (a3 ) + A1

56

96

13111141

(100; 93; ¡ 84; ¡ 89; 99; ¡ 81) + (¡ 83)

E6 + A1

57

97

13103041

(¡ 81; 93; 95; ¡ 85; ¡ 88; 103; ¡ 84)

E7 (a2 )

(¡ 86; 103; ¡ 88; ¡ 85; 93; 95; ¡ 81)

E7 (a2 )0

(88; ¡ 70; 89; ¡ 80; ¡ 87; 98; ¡ 81; 95)

A8

(¡ 81; 94; ¡ 83; ¡ 85; 93; ¡ 70; 92; 100)

D8 (a2 )

(¡ 80; 93; ¡ 82; ¡ 81; 94; ¡ 83; 92; 95)

E8 (a6 )

58

98

00400400

58

99

22202022

(¡ 87; 98; ¡ 81; 95; ¡ 84; ¡ 89; 103; ¡ 85)

D8 (a2 )

59

100

31131211

(¡ 86; 98; ¡ 87; 96; ¡ 88; 99; ¡ 85)

D7

60

101

22202042

(¡ 81; 93; 95; ¡ 85; ¡ 88; 103; ¡ 84) + (¡ 87)

E7 (a2 ) + A1

(¡ 81; 99; ¡ 88; ¡ 85; 93; 100; ¡ 86) + (¡ 87)

E7 (a2 )0 + A1


0

= E VIII)


597

k

i

j (H i )

Support of E i

Type of E i

60

102

04004040

(¡ 77; 93; 87; ¡ 82; ¡ 86; 103; ¡ 78) + (95)

E7 (a2 ) + A1

(99; ¡ 88; 96; ¡ 82; 93; ¡ 86; 94) + (¡ 72)

E7 (a2 )0 + A1

(103; 93; ¡ 86; ¡ 82; 87; ¡ 77) + (95; ¡ 84)

E 6 + A2

(¡ 87; 96; ¡ 88; ¡ 85; 93; 95; 99)

E7 (a1 )

(93; ¡ 85; 95; ¡ 84; 96; ¡ 87; ¡ 88)

E7 (a1 )0

(¡ 87; 96; ¡ 84; 95; ¡ 85; 93; ¡ 89)

E7 (a1 )00

(¡ 86; 98; ¡ 87; 96; ¡ 84; ¡ 89; 95; 99)

D8 (a1 )

(¡ 86; 98; ¡ 87; 96; ¡ 84; 95; ¡ 85; ¡ 88)

D8 (a1 )0

(89; 98; ¡ 76; ¡ 90; 100; ¡ 91; ¡ 92; 96)

E8 (a5 )

(¡ 85; 89; ¡ 76; 96; ¡ 87; 98; ¡ 91; ¡ 86)

D8 (a1 )

(¡ 86; 98; ¡ 87; 96; ¡ 76; 89; ¡ 85; ¡ 88)

D8 (a1 )0

(¡ 87; 96; ¡ 88; ¡ 85; 93; 95; 99) + (¡ 86)

E7 (a1 ) + A1

(93; ¡ 85; 95; ¡ 84; 96; ¡ 87; ¡ 88) + (¡ 86)

E7 (a1 )0 + A1

(¡ 87; 96; ¡ 88; 99; ¡ 89; 93; ¡ 85) + (¡ 86)

E7 (a1 )00 + A1

(94; ¡ 87; 96; ¡ 88; ¡ 85; 93; 95; ¡ 81)

E8 (b4 )

(94; ¡ 87; 96; ¡ 83; ¡ 85; 93; 95; 99)

E8 (b4 )0

(¡ 88; 96; ¡ 78; ¡ 89; 93; 94; 86) + (95)

E7 (a1 ) + A1

(93; ¡ 80; 86; ¡ 78; 96; ¡ 88; ¡ 87) + (95)

E7 (a1 )0 + A1

(¡ 88; 96; ¡ 78; 86; ¡ 80; 93; ¡ 89) + (95)

E7 (a1 )00 + A1

61

62

62

63

63

103

104

105

106

107

13131043

22222022

40040400

22222042

04040044

64

108

34131341

(93; ¡ 86; ¡ 89; 94; ¡ 87; 96; ¡ 88)

E7

65

109

22222222

(¡ 86; 94; ¡ 87; 96; ¡ 88; ¡ 85; 93; 95)

D8

65

110

44040400

(¡ 89; 94; ¡ 83; ¡ 95; 100; 102; 96; ¡ 93)

E8 (a4 )

(¡ 91; 102; ¡ 92; 96; ¡ 83; 94; ¡ 89; ¡ 90)

D8

66

111

24222242

(93; ¡ 86; ¡ 89; 94; ¡ 87; 96; ¡ 88) + (95)

E 7 + A1

66

112

44040440

(93; ¡ 86; ¡ 89; 94; ¡ 87; 96; ¡ 88; ¡ 91)

E8 (a3 )

(93; ¡ 90; ¡ 89; 94; ¡ 83; 96; ¡ 92) + (¡ 91)

E 7 + A1

(95; 96; ¡ 91; ¡ 87; 94; ¡ 89; 93; ¡ 86)

E8 (a2 )

(94; 96; ¡ 86; ¡ 92; 95; ¡ 85; 93; ¡ 89)

E8 (a2 )0

(93; ¡ 89; 99; ¡ 91; ¡ 87; 90; 96; ¡ 92)

E8 (a2 )00

67

113

44044044


0

= E VIII)


598

k

i

j (H i )

68

114

44440444

69

115

84444444

Support of E i

Type of E i

(93; ¡ 89; 94; ¡ 90; ¡ 88; 96; 91; 95)

E8 (a1 )

(93; ¡ 89; 94; ¡ 90; ¡ 88; 96; 91; ¡ 86)

E8 (a1 )0

(93; ¡ 89; 94; ¡ 90; 95; ¡ 92; 96; ¡ 88)

E8 (a1 )00

(96; ¡ 91; ¡ 92; 95; ¡ 90; 94; ¡ 89; 93)

E8


0

= E VIII)


i!j

i!j

i!j

i!j

0

1!0

2!1

3!2

4!3

5!3

6!3

7 ! 4; 6

8 ! 5; 6

9!7

10 ! 7; 8

11 ! 10

12 ! 9

13 ! 9; 10

14 ! 12

15 ! 12; 13

16 ! 13

17 ! 16

18 ! 11; 17

19 ! 18

20 ! 18

21 ! 19; 20

22 ! 15; 17

23 ! 18; 22

24 ! 19; 23

25 ! 20; 23

26 ! 14; 24

27 ! 25

28 ! 24; 25

29 ! 26; 28

30 ! 27; 28

31 ! 27

32 ! 27; 28

33 ! 21; 32

34 ! 31

35 ! 26; 32

36 ! 31; 32

37 ! 29; 35

38 ! 30; 36

39 ! 35; 36

40 ! 33; 37

41 ! 33; 38

42 ! 37

43 ! 37; 39

44 ! 34; 38; 39

45 ! 43

46 ! 43; 44

47 ! 46

48 ! 40; 42

49 ! 40; 41; 46

50 ! 42; 46

51 ! 48

52 ! 49; 50

53 ! 48; 49; 50

54 ! 45; 47

55 ! 47; 49

56 ! 54; 55

57 ! 45; 50

58 ! 47; 57

59 ! 52; 57

60 ! 51; 53; 59

61 ! 53; 58; 59

62 ! 54; 58

63 ! 52; 55; 58

64 ! 61; 63

65 ! 61; 62

66 ! 56; 64; 65

67 ! 60; 65

68 ! 64

69 ! 64; 65

70 ! 67

71 ! 67; 68; 69

72 ! 66; 67; 69

73 ! 66; 68; 69

74 ! 71

75 ! 73

76 ! 71; 73

77 ! 70; 76

78 ! 75; 76

79 ! 75

80 ! 74; 76

81 ! 72; 74; 75; 76

82 ! 77; 78

83 ! 80; 81

84 ! 70; 81

85 ! 79; 80; 81

86 ! 84; 85

87 ! 77; 80; 84

88 ! 78; 85

89 ! 83; 88

90 ! 83; 87

91 ! 86; 88

92 ! 86; 87

93 ! 89; 91

94 ! 90

95 ! 89; 90; 91

96 ! 82; 91; 92

97 ! 93; 95; 96

98 ! 93; 95

99 ! 92; 94; 95

100 ! 98; 99

101 ! 97; 99

102 ! 97; 98

103 ! 101; 102

104 ! 100

105 ! 100; 101; 102

106 ! 103; 104; 105

107 ! 103; 105

108 ! 106; 107

109 ! 106; 107

110 ! 106

111 ! 108; 109

112 ! 108; 110

113 ! 111; 112

114 ! 113

115 ! 114 j

Table 2 The irreducible components O1j of @O1i

599


600

i

j

7

:

5

9

:

12

i

j

8

:

4

85

11

:

9

:

10 8 5

14

:

13 10 8 5

16

:

12

17

:

15 14 12

18

:

15 14 12

19

:

15 14 12

20

:

15 14 12

21

:

15 14 12

22

:

14

23

:

14

24

:

20 14

25

:

19 14

26

:

25 20

27

:

24 19 14

28

:

14

29

:

27

30

:

26 14

31

:

28 26 24 19 14

32

:

26 14

33

:

31 26 14

34

:

32 28 26 24 19 14

35

:

31

36

:

26 14

37

:

36 34 31 30

38

:

35 34 29 26 14

39

:

34

40

:

38 36 34 31 30

41

:

37 35 34 29 26 14

42

:

39 38 36 34 31 30

43

:

38 34 30

44

:

37 29

45

:

44 42 38 34 30

46

:

42

47

:

45 42

48

:

46 45 44 43 41 39

49

:

45 42

51

:

50 49 46 45 44 43 41

38 36 34 31 30 50

:

45

39 38 36 34 31 30 52

:

48 45

53

:

45

54

:

49 48 42 41 40 33 21

55

:

48 45 42

56

:

48 42

59

:

53 51 48

60

:

58 47

61

:

51

62

:

59 55 53 52 51 49

63

:

59 54 53 51 48

65

:

63 60 55 51

48 41 40 33 21 64

:

62 60 54 51

Table 3 Orbits O1j » G ¢ O1i n O1i with dim(O1j ) < dim(O1i )


i

j

i

j

66

:

60 51

67

:

64 63 55

68

:

65 62 60 54 51

69

:

60 51

70

:

69 68 64 63 55

72

:

68

73

:

67 60 51

74

:

70

75

:

72 71 70 67 60 51

76

:

72 70

77

:

75 72

78

:

72 70

79

:

76 74 72 71 70 67 60 51

80

:

75 72 70

81

:

70

82

:

72

83

:

79 70

84

:

80 79

85

:

70

87

:

85 79 78

88

:

84 77 70

89

:

87 84 77 70

90

:

88 85 79 78

91

:

87 77

92

:

88 78

93

:

92 90 87 77

94

:

92 91 89 88 86 85 79 78

95

:

92

97

:

94

98

:

92

99

:

93

101

:

98

102

:

99 94

104

:

102 101 97 96 82

107

:

104

110

:

107

111

:

110

112

:

109

Table 3 Orbits O1j » G ¢ O1i n O1i with dim(O1j ) < dim(O1i )

601


602

i

j

Type of E

E

2

1

A1

E1

3

2

2A1

E2

4,5,6

3

3A1

E3

7

4

A2

E4

8

5

(4A1 )00

E5

7,8

6

(4A1 )0

E6

9,10

7

A2 + A1

E7

10

8

5A1

E8

12,13

9

A2 + 2A1

E9

11

10

A2 + 2A1

E 10

13

10

6A1

E 10

18

11

A3

E 11

14,15

12

A2 + 3A1

E 12

15,16

13

A2 + 3A1

E 13

26

14

2A2

E 14

22

15

A2 + 4A1

E 15

17

16

2A2

E 16

18

17

2A2 + A1

E 17

22

17

2A2 + A1

(110; ¡ 63) + (¡ 58; ¡ 67) + (¡ 66)

19,20,23

18

A3 + A1

E 18

21,24

19

(A3 + 2A1 )00

E 19

21

20

(A3 + 2A1 )00

E 20

25

20

3A2

E 20

33

21

D4

E 21

23

22

2A2 + 2A1

E 22

24,25

23

(A3 + 2A1 )0

E 23

26

24

A3 + 3A1

E 24

28

24

A3 + 3A1

(108; ¡ 71; ¡ 67) + (¡ 63) + (¡ 65) + (¡ 75)

27

25

A3 + 3A1

E 25

28

25

A3 + 3A1

(103; ¡ 66; ¡ 65) + (108) + (¡ 63) + (¡ 67) j

Table 4 Elements E 2 p2 (H i ) \ O1j


i

j

Type of E

603

E

29

26

A3 + A2

(112; ¡ 93; ¡ 54) + (96; ¡ 66)

35

26

A3 + A2

E 26

30

27

A3 + A2

(¡ 51; 93; ¡ 64) + (96; ¡ 66)

31,32

27

A3 + A2

E 27

29,30

28

A3 + 4A1

E 28

32

28

D4 (a1 ) + 2A1

E 28

37

29

A4

E 29

38

30

(2A3 )00

E 30

34,36

31

A3 + A2 + A1

E 31

33

32

A3 + A2 + A1

(¡ 65; 105; ¡ 71) + (¡ 69; ¡ 79) + (¡ 70)

35,36

32

A3 + A2 + A1

E 32

40,41

33

D4 + A1

E 33

44

34

A3 + A2 + 2A1

E 34

37

35

D4 (a1 ) + A2

E 35

39

35

D4 (a1 ) + A2

(106; ¡ 73; ¡ 59; ¡ 75) + (¡ 71; ¡ 77)

38,39

36

A3 + A2 + 2A1

E 36

40

37

A4 + A1

(¡ 65; ¡ 88; 93; ¡ 69) + (¡ 46)

42,43

37

A4 + A1

E 37

41

38

2A3 + A1

E 38

44

38

2A3 + A1

(106; ¡ 79; ¡ 69) + (¡ 70; 102; ¡ 72) + (¡ 71)

43

39

(2A3 )0

(105; ¡ 80; ¡ 72) + (¡ 60; 99; ¡ 70)

¤

44

39

(2A3 )0

(102; ¡ 77; ¡ 71) + (106; ¡ 79; ¡ 69)

¤

48,49

40

D4 + 2A1

E 40

49

41

D4 + 2A1

E 41

48

42

A4 + 2A1

(¡ 80; 93; ¡ 57; ¡ 88) + (¡ 59) + (¡ 81)

¤

50

42

A4 + 2A1

(¡ 69; ¡ 84; 108; ¡ 82) + (¡ 73) + (¡ 77)

¤

45,46

43

A4 + 2A1

E 43

46

44

D4 (a1 ) + A3

E 44

54

45

A4 + A2

E 45

57

45

A4 + A2

(¡ 72; 102; ¡ 75; ¡ 83) + (106; ¡ 79) j


¤

¤

¤

¤


604

i

j

Type of E

E

47

46

A4 + A2

(¡ 71; 96; ¡ 73; ¡ 70) + (98; ¡ 65)

49

46

A4 + A2

E 46

50

46

A4 + A2

(¡ 69; ¡ 84; 108; ¡ 82) + (103; ¡ 73)

54,55,58

47

A5

E 47

51,53

48

D4 + 3A1

E 48

52,53,55

49

D5 (a1 ) + A1

E 49

52,57

50

A4 + A2 + A1

E 50

53

50

A4 + A2 + A1

(¡ 69; ¡ 84; 105; ¡ 76) + (106; ¡ 73) + (¡ 77)

60

51

D 4 + A2

E 51

59,63

52

D 4 + A2

E 52

60,61

53

D4 + 4A1

E 53

56,62

54

(A5 + A1 )00

E 54

56

55

E6 (a3 )

E 55

63

55

(A5 + A1 )00

E 55

66

56

D5

E 56

58

57

A4 + A3

(¡ 64; 102; ¡ 81; ¡ 76) + (¡ 78; 100; ¡ 79)

59

57

A4 + A3

E 57

61,62

58

(A5 + A1 )0

(102; ¡ 81; ¡ 76; 103; ¡ 79) + (¡ 78)

63

58

(A5 + A1 )0

E 58

60

59

D5 (a1 ) + A2

(¡ 78; 105; ¡ 80; ¡ 88; ¡ 77) + (106; ¡ 82)

61

59

D5 (a1 ) + A2

E 59

67

60

D 4 + A3

E 60

64,65

61

D6 (a2 )

E 61

65

62

A5 + 2A1

E 62

64

63

A5 + 2A1

E 63

66

64

D6 (a2 ) + A1

(103; ¡ 76; ¡ 81; 98; ¡ 78; 93) + (¡ 52)

68,69

64

A5 + A2

E 64

66

65

D6 (a2 ) + A1

(¡ 73; 103; ¡ 76; ¡ 81; 98; ¡ 78) + (¡ 85)

67,69

65

D6 (a2 ) + A1

E 65

72

66

D 5 + A1

E 66 j


¤

¤

¤

¤

¤

¤


i

j

Type of E

605

E

73

66

D5 + A1

(¡ 82; 93; ¡ 80; 103; ¡ 77) + (¡ 78)

70

67

D4 + D4 (a1 )

E 67

71

67

D4 + D4 (a1 )

(94; ¡ 77; 102; ¡ 85) + (96; ¡ 78; ¡ 76; ¡ 53)

72

67

D6 (a2 ) + 2A1

E 67

71

68

D4 + D4 (a1 )

E 68

73

68

D6 (a2 ) + 2A1

E 68

71

69

D6 (a2 ) + 2A1

E 69

72

69

D6 (a2 ) + 2A1

(106; ¡ 82; 93; ¡ 69; ¡ 88; ¡ 81) + (¡ 78)

¤

¤

+(¡ 46) 73

69

D6 (a2 ) + 2A1

(¡ 67; 95; ¡ 75; ¡ 78; 98; ¡ 77) + (103) + (¡ 82)

77

70

A6

(96; ¡ 78; ¡ 89; 94; ¡ 77; 102)

¤

84

70

A6

(¡ 81; 98; ¡ 83; ¡ 85; 106; ¡ 92)

¤

74

71

A6

E 71

76

71

A6

(¡ 70; 99; ¡ 81; 98; ¡ 87; ¡ 80)

81

72

D5 + 2A1

E 72

75,76

73

D6 (a1 )

E 73

80

74

A6 + A1

E 74

81

74

A6 + A1

(¡ 83; 103; ¡ 80; 93; ¡ 70; ¡ 90) + (¡ 84)

78

75

D6 (a1 ) + A1

(¡ 82; 93; ¡ 80; ¡ 77; 103; 99) + (¡ 84)

79,81

75

D6 (a1 ) + A1

E 75

77

76

D6 (a1 ) + A1

(96; ¡ 73; 94; ¡ 77; ¡ 89; ¡ 85) + (¡ 51)

¤

78

76

E7 (a4 )

(94; ¡ 77; ¡ 85; 93; ¡ 76; 96; ¡ 73)

¤

80,81

76

D6 (a1 ) + A1

E 76

82

77

(A7 )00

E 77

87

77

E6 (a1 )

E 77

82,88

78

(A7 )00

E 78

85

79

D5 + A2

(¡ 83; 98; ¡ 81; ¡ 82; 99) + (100; ¡ 84)

¤

83

80

D5 + A2

(98; ¡ 78; 96; ¡ 82; ¡ 88) + (¡ 67; ¡ 80)

¤

85

80

D5 + A2

E 80

87

80

D6 (a1 ) + 2A1

E 80 j


¤

¤


606

i

j

Type of E

E

83

81

E7 (a4 )0 + A1

E 81

84

81

D6 (a1 ) + 2A1

E 81

85

81

D6 (a1 ) + 2A1

(¡ 83; 98; ¡ 81; ¡ 82; 99; 106) + (¡ 80) + (¡ 84)

96

82

E6

E 82

89,90

83

D6

E 83

86

84

D7 (a2 )

(¡ 81; 98; ¡ 83; ¡ 85; 100; ¡ 84; ¡ 83)

¤

87

84

D7 (a2 )

(¡ 81; 99; ¡ 79; ¡ 93; 105; ¡ 83; 100)

¤

86,88

85

D5 + A3

E 85

91

86

(A7 )0

(95; ¡ 75; 93; ¡ 82; ¡ 86; 103; ¡ 88)

92

86

(A7 )0

E 86

90

87

E6 (a1 ) + A1

(93; ¡ 76; 96; ¡ 84; ¡ 87; ¡ 90) + (¡ 59)

92

87

E6 (a1 ) + A1

E 87

89,91

88

A7 + A1

E 88

93,95

89

D6 + A1

E 89

94,95

90

D6 + A1

E 90

93

91

D8 (a3 )

E 91

95

91

D8 (a3 )

(¡ 84; 100; ¡ 80; ¡ 87; 98; ¡ 81; 99; 93)

96

91

D8 (a3 )

(¡ 84; 100; ¡ 85; ¡ 83; 98; ¡ 81; 99; 93)

96

92

D8 (a3 )

E 92

99

92

D8 (a3 )

(98; ¡ 81; 95; ¡ 84; ¡ 89; 103; ¡ 73; ¡ 85)

97

93

D7 (a1 )

(¡ 81; 95; ¡ 84; ¡ 89; 93; 103; ¡ 57)

98

93

D7 (a1 )

E 93

99

94

D7 (a1 )

E 94

97

95

E7 (a3 ) + A1

E 95

98,99

95

D6 + 2A1

E 95

97

96

E6 + A1

E 96

101,102

97

E7 (a2 )

E 97

100

98

E8 (a6 )

(¡ 86; 98; ¡ 87; 96; ¡ 88; 99; ¡ 80; ¡ 71)

102

98

D8 (a2 )

E 98

100

99

D8 (a2 )

(¡ 87; 98; ¡ 86; 100; ¡ 85; ¡ 88; 99; ¡ 84) j


¤

¤

¤

¤

¤


i

j

Type of E

607

E

D8 (a2 )

E 99

100

D7

E 100

103

101

E7 (a2 ) + A1

E 101

105

101

E7 (a2 )0 + A1

E 101

103

102

E7 (a2 ) + A1

(93; 95; ¡ 89; ¡ 84; 96; ¡ 87; 103) + (¡ 38)

105

102

E7 (a2 )0 + A1

E 102

106,107

103

E7 (a1 )

E 103

106

104

D8 (a1 )

E 104

106

105

E8 (a5 )

E 105

107

105

E8 (a5 )

(86; ¡ 78; 96; ¡ 88; ¡ 85; 93; 95; ¡ 63)

108

106

E7 (a1 )00 + A1

E 106

109

106

E7 (a1 ) + A1

E 106

110

106

E7 (a1 ) + A1

(¡ 89; 94; ¡ 83; ¡ 95; 102; 100; 96) + (¡ 57)

¤

108

107

E7 (a1 ) + A1

(100; ¡ 86; 94; ¡ 89; 93; ¡ 72; ¡ 87) + (¡ 88)

¤

109

107

E8 (b4 )

E 107

111,112

108

E7

E 108

111

109

D8

E 109

112

110

D8

E 110

113

111

E7 + A1

E 111

113

112

E8 (a3 )

E 112

114

113

E8 (a2 )0

E 113

115

114

E8 (a1 )00

E 114

101

99

104,105

j


¤

¤


608

(8; 4)y

(14; 5)y

(11; 9)

(22; 11)*

(21; 12)

(34; 14)

(41; 14)

(34; 19)y

(26; 20)y

(62; 21)

(29; 27)

(44; 29)y

(45; 30)y

(51; 30)

(51; 31)

(41; 34)

(45; 34)

(56; 42)

(52; 45)

(53; 45)

(55; 45)

(60; 47)y

(59; 48)

(63; 48)

(79; 51)

(68; 54)y

(70; 55)

(74; 56)*

(63; 59)

(72; 68)

(89; 70)

(80; 72)y

(82; 72)

(82; 74)*

(77; 75)

(80; 75)y

(93; 77)

(92; 78)y

(94; 78)

(82; 79)*

(94; 79)

(84; 80)

(104; 82)

(96; 83)*

(98; 92)

(99; 93)

(102; 94)

(101; 98)

(103; 100)¤

(107; 104)

(110; 107)

(112; 109)

(111; 110)

Table 5 The list of critical pairs


(8; 4)

(14; 5)

(17; 12)

(27; 14)

(28; 14)

(34; 19)

(26; 20)

(34; 24)

(26; 25)

(32; 26)

(34; 26)

(34; 28)

(44; 29)

(42; 30)

(45; 30)

(35; 31)

(34; 32)

(38; 34)

(39; 34)

(42; 34)

(44; 37)

(42; 38)

(45; 38)

(42; 39)

(51; 41)

(45; 42)

(45; 44)

(49; 45)

(50; 45)

(60; 47)

(52; 48)

(51; 49)

(59; 51)

(62; 51)

(63; 51)

(68; 54)

(62; 55)

(60; 58)

(68; 60)

(68; 62)

(67; 64)

(68; 65)

(79; 70)

(80; 70)

(81; 70)

(79; 72)

(80; 72)

(80; 75)

(79; 76)

(91; 77)

(92; 78)

(91; 87)

(92; 88)

(94; 89)

(93; 90) j

Table 6 Some pairs (i; j) such that O1i ¦ O1j

609


610

p2 (H i )

i 1

¡ 1;

2

¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1;

3

¡ 66; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

4

¡ 93; ¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 66; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1;

5

93; 98; ¡ 79; 102; ¡ 73; 105; ¡ 66; ¡ 67; 108; ¡ 59; ¡ 60; 110; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

6

¡ 65; ¡ 58; ¡ 60; ¡ 51; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

7

¡ 47; ¡ 82; ¡ 76; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 66; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

8

¡ 44; 105; ¡ 67; 108; ¡ 59; ¡ 60; 110; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

9

¡ 81; ¡ 75; ¡ 77; ¡ 69; ¡ 70; ¡ 72; ¡ 63; ¡ 64; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

10

¡ 71; ¡ 64; ¡ 66; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 44; ¡ 45; ¡ 37; 110; ¡ 54; 112; ¡ 47; ¡ 46; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

11

¡ 44; 93; 98; ¡ 79; 102; ¡ 73; ¡ 66; 105; ¡ 67; 108; ¡ 59; ¡ 60; 110; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

12

¡ 96; ¡ 92; ¡ 87; ¡ 88; ¡ 82; ¡ 83; ¡ 84; ¡ 76; ¡ 77; ¡ 78; ¡ 79; ¡ 70; ¡ 71; ¡ 72; ¡ 73; ¡ 63; ¡ 64; ¡ 66; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

13

¡ 77; ¡ 70; ¡ 72; ¡ 63; ¡ 64; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 38; 112; ¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 14

¡ 112; ¡ 110; ¡ 108; ¡ 105; ¡ 106; ¡ 102; ¡ 103; ¡ 98; ¡ 99; ¡ 100; ¡ 93; ¡ 94; ¡ 95; ¡ 96; ¡ 89; ¡ 90; ¡ 91; ¡ 92; ¡ 85; ¡ 86; ¡ 87; ¡ 88; ¡ 80; ¡ 82; ¡ 81; ¡ 83; ¡ 84; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 79; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 73; ¡ 63; ¡ 64; ¡ 66; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1;

15

¡ 96; ¡ 92; ¡ 87; ¡ 88; ¡ 82; ¡ 83; ¡ 84; ¡ 76; ¡ 77; ¡ 78; ¡ 79; ¡ 70; ¡ 71; ¡ 72; ¡ 73; ¡ 63; ¡ 64; ¡ 66; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

16

96; ¡ 81; 100; ¡ 75; ¡ 77; 103; ¡ 69; ¡ 70; ¡ 72; 106; 105; ¡ 63; ¡ 64; ¡ 65; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 51; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

17

¡ 66; ¡ 75; ¡ 69; ¡ 70; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 105; ¡ 67; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 44; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

18

¡ 71; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 44; 98; ¡ 79; 102; ¡ 73; 105; ¡ 67; 108; ¡ 60; ¡ 66; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 37; 110; ¡ 54; 112; ¡ 47; ¡ 46; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

19

¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 44; 98; ¡ 79; 102; ¡ 73; 105; ¡ 67; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 66; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

20

89; 94; 93; ¡ 84; 99; 98; ¡ 78; ¡ 79; 103; 102; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 44; ¡ 45; ¡ 37; 110; ¡ 54; 112; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

21

¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 44; 93; 98; ¡ 79; 102; ¡ 73; 105; ¡ 67; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 66; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

611


612

i

p2 (H i )

22

¡ 69; ¡ 63; ¡ 78; ¡ 71; ¡ 72; ¡ 73; ¡ 64; ¡ 66; ¡ 65; ¡ 67; ¡ 58; ¡ 59; ¡ 60; ¡ 53; 110; 112; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

23

¡ 77; ¡ 70; ¡ 72; ¡ 63; ¡ 64; ¡ 65; ¡ 57; ¡ 58; ¡ 51; ¡ 79; ¡ 73; ¡ 66; 105; 108; 110; ¡ 44; ¡ 67; ¡ 59; ¡ 60; ¡ 52; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 38; 112; ¡ 37; ¡ 30; ¡ 23; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

24

¡ 65; ¡ 79; ¡ 73; ¡ 67; ¡ 80; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 57; 108; 110; 112; ¡ 58; ¡ 51; ¡ 44; ¡ 60; ¡ 54; ¡ 47; ¡ 66; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

25

¡ 63; 99; ¡ 78; 103; 102; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 58; ¡ 59; ¡ 60; ¡ 53; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; 110; ¡ 54; ¡ 46; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

26

¡ 69; ¡ 79; ¡ 87; ¡ 82; ¡ 83; ¡ 76; ¡ 77; ¡ 78; ¡ 70; ¡ 71; ¡ 72; ¡ 64; ¡ 65; ¡ 58; 112; ¡ 63; ¡ 57; ¡ 51; ¡ 44; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 60; ¡ 53; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

27

¡ 88; ¡ 83; ¡ 84; ¡ 76; ¡ 77; ¡ 78; ¡ 79; ¡ 70; ¡ 71; ¡ 72; ¡ 73; ¡ 63; ¡ 64; ¡ 66; ¡ 67; ¡ 57; ¡ 59; ¡ 52; 106; 108; 110; ¡ 65; ¡ 58; ¡ 60; ¡ 51; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

28

¡ 71; ¡ 66; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 103; ¡ 72; 106; 105; ¡ 65; ¡ 67; 108; ¡ 60; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; 110; ¡ 54; 112; ¡ 47; ¡ 44; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

29

¡ 93; ¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 76; ¡ 71; ¡ 66; 96; ¡ 81; 100; ¡ 77; 103; ¡ 72; 106; 105; ¡ 65; ¡ 67; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 75; ¡ 69; ¡ 70; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

30

93; 98; ¡ 79; 102; ¡ 73; ¡ 66; 96; ¡ 81; 100; ¡ 75; ¡ 77; 103; ¡ 69; ¡ 70; ¡ 72; 106; ¡ 63; ¡ 64; ¡ 65; ¡ 57; ¡ 58; ¡ 51; ¡ 44; 105; ¡ 67; 108; ¡ 59; ¡ 60; 110; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 31

91; ¡ 87; 95; ¡ 82; ¡ 83; 100; 99; ¡ 76; ¡ 77; ¡ 78; 103; 102; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 53; ¡ 54; ¡ 46; ¡ 63; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

32

¡ 69; ¡ 83; ¡ 76; ¡ 77; ¡ 78; ¡ 70; ¡ 71; ¡ 72; ¡ 73; ¡ 64; ¡ 66; ¡ 67; ¡ 59; 106; 108; 110; ¡ 63; ¡ 57; ¡ 52; ¡ 65; ¡ 58; ¡ 60; ¡ 53; ¡ 54; ¡ 46; 112; ¡ 47; ¡ 39; ¡ 51; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

33

¡ 65; ¡ 80; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 57; 93; ¡ 58; ¡ 51; ¡ 44; 98; 102; 105; ¡ 79; ¡ 73; ¡ 67; 108; 110; 112; ¡ 60; ¡ 54; ¡ 47; ¡ 66; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

34

69; 75; 81; 80; ¡ 96; 86; 85; ¡ 92; 91; 90; 89; ¡ 87; ¡ 88; 95; 94; 93; ¡ 82; ¡ 83; ¡ 84; 100; 99; 98; ¡ 76; ¡ 77; ¡ 78; ¡ 79; 103; 102; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 63; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

35

¡ 91; ¡ 86; ¡ 88; ¡ 80; ¡ 81; ¡ 83; ¡ 84; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 79; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 73; ¡ 63; ¡ 64; ¡ 66; ¡ 67; ¡ 57; ¡ 59; ¡ 52; 106; 108; 110; 112; ¡ 65; ¡ 58; ¡ 60; ¡ 51; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

36

¡ 69; 91; ¡ 87; 95; ¡ 82; ¡ 83; 100; 99; ¡ 76; ¡ 77; ¡ 78; 103; 102; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 53; ¡ 54; ¡ 46; ¡ 63; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

37

¡ 73; ¡ 81; ¡ 77; ¡ 72; ¡ 65; ¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 76; ¡ 71; 105; 108; 110; 112; ¡ 66; ¡ 67; ¡ 60; ¡ 54; ¡ 47; ¡ 75; ¡ 70; ¡ 64; ¡ 58; ¡ 59; ¡ 53; ¡ 46; ¡ 39; ¡ 69; ¡ 63; ¡ 57; ¡ 51; ¡ 44; ¡ 52; ¡ 45; ¡ 38; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

38

¡ 71; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 98; ¡ 79; 102; ¡ 73; 103; ¡ 72; 106; ¡ 65; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 66; 105; ¡ 67; 108; ¡ 60; ¡ 44; ¡ 59; ¡ 52; ¡ 53; ¡ 45; 110; ¡ 54; 112; ¡ 47; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

613


614

p2 (H i )

i 39

¡ 78; ¡ 71; ¡ 73; ¡ 66; ¡ 81; ¡ 75; ¡ 77; ¡ 70; 103; 106; 105; 108; ¡ 69; ¡ 63; ¡ 72; ¡ 64; ¡ 65; ¡ 67; ¡ 58; ¡ 59; ¡ 60; ¡ 53; 110; 112; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 57; ¡ 51; ¡ 52; ¡ 45; ¡ 38; ¡ 31; ¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

40

¡ 69; ¡ 87; ¡ 82; ¡ 83; ¡ 76; ¡ 77; ¡ 78; ¡ 70; ¡ 71; ¡ 72; ¡ 64; ¡ 65; ¡ 58; 93; 98; ¡ 63; ¡ 57; ¡ 51; ¡ 44; 102; 105; 108; 110; ¡ 79; 112; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 60; ¡ 53; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

41

¡ 71; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 93; 103; ¡ 72; 106; ¡ 65; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 98; ¡ 79; 102; ¡ 73; ¡ 44; ¡ 66; 105; ¡ 67; 108; ¡ 60; ¡ 59; ¡ 52; ¡ 53; ¡ 45; 110; ¡ 54; 112; ¡ 47; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

42

¡ 81; ¡ 77; ¡ 84; ¡ 78; ¡ 79; ¡ 73; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 110; 112; ¡ 71; ¡ 66; ¡ 72; ¡ 65; ¡ 67; ¡ 60; ¡ 75; ¡ 69; ¡ 70; ¡ 63; ¡ 54; ¡ 47; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; ¡ 44; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

43

¡ 85; ¡ 80; ¡ 82; ¡ 75; ¡ 76; ¡ 70; 99; ¡ 78; 103; 102; ¡ 72; ¡ 73; 106; 105; ¡ 65; ¡ 67; 108; ¡ 60; ¡ 71; ¡ 64; ¡ 66; ¡ 58; ¡ 59; ¡ 53; 110; ¡ 54; 112; ¡ 47; ¡ 46; ¡ 39; ¡ 69; ¡ 63; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

44

¡ 69; ¡ 79; ¡ 83; ¡ 76; ¡ 77; ¡ 78; ¡ 70; ¡ 71; ¡ 72; ¡ 64; 102; 105; 106; ¡ 63; ¡ 57; ¡ 65; ¡ 58; ¡ 73; ¡ 66; ¡ 67; ¡ 59; 108; 110; ¡ 51; ¡ 44; ¡ 52; ¡ 60; ¡ 53; ¡ 54; ¡ 46; 112; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 47; ¡ 39; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

45

79; ¡ 98; 84; ¡ 93; ¡ 94; 88; ¡ 89; ¡ 90; 92; 91; ¡ 85; ¡ 86; ¡ 87; 96; 95; ¡ 80; ¡ 82; ¡ 81; ¡ 83; 100; 99; ¡ 75; ¡ 76; ¡ 77; ¡ 78; 103; 102; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 53; ¡ 54; 112; ¡ 46; ¡ 47; ¡ 39; ¡ 69; ¡ 63; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 46

¡ 85; ¡ 80; ¡ 82; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 63; 94; ¡ 84; 99; 98; ¡ 78; ¡ 79; 103; 102; ¡ 72; ¡ 73; 106; 105; ¡ 65; ¡ 67; 108; ¡ 60; ¡ 71; ¡ 64; ¡ 66; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 44; ¡ 45; ¡ 37; 110; ¡ 54; 112; ¡ 47; ¡ 46; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

47

¡ 71; 96; ¡ 81; 100; ¡ 77; 98; ¡ 79; 102; ¡ 73; ¡ 66; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 103; ¡ 72; 106; ¡ 65; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 105; ¡ 67; 108; ¡ 60; ¡ 59; ¡ 52; ¡ 53; ¡ 45; 110; ¡ 54; 112; ¡ 47; ¡ 44; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

48

¡ 91; ¡ 86; ¡ 88; ¡ 80; ¡ 81; ¡ 83; ¡ 84; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 63; ¡ 64; ¡ 57; 93; 98; 102; 105; ¡ 65; ¡ 58; ¡ 51; ¡ 44; 108; 110; 112; ¡ 79; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 52; ¡ 60; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

49

¡ 69; ¡ 83; ¡ 76; ¡ 77; ¡ 78; ¡ 70; ¡ 71; ¡ 72; ¡ 64; 93; 98; 106; ¡ 63; ¡ 57; ¡ 65; ¡ 58; ¡ 79; 102; 105; ¡ 51; ¡ 44; ¡ 73; ¡ 66; ¡ 67; ¡ 59; 108; 110; ¡ 52; ¡ 60; ¡ 53; ¡ 54; ¡ 46; 112; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 47; ¡ 39; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

50

¡ 75; ¡ 69; ¡ 77; ¡ 82; ¡ 76; ¡ 84; ¡ 78; ¡ 79; ¡ 73; 103; 106; 105; 108; ¡ 70; ¡ 63; ¡ 71; ¡ 66; ¡ 72; ¡ 65; ¡ 67; ¡ 60; 110; ¡ 54; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; 112; ¡ 44; ¡ 37; ¡ 46; ¡ 38; ¡ 47; ¡ 30; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

51

93; 98; 102; 105; 108; 110; 112; ¡ 106; ¡ 103; ¡ 99; ¡ 100; ¡ 94; ¡ 95; ¡ 96; ¡ 89; ¡ 90; ¡ 91; ¡ 92; ¡ 85; ¡ 86; ¡ 87; ¡ 88; ¡ 80; ¡ 82; ¡ 81; ¡ 83; ¡ 84; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 63; ¡ 64; ¡ 65; ¡ 57; ¡ 58; ¡ 51; ¡ 44; ¡ 79; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 60; ¡ 52; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

52

¡ 69; 91; ¡ 87; 95; ¡ 82; ¡ 83; 100; 99; ¡ 76; ¡ 77; ¡ 78; 103; ¡ 70; ¡ 71; ¡ 72; 106; ¡ 64; ¡ 65; ¡ 58; 93; 98; ¡ 79; ¡ 63; ¡ 57; ¡ 51; ¡ 44; 102; ¡ 73; 105; ¡ 66; ¡ 67; 108; ¡ 59; ¡ 60; 110; ¡ 53; ¡ 54; ¡ 46; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

615


616

i

p2 (H i )

53

¡ 91; ¡ 86; ¡ 88; ¡ 80; ¡ 81; ¡ 83; ¡ 84; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 63; ¡ 64; ¡ 57; 93; 98; 102; 105; 106; ¡ 65; ¡ 58; ¡ 51; ¡ 44; ¡ 79; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 52; 108; 110; 112; ¡ 60; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

54

¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 76; ¡ 71; 96; ¡ 81; 100; ¡ 77; 103; ¡ 72; 106; ¡ 65; 98; ¡ 79; 102; ¡ 73; ¡ 66; ¡ 75; ¡ 69; ¡ 70; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 105; ¡ 67; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 44; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

55

89; 94; 93; ¡ 84; 99; 98; ¡ 78; ¡ 79; 102; ¡ 71; ¡ 73; ¡ 66; 96; ¡ 81; 100; ¡ 75; ¡ 77; ¡ 69; ¡ 70; ¡ 63; 103; ¡ 72; 106; 105; ¡ 64; ¡ 65; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 45; ¡ 44; ¡ 37; 110; ¡ 54; 112; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

56

¡ 89; ¡ 85; ¡ 80; ¡ 82; ¡ 76; ¡ 71; 93; 96; ¡ 81; 100; ¡ 77; 103; ¡ 72; 106; ¡ 65; ¡ 75; ¡ 69; ¡ 70; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 98; ¡ 79; 102; ¡ 73; ¡ 44; ¡ 66; 105; ¡ 67; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

57

¡ 79; ¡ 80; ¡ 75; ¡ 82; ¡ 83; ¡ 77; ¡ 78; ¡ 72; 102; 105; 106; ¡ 65; ¡ 69; ¡ 73; ¡ 67; ¡ 76; ¡ 70; ¡ 71; ¡ 64; 108; 110; ¡ 58; ¡ 60; ¡ 54; ¡ 63; ¡ 57; ¡ 66; ¡ 59; 112; ¡ 47; ¡ 51; ¡ 44; ¡ 52; ¡ 53; ¡ 46; ¡ 39; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

58

¡ 79; ¡ 81; ¡ 75; ¡ 83; ¡ 76; ¡ 78; ¡ 71; 100; 103; 102; ¡ 69; ¡ 73; ¡ 66; ¡ 77; ¡ 70; ¡ 72; ¡ 64; 105; 106; ¡ 63; ¡ 57; ¡ 65; ¡ 58; ¡ 67; ¡ 59; 108; 110; ¡ 51; ¡ 52; ¡ 60; ¡ 53; ¡ 54; ¡ 46; 112; ¡ 44; ¡ 45; ¡ 38; ¡ 47; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

59

¡ 75; ¡ 69; ¡ 77; ¡ 82; ¡ 76; ¡ 84; ¡ 78; 98; 102; 103; 106; ¡ 70; ¡ 63; ¡ 71; ¡ 72; ¡ 65; ¡ 79; ¡ 73; 105; 108; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 66; ¡ 67; ¡ 60; 110; ¡ 44; ¡ 54; ¡ 59; ¡ 52; ¡ 53; ¡ 45; 112; ¡ 37; ¡ 46; ¡ 38; ¡ 47; ¡ 30; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 60

¡ 80; ¡ 81; ¡ 82; ¡ 88; ¡ 83; ¡ 84; ¡ 78; 105; 106; ¡ 75; ¡ 69; ¡ 76; ¡ 71; ¡ 77; ¡ 72; ¡ 79; ¡ 73; 108; 110; ¡ 65; ¡ 66; ¡ 67; ¡ 70; ¡ 63; ¡ 64; ¡ 57; 112; ¡ 58; ¡ 51; ¡ 59; ¡ 52; ¡ 60; ¡ 54; ¡ 44; ¡ 47; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 37; ¡ 30; ¡ 39; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

61

¡ 84; ¡ 79; ¡ 86; ¡ 80; ¡ 81; ¡ 83; ¡ 75; ¡ 76; ¡ 77; ¡ 70; 99; 103; 102; 105; ¡ 69; ¡ 63; ¡ 78; ¡ 71; ¡ 72; ¡ 73; ¡ 64; ¡ 66; ¡ 67; ¡ 59; ¡ 57; ¡ 52; 106; 108; ¡ 65; ¡ 58; ¡ 60; ¡ 53; 110; 112; ¡ 51; ¡ 44; ¡ 45; ¡ 37; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 38; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

62

¡ 78; ¡ 79; ¡ 81; ¡ 77; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 102; 103; 106; ¡ 71; ¡ 72; ¡ 65; ¡ 73; ¡ 75; ¡ 70; 105; 108; ¡ 64; ¡ 58; ¡ 66; ¡ 67; ¡ 60; ¡ 69; ¡ 63; 110; 112; ¡ 54; ¡ 47; ¡ 57; ¡ 51; ¡ 59; ¡ 53; ¡ 44; ¡ 46; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

63

¡ 75; ¡ 69; ¡ 76; 94; ¡ 84; 99; 98; ¡ 78; ¡ 79; 102; ¡ 73; 100; ¡ 77; ¡ 70; ¡ 63; ¡ 71; ¡ 66; 103; ¡ 72; 105; ¡ 67; ¡ 64; ¡ 57; ¡ 59; ¡ 52; 106; ¡ 65; 108; ¡ 60; ¡ 58; ¡ 51; ¡ 53; ¡ 45; 110; ¡ 54; ¡ 44; ¡ 37; ¡ 46; ¡ 38; 112; ¡ 47; ¡ 30; ¡ 39; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

64

¡ 80; ¡ 75; ¡ 69; 90; ¡ 88; 95; 94; ¡ 83; ¡ 84; 100; 99; 98; ¡ 77; ¡ 78; ¡ 79; 103; 102; ¡ 72; ¡ 73; 105; ¡ 67; ¡ 76; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 66; ¡ 57; ¡ 59; ¡ 52; 106; ¡ 65; 108; ¡ 60; 110; ¡ 54; ¡ 58; ¡ 51; ¡ 53; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; 112; ¡ 47; ¡ 39; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

65

¡ 80; ¡ 75; ¡ 82; ¡ 83; ¡ 77; ¡ 84; ¡ 79; 99; 103; 102; 105; ¡ 69; ¡ 76; ¡ 70; ¡ 78; ¡ 72; ¡ 73; ¡ 67; 106; 108; ¡ 63; ¡ 65; ¡ 60; ¡ 71; ¡ 64; ¡ 66; ¡ 59; 110; ¡ 54; ¡ 57; ¡ 52; ¡ 58; ¡ 53; 112; ¡ 46; ¡ 47; ¡ 51; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 39; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

66

¡ 78; ¡ 81; ¡ 77; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 93; 103; 106; ¡ 71; ¡ 72; ¡ 65; ¡ 75; ¡ 70; 98; ¡ 64; ¡ 58; ¡ 69; ¡ 63; ¡ 79; 102; ¡ 57; ¡ 51; ¡ 73; 105; 108; ¡ 44; ¡ 66; ¡ 67; ¡ 60; 110; 112; ¡ 54; ¡ 47; ¡ 59; ¡ 53; ¡ 46; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

617


618

i

p2 (H i )

67

¡ 84; ¡ 79; ¡ 90; ¡ 85; ¡ 86; ¡ 87; ¡ 80; ¡ 82; ¡ 81; ¡ 83; ¡ 75; ¡ 76; ¡ 77; ¡ 70; 99; 103; 102; 106; 105; 108; ¡ 69; ¡ 63; ¡ 78; ¡ 71; ¡ 72; ¡ 73; ¡ 64; ¡ 66; ¡ 65; ¡ 67; ¡ 58; ¡ 59; ¡ 60; ¡ 53; 110; 112; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

68

82; 87; 85; ¡ 91; 92; 90; 89; ¡ 86; ¡ 88; 96; 95; 94; 93; ¡ 80; ¡ 81; ¡ 83; ¡ 84; 100; 99; 98; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 79; 103; 102; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 105; ¡ 63; ¡ 64; ¡ 66; ¡ 67; ¡ 57; ¡ 59; ¡ 52; 106; ¡ 65; 108; ¡ 58; ¡ 60; 110; ¡ 51; ¡ 53; ¡ 54; 112; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

69

¡ 80; ¡ 75; ¡ 69; ¡ 82; 90; ¡ 88; 95; 94; ¡ 83; ¡ 84; 100; 99; 98; ¡ 77; ¡ 78; ¡ 79; 103; 102; ¡ 72; ¡ 73; 105; ¡ 67; ¡ 76; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 66; ¡ 57; ¡ 59; ¡ 52; 106; ¡ 65; 108; ¡ 60; 110; ¡ 54; ¡ 58; ¡ 51; ¡ 53; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; 112; ¡ 47; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

70

96; 100; 103; 106; 105; 108; 110; 112; ¡ 102; ¡ 98; ¡ 99; ¡ 93; ¡ 94; ¡ 95; ¡ 89; ¡ 90; ¡ 91; ¡ 92; ¡ 85; ¡ 86; ¡ 87; ¡ 88; ¡ 80; ¡ 82; ¡ 83; ¡ 84; ¡ 76; ¡ 78; ¡ 79; ¡ 71; ¡ 73; ¡ 66; ¡ 81; ¡ 75; ¡ 77; ¡ 69; ¡ 70; ¡ 72; ¡ 63; ¡ 64; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 44; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

71

¡ 85; ¡ 80; ¡ 82; ¡ 76; 94; ¡ 84; 99; 98; ¡ 78; ¡ 79; 102; ¡ 73; 96; ¡ 81; 100; ¡ 77; ¡ 71; ¡ 66; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 103; ¡ 72; 106; 105; ¡ 65; ¡ 67; 108; ¡ 60; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; 110; ¡ 54; 112; ¡ 47; ¡ 44; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

72

¡ 80; ¡ 81; ¡ 82; ¡ 88; ¡ 83; ¡ 84; ¡ 78; 93; 106; ¡ 75; ¡ 69; ¡ 76; ¡ 71; ¡ 77; ¡ 72; 98; 102; ¡ 65; ¡ 70; ¡ 63; ¡ 64; ¡ 57; 105; ¡ 58; ¡ 51; ¡ 79; ¡ 73; 108; 110; ¡ 44; ¡ 66; ¡ 67; 112; ¡ 59; ¡ 52; ¡ 60; ¡ 54; ¡ 47; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 37; ¡ 30; ¡ 39; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

73

¡ 80; ¡ 75; ¡ 82; 93; 95; ¡ 83; 100; 99; ¡ 77; ¡ 78; 103; ¡ 72; ¡ 76; ¡ 70; ¡ 71; ¡ 64; 98; ¡ 79; 106; ¡ 65; ¡ 58; ¡ 69; ¡ 63; ¡ 57; 102; ¡ 73; 105; ¡ 67; ¡ 51; ¡ 44; ¡ 66; ¡ 59; 108; ¡ 60; 110; ¡ 54; ¡ 53; ¡ 46; 112; ¡ 47; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 74

¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 84; ¡ 79; 99; 102; 100; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 73; 103; 105; ¡ 69; ¡ 70; ¡ 71; ¡ 66; ¡ 72; ¡ 67; 106; 108; ¡ 63; ¡ 64; ¡ 59; ¡ 65; ¡ 60; 110; ¡ 54; ¡ 57; ¡ 52; ¡ 58; ¡ 53; 112; ¡ 46; ¡ 47; ¡ 51; ¡ 45; ¡ 38; ¡ 39; ¡ 44; ¡ 37; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

75

¡ 80; ¡ 75; ¡ 82; ¡ 83; ¡ 77; ¡ 84; 93; 99; 103; ¡ 69; ¡ 76; ¡ 70; ¡ 78; ¡ 72; 98; 106; ¡ 63; ¡ 65; ¡ 71; ¡ 64; ¡ 79; 102; 105; ¡ 57; ¡ 58; ¡ 73; ¡ 67; 108; ¡ 51; ¡ 44; ¡ 60; ¡ 66; ¡ 59; 110; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 37; ¡ 46; ¡ 47; ¡ 38; ¡ 30; ¡ 39; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

76

¡ 90; ¡ 85; ¡ 86; ¡ 87; ¡ 80; ¡ 82; ¡ 81; ¡ 83; ¡ 75; ¡ 76; ¡ 77; ¡ 70; 93; 98; 99; 103; 106; ¡ 78; ¡ 71; ¡ 72; ¡ 64; ¡ 65; ¡ 58; ¡ 79; ¡ 69; ¡ 63; 102; 105; 108; ¡ 57; ¡ 51; ¡ 44; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 60; ¡ 53; 110; 112; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

77

¡ 93; ¡ 89; 94; ¡ 84; 99; 98; ¡ 78; ¡ 79; 102; ¡ 73; 96; ¡ 81; 100; ¡ 77; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 103; ¡ 72; 106; 105; ¡ 65; ¡ 67; 108; ¡ 60; ¡ 71; ¡ 66; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 110; ¡ 54; 112; ¡ 47; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; ¡ 44; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

78

¡ 85; ¡ 80; ¡ 82; ¡ 76; 93; 94; ¡ 84; 99; ¡ 78; 96; ¡ 81; 100; ¡ 77; ¡ 71; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 98; ¡ 79; 102; ¡ 73; 103; ¡ 72; 106; ¡ 65; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 66; 105; ¡ 67; 108; ¡ 60; ¡ 44; ¡ 59; ¡ 52; ¡ 53; ¡ 45; 110; ¡ 54; 112; ¡ 47; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

79

¡ 80; ¡ 75; ¡ 69; ¡ 82; 90; ¡ 88; 95; 94; ¡ 83; ¡ 84; 100; 99; ¡ 77; ¡ 78; 103; ¡ 72; 93; ¡ 76; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 57; 98; ¡ 79; 102; ¡ 73; 105; ¡ 67; 106; ¡ 65; ¡ 58; ¡ 51; ¡ 44; ¡ 66; ¡ 59; ¡ 52; 108; ¡ 60; 110; ¡ 54; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 30; 112; ¡ 47; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

80

84; ¡ 94; 88; ¡ 89; ¡ 90; 92; 91; ¡ 85; ¡ 86; ¡ 87; 96; 95; ¡ 80; ¡ 82; ¡ 81; ¡ 83; 100; 99; ¡ 75; ¡ 76; ¡ 77; ¡ 78; 103; ¡ 70; ¡ 71; ¡ 72; 106; ¡ 64; ¡ 65; ¡ 58; 93; 98; ¡ 79; ¡ 69; ¡ 63; ¡ 57; ¡ 51; ¡ 44; 102; ¡ 73; 105; ¡ 66; ¡ 67; 108; ¡ 59; ¡ 60; 110; ¡ 53; ¡ 54; 112; ¡ 46; ¡ 47; ¡ 39; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

619


620

i 81

p2 (H i )

¡ 84; ¡ 90; ¡ 85; ¡ 86; ¡ 87; ¡ 80; ¡ 82; ¡ 81; ¡ 83; ¡ 75; ¡ 76; ¡ 77; ¡ 70; 93; 98; 99; 103; 106; ¡ 69; ¡ 63; ¡ 78; ¡ 71; ¡ 72; ¡ 64; ¡ 65; ¡ 58; ¡ 79; 102; 105; 108; ¡ 57; ¡ 51; ¡ 44; ¡ 73; ¡ 66; ¡ 67; ¡ 59; ¡ 60; ¡ 53; 110; 112; ¡ 52; ¡ 45; ¡ 37; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

82

¡ 89; 93; 94; ¡ 84; 99; ¡ 78; 96; ¡ 81; 100; ¡ 77; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 103; ¡ 72; 106; ¡ 65; ¡ 71; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 98; ¡ 79; 102; ¡ 73; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 105; ¡ 67; 108; ¡ 60; ¡ 44; ¡ 66; 110; ¡ 54; 112; ¡ 47; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

83

¡ 80; ¡ 82; ¡ 88; ¡ 83; ¡ 84; ¡ 78; 93; 96; ¡ 76; ¡ 71; 98; 102; 100; 103; ¡ 81; 106; ¡ 75; ¡ 69; ¡ 77; ¡ 72; ¡ 79; ¡ 73; ¡ 65; ¡ 66; ¡ 70; ¡ 63; ¡ 64; ¡ 57; 105; ¡ 58; ¡ 51; 108; 110; ¡ 67; 112; ¡ 59; ¡ 52; ¡ 60; ¡ 54; ¡ 44; ¡ 47; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

84

¡ 81; ¡ 85; ¡ 80; ¡ 92; ¡ 87; ¡ 88; ¡ 83; 98; 102; 103; 106; ¡ 75; ¡ 69; ¡ 77; ¡ 82; ¡ 76; 105; 108; ¡ 70; ¡ 63; ¡ 84; ¡ 78; ¡ 71; ¡ 72; ¡ 65; ¡ 79; ¡ 73; 110; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 66; ¡ 67; ¡ 60; 112; ¡ 44; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 37; ¡ 54; ¡ 46; ¡ 38; ¡ 47; ¡ 30; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

85

¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 84; 98; 99; 100; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 79; 102; 103; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 105; 106; ¡ 63; ¡ 64; ¡ 65; ¡ 66; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 51; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

86

¡ 81; ¡ 83; ¡ 84; ¡ 85; 98; 99; 100; ¡ 77; ¡ 78; ¡ 79; ¡ 80; ¡ 82; 102; 103; ¡ 72; ¡ 73; ¡ 75; ¡ 76; 105; 106; ¡ 65; ¡ 67; ¡ 69; ¡ 70; ¡ 71; 108; ¡ 60; ¡ 63; ¡ 64; ¡ 66; 110; ¡ 54; ¡ 57; ¡ 58; ¡ 59; 112; ¡ 47; ¡ 51; ¡ 52; ¡ 53; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

87

¡ 81; ¡ 83; ¡ 84; ¡ 79; ¡ 93; ¡ 89; 99; 102; 100; ¡ 77; ¡ 78; ¡ 73; ¡ 85; 103; 105; ¡ 72; ¡ 67; ¡ 80; ¡ 82; 106; 108; ¡ 65; ¡ 60; ¡ 75; ¡ 76; 110; ¡ 54; ¡ 69; ¡ 70; ¡ 71; ¡ 66; 112; ¡ 47; ¡ 63; ¡ 64; ¡ 59; ¡ 57; ¡ 52; ¡ 58; ¡ 53; ¡ 46; ¡ 51; ¡ 45; ¡ 38; ¡ 39; ¡ 44; ¡ 37; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 88

¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 84; 93; 99; 100; ¡ 75; ¡ 76; ¡ 77; ¡ 78; 98; 103; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 79; 102; 106; ¡ 63; ¡ 64; ¡ 65; ¡ 73; 105; ¡ 57; ¡ 58; ¡ 66; ¡ 67; 108; ¡ 51; ¡ 59; ¡ 60; 110; ¡ 44; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

89

¡ 80; ¡ 82; ¡ 83; ¡ 84; 93; 96; 99; ¡ 76; ¡ 78; ¡ 81; 98; 100; ¡ 71; ¡ 75; ¡ 77; ¡ 79; 102; 103; ¡ 69; ¡ 70; ¡ 72; ¡ 73; 106; ¡ 63; ¡ 64; ¡ 65; ¡ 66; 105; ¡ 57; ¡ 58; ¡ 67; 108; ¡ 51; ¡ 59; ¡ 60; 110; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

90

¡ 84; ¡ 90; ¡ 85; ¡ 86; ¡ 87; ¡ 80; ¡ 82; ¡ 83; ¡ 76; 93; 98; 96; 100; ¡ 78; ¡ 71; 102; 103; 106; ¡ 79; ¡ 81; ¡ 75; ¡ 77; ¡ 70; ¡ 69; ¡ 63; ¡ 72; ¡ 64; ¡ 65; ¡ 58; ¡ 73; ¡ 66; 105; 108; ¡ 57; ¡ 51; 110; 112; ¡ 67; ¡ 59; ¡ 60; ¡ 53; ¡ 44; ¡ 52; ¡ 45; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 38; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

91

¡ 86; ¡ 80; ¡ 81; ¡ 75; ¡ 87; ¡ 82; ¡ 88; ¡ 84; 93; 98; 95; 100; 99; 103; ¡ 69; ¡ 79; ¡ 83; ¡ 76; ¡ 77; ¡ 78; ¡ 70; ¡ 71; ¡ 72; ¡ 64; 102; 105; 106; ¡ 63; ¡ 57; ¡ 65; ¡ 58; ¡ 73; ¡ 66; ¡ 67; ¡ 59; 108; 110; ¡ 51; ¡ 44; ¡ 52; ¡ 60; ¡ 53; ¡ 54; ¡ 46; 112; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 47; ¡ 39; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

92

87; ¡ 91; 92; 90; ¡ 86; ¡ 88; 96; 95; 94; ¡ 81; ¡ 83; ¡ 84; 100; 99; 98; ¡ 77; ¡ 78; ¡ 79; 103; 102; ¡ 72; ¡ 73; 105; ¡ 67; ¡ 93; ¡ 89; ¡ 85; ¡ 82; ¡ 80; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 66; ¡ 57; ¡ 59; ¡ 52; 106; ¡ 65; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 58; ¡ 51; ¡ 53; ¡ 44; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

93

¡ 80; ¡ 82; 90; ¡ 88; 95; 94; ¡ 83; ¡ 84; 99; ¡ 78; 93; 96; ¡ 81; ¡ 75; ¡ 69; ¡ 76; ¡ 71; 98; ¡ 79; 102; ¡ 73; 100; ¡ 77; 103; ¡ 72; ¡ 66; ¡ 70; ¡ 63; ¡ 64; ¡ 57; 105; ¡ 67; 106; ¡ 65; ¡ 58; ¡ 51; ¡ 59; ¡ 52; 108; ¡ 60; 110; ¡ 54; ¡ 44; ¡ 53; ¡ 45; ¡ 46; ¡ 38; 112; ¡ 47; ¡ 37; ¡ 30; ¡ 39; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

94

93; 98; 102; 96; 100; 103; 106; ¡ 99; ¡ 94; ¡ 95; ¡ 89; ¡ 90; ¡ 91; ¡ 92; ¡ 85; ¡ 86; ¡ 87; ¡ 88; ¡ 80; ¡ 82; ¡ 83; ¡ 84; ¡ 76; ¡ 78; ¡ 71; ¡ 79; ¡ 73; ¡ 66; ¡ 81; ¡ 75; ¡ 77; ¡ 69; ¡ 70; ¡ 72; ¡ 63; ¡ 64; ¡ 65; ¡ 57; ¡ 58; ¡ 51; 105; 108; 110; 112; ¡ 44; ¡ 67; ¡ 59; ¡ 60; ¡ 52; ¡ 53; ¡ 54; ¡ 45; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

621


622

p2 (H i )

i 95

¡ 84; ¡ 90; ¡ 85; ¡ 86; ¡ 87; ¡ 80; ¡ 82; ¡ 83; ¡ 76; 93; 98; 96; 100; 99; ¡ 78; ¡ 71; ¡ 79; ¡ 81; ¡ 75; ¡ 77; ¡ 70; 102; 103; 106; ¡ 69; ¡ 63; ¡ 72; ¡ 64; ¡ 65; ¡ 58; ¡ 73; ¡ 66; 105; 108; ¡ 57; ¡ 51; ¡ 67; ¡ 59; ¡ 60; ¡ 53; 110; 112; ¡ 44; ¡ 52; ¡ 45; ¡ 54; ¡ 46; ¡ 47; ¡ 39; ¡ 37; ¡ 38; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

96

¡ 81; ¡ 83; ¡ 84; ¡ 89; 93; 99; 100; ¡ 77; ¡ 78; ¡ 85; 103; ¡ 72; ¡ 80; ¡ 82; 106; ¡ 65; ¡ 75; ¡ 76; 98; ¡ 69; ¡ 70; ¡ 71; ¡ 79; 102; ¡ 63; ¡ 64; ¡ 73; 105; ¡ 57; ¡ 58; ¡ 67; 108; ¡ 51; ¡ 60; 110; ¡ 44; ¡ 54; ¡ 66; 112; ¡ 47; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

97

¡ 86; ¡ 81; ¡ 88; ¡ 84; ¡ 89; ¡ 85; 93; 95; 100; 99; 103; ¡ 82; ¡ 83; ¡ 77; ¡ 78; ¡ 72; ¡ 80; ¡ 75; 98; 106; ¡ 65; ¡ 69; ¡ 76; ¡ 70; ¡ 71; ¡ 64; ¡ 79; 102; 105; ¡ 63; ¡ 57; ¡ 73; ¡ 67; ¡ 58; 108; 110; ¡ 51; ¡ 44; ¡ 60; ¡ 54; ¡ 66; ¡ 59; 112; ¡ 47; ¡ 52; ¡ 53; ¡ 46; ¡ 39; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

98

88; ¡ 90; 92; 91; ¡ 85; ¡ 86; ¡ 87; 96; 95; ¡ 80; ¡ 82; ¡ 81; ¡ 83; 100; ¡ 75; ¡ 76; ¡ 77; ¡ 70; 89; 94; 93; ¡ 84; 98; ¡ 79; ¡ 69; ¡ 63; 99; ¡ 78; 103; 102; ¡ 71; ¡ 72; ¡ 73; 106; 105; ¡ 64; ¡ 66; ¡ 65; ¡ 67; 108; ¡ 58; ¡ 59; ¡ 60; ¡ 53; ¡ 57; ¡ 51; ¡ 52; ¡ 44; ¡ 45; ¡ 37; 110; ¡ 54; 112; ¡ 46; ¡ 47; ¡ 39; ¡ 38; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

99

¡ 86; ¡ 81; ¡ 87; ¡ 88; ¡ 84; ¡ 89; ¡ 85; 95; 100; 99; 103; 98; ¡ 79; ¡ 80; ¡ 75; ¡ 82; ¡ 83; ¡ 77; ¡ 78; ¡ 72; 102; 105; 106; ¡ 65; ¡ 69; ¡ 73; ¡ 67; ¡ 76; ¡ 70; ¡ 71; ¡ 64; 108; 110; ¡ 58; ¡ 60; ¡ 54; ¡ 63; ¡ 57; ¡ 66; ¡ 59; 112; ¡ 47; ¡ 51; ¡ 44; ¡ 52; ¡ 53; ¡ 46; ¡ 39; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

100

¡ 85; ¡ 86; ¡ 87; ¡ 88; 96; 98; 99; ¡ 80; ¡ 82; ¡ 83; 100; 102; ¡ 76; ¡ 84; ¡ 78; ¡ 79; ¡ 81; 103; ¡ 71; ¡ 73; ¡ 75; ¡ 77; 105; 106; ¡ 66; ¡ 69; ¡ 70; 108; ¡ 63; ¡ 72; ¡ 64; ¡ 65; ¡ 67; 110; ¡ 57; ¡ 58; ¡ 59; ¡ 60; 112; ¡ 51; ¡ 52; ¡ 53; ¡ 45; ¡ 54; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 44; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


p2 (H i )

i 101

¡ 86; ¡ 81; ¡ 87; ¡ 88; ¡ 84; ¡ 89; ¡ 85; 93; 95; 100; 99; 103; ¡ 80; ¡ 75; ¡ 82; ¡ 83; ¡ 77; ¡ 78; ¡ 72; 98; 106; ¡ 65; ¡ 69; ¡ 76; ¡ 70; ¡ 71; ¡ 64; ¡ 79; 102; 105; ¡ 58; ¡ 63; ¡ 57; ¡ 73; ¡ 67; 108; 110; ¡ 51; ¡ 44; ¡ 60; ¡ 54; ¡ 66; ¡ 59; 112; ¡ 47; ¡ 52; ¡ 53; ¡ 46; ¡ 39; ¡ 45; ¡ 37; ¡ 38; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

102

87; ¡ 91; 92; 90; ¡ 86; ¡ 88; 96; 95; 94; ¡ 81; ¡ 83; ¡ 84; 100; 99; ¡ 77; ¡ 78; 103; ¡ 72; ¡ 89; ¡ 85; ¡ 82; 93; ¡ 80; ¡ 75; ¡ 76; ¡ 69; ¡ 70; ¡ 71; ¡ 63; ¡ 64; ¡ 57; 98; ¡ 79; 102; ¡ 73; 105; ¡ 67; 106; ¡ 65; ¡ 58; ¡ 51; ¡ 44; ¡ 66; ¡ 59; ¡ 52; 108; ¡ 60; 110; ¡ 54; 112; ¡ 47; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 24; ¡ 23; ¡ 16; ¡ 9; ¡ 1

103

¡ 87; ¡ 88; ¡ 84; ¡ 89; ¡ 85; 93; 95; 99; 96; ¡ 80; ¡ 81; ¡ 83; ¡ 78; ¡ 82; 98; 100; 103; ¡ 75; ¡ 76; ¡ 71; ¡ 77; ¡ 72; ¡ 79; 102; 106; ¡ 65; ¡ 69; ¡ 73; ¡ 70; ¡ 64; 105; ¡ 58; ¡ 63; ¡ 57; ¡ 66; ¡ 67; 108; 110; ¡ 51; ¡ 60; ¡ 54; ¡ 59; 112; ¡ 44; ¡ 47; ¡ 52; ¡ 53; ¡ 46; ¡ 45; ¡ 38; ¡ 39; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

104

¡ 86; ¡ 87; ¡ 88; ¡ 84; ¡ 89; ¡ 85; 95; 99; 96; 98; ¡ 79; ¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 78; 100; 103; 102; ¡ 73; ¡ 75; ¡ 76; ¡ 71; ¡ 77; ¡ 72; 105; 106; ¡ 65; ¡ 66; ¡ 67; ¡ 69; ¡ 70; ¡ 64; 108; 110; ¡ 58; ¡ 59; ¡ 60; ¡ 54; ¡ 63; ¡ 57; 112; ¡ 47; ¡ 51; ¡ 52; ¡ 53; ¡ 46; ¡ 39; ¡ 44; ¡ 45; ¡ 38; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

105

89; 94; 93; 99; 98; 102; ¡ 95; ¡ 90; ¡ 91; ¡ 92; ¡ 85; ¡ 86; ¡ 87; ¡ 88; ¡ 80; ¡ 82; ¡ 83; ¡ 76; 96; 100; ¡ 81; ¡ 75; ¡ 77; ¡ 69; ¡ 70; ¡ 63; ¡ 84; ¡ 78; ¡ 79; ¡ 71; ¡ 73; ¡ 66; 103; 106; 105; 108; ¡ 72; ¡ 64; ¡ 65; ¡ 67; ¡ 57; ¡ 58; ¡ 59; ¡ 60; ¡ 51; ¡ 52; ¡ 53; ¡ 45; 110; 112; ¡ 44; ¡ 37; ¡ 54; ¡ 46; ¡ 47; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

106

¡ 86; ¡ 87; ¡ 88; ¡ 84; ¡ 89; ¡ 85; 93; 95; 99; 96; ¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 78; 98; 100; 103; ¡ 75; ¡ 76; ¡ 71; ¡ 77; ¡ 72; ¡ 79; 102; 106; ¡ 65; ¡ 69; ¡ 70; ¡ 64; ¡ 73; 105; ¡ 58; ¡ 63; ¡ 57; ¡ 66; ¡ 67; 108; 110; ¡ 51; ¡ 59; ¡ 60; ¡ 54; 112; ¡ 44; ¡ 47; ¡ 52; ¡ 53; ¡ 46; ¡ 39; ¡ 45; ¡ 38; ¡ 31; ¡ 37; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

623


624

p2 (H i )

i 107

86; ¡ 92; 91; 90; ¡ 87; ¡ 88; 95; 94; ¡ 83; ¡ 84; 99; ¡ 78; ¡ 89; ¡ 85; ¡ 80; 93; 96; ¡ 81; ¡ 75; ¡ 69; ¡ 82; ¡ 76; ¡ 71; 98; ¡ 79; 102; ¡ 73; 100; ¡ 77; 103; ¡ 72; 106; ¡ 65; ¡ 66; ¡ 70; ¡ 63; ¡ 64; ¡ 57; ¡ 58; ¡ 51; 105; ¡ 67; 108; ¡ 60; 110; ¡ 54; ¡ 44; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 38; 112; ¡ 47; ¡ 37; ¡ 30; ¡ 23; ¡ 39; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

108

¡ 86; ¡ 87; ¡ 88; ¡ 89; 93; 94; 96; ¡ 83; 99; 100; ¡ 81; ¡ 84; ¡ 85; ¡ 77; ¡ 78; ¡ 80; ¡ 82; 98; 103; ¡ 76; 102; 106; ¡ 72; ¡ 75; ¡ 79; ¡ 65; ¡ 69; ¡ 70; ¡ 71; ¡ 73; 105; ¡ 63; 108; ¡ 64; ¡ 67; ¡ 57; ¡ 58; ¡ 60; ¡ 66; 110; ¡ 51; 112; ¡ 54; ¡ 59; ¡ 44; ¡ 47; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

109

¡ 85; ¡ 86; ¡ 87; ¡ 88; 93; 94; 95; 96; ¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 84; 98; 99; 100; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 79; 102; 103; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 105; 106; ¡ 63; ¡ 64; ¡ 65; ¡ 66; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 59; ¡ 60; 110; ¡ 51; ¡ 52; ¡ 53; ¡ 54; 112; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

110

¡ 93; ¡ 89; 94; 99; 98; 102; ¡ 95; ¡ 90; ¡ 91; ¡ 92; ¡ 86; ¡ 87; ¡ 88; ¡ 83; 96; 100; ¡ 81; ¡ 77; ¡ 84; ¡ 78; ¡ 79; ¡ 73; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 103; 106; 105; 108; ¡ 71; ¡ 66; ¡ 72; ¡ 65; ¡ 67; ¡ 60; ¡ 75; ¡ 69; ¡ 70; ¡ 63; 110; 112; ¡ 54; ¡ 47; ¡ 64; ¡ 57; ¡ 58; ¡ 59; ¡ 51; ¡ 52; ¡ 53; ¡ 45; ¡ 44; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

111

¡ 86; ¡ 87; ¡ 88; ¡ 89; 93; 94; 95; 96; ¡ 81; ¡ 83; ¡ 84; ¡ 85; 99; 100; ¡ 77; ¡ 78; ¡ 80; ¡ 82; 98; 103; ¡ 72; ¡ 75; ¡ 76; ¡ 79; 102; 106; ¡ 65; ¡ 69; ¡ 70; ¡ 71; ¡ 73; 105; ¡ 63; ¡ 64; ¡ 67; 108; ¡ 57; ¡ 58; ¡ 60; ¡ 66; 110; ¡ 51; ¡ 54; ¡ 59; 112; ¡ 44; ¡ 47; ¡ 52; ¡ 53; ¡ 45; ¡ 46; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

112

¡ 89; 93; 94; 99; ¡ 95; ¡ 90; ¡ 91; ¡ 92; ¡ 86; ¡ 87; ¡ 88; ¡ 83; 96; 100; ¡ 81; ¡ 77; ¡ 84; ¡ 78; ¡ 85; ¡ 80; ¡ 82; ¡ 76; 98; 102; 103; 106; ¡ 71; ¡ 72; ¡ 65; ¡ 75; ¡ 69; ¡ 70; ¡ 63; ¡ 79; ¡ 73; 105; 108; ¡ 64; ¡ 57; ¡ 58; ¡ 51; ¡ 66; ¡ 67; ¡ 60; 110; 112; ¡ 44; ¡ 54; ¡ 47; ¡ 59; ¡ 52; ¡ 53; ¡ 45; ¡ 37; ¡ 46; ¡ 38; ¡ 39; ¡ 31; ¡ 30; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )


i

p2 (H i )

113

¡ 89; ¡ 85; 90; 95; 94; 99; ¡ 91; ¡ 86; ¡ 92; ¡ 87; 93; 96; ¡ 80; ¡ 81; ¡ 82; ¡ 88; ¡ 83; ¡ 84; ¡ 78; 98; 102; 100; 103; ¡ 75; ¡ 69; ¡ 76; ¡ 71; ¡ 77; ¡ 72; ¡ 79; ¡ 73; 105; 106; ¡ 65; ¡ 66; ¡ 67; ¡ 70; ¡ 63; ¡ 64; ¡ 57; 108; 110; ¡ 58; ¡ 51; ¡ 59; ¡ 52; ¡ 60; ¡ 54; 112; ¡ 44; ¡ 47; ¡ 53; ¡ 45; ¡ 46; ¡ 38; ¡ 37; ¡ 30; ¡ 39; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1

114

¡ 89; ¡ 90; ¡ 86; 91; 95; ¡ 92; ¡ 88; 93; 94; 96; ¡ 81; ¡ 84; ¡ 85; ¡ 80; ¡ 87; ¡ 83; 98; 99; 100; ¡ 75; ¡ 77; ¡ 78; ¡ 79; ¡ 82; ¡ 76; 102; 103; 106; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 65; ¡ 73; 105; 108; ¡ 63; ¡ 64; ¡ 58; ¡ 66; ¡ 67; ¡ 60; 110; ¡ 54; ¡ 57; ¡ 51; ¡ 59; ¡ 53; 112; ¡ 44; ¡ 46; ¡ 47; ¡ 52; ¡ 45; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 23; ¡ 31; ¡ 24; ¡ 16; ¡ 9; ¡ 1

115

¡ 89; ¡ 90; ¡ 91; ¡ 92; 93; 94; 95; 96; ¡ 85; ¡ 86; ¡ 87; ¡ 88; 98; 99; 100; ¡ 80; ¡ 81; ¡ 82; ¡ 83; ¡ 84; 102; 103; ¡ 75; ¡ 76; ¡ 77; ¡ 78; ¡ 79; 105; 106; ¡ 69; ¡ 70; ¡ 71; ¡ 72; ¡ 73; 108; ¡ 63; ¡ 64; ¡ 65; ¡ 66; ¡ 67; 110; ¡ 57; ¡ 58; ¡ 59; ¡ 60; 112; ¡ 51; ¡ 52; ¡ 53; ¡ 54; ¡ 44; ¡ 45; ¡ 46; ¡ 47; ¡ 37; ¡ 38; ¡ 39; ¡ 30; ¡ 31; ¡ 23; ¡ 24; ¡ 16; ¡ 9; ¡ 1 Table 7 Root spaces in p2 (H i )

625


626

k

d0 (j; k); j = 1; 2; : : :

d1 (j; k); j = 1; 2; : : :

1

91 120

99 127 128

2

74 114 120

82 120 128

3

64 106 119 120

72 111 127 128

4

63 94 119 119 120

71 96 127 128

5

63 92 119 120

71 98 127 127 128

6

56 100 116 120

64 104 124 128

7

52 94 113 119 120

60 95 121 128

8

52 92 113 120

60 97 121 127 128

9

47 92 109 117 120

55 88 117 128

10

47 88 109 119 120

55 92 117 126 128

11

46 70 91 114 119 120

54 75 99 120 127 127 128

12

43 92 106 113 120

51 79 114 128

13

43 84 106 117 120

51 87 114 124 128

14

42 92 106 106 120

50 64 114 128

15

42 78 106 113 120

50 78 114 121 128

16

42 76 106 114 120

50 80 114 120 128

17

39 76 99 114 119 120

47 79 107 120 127 128

18

38 70 91 110 118 120

46 74 99 116 126 127 128

19

37 67 91 106 118 119 120

45 69 99 111 126 127 128

20

37 66 91 105 118 120

45 70 99 112 126 126 128

21

36 52 64 79 91 106 118 119 119 120

44 56 72 84 99 111 126 126 127 127 128

22

36 74 96 112 118 120

44 76 104 118 126 128

23

34 68 90 108 116 120

42 71 98 114 124 127 128

24

32 67 88 106 115 119 120

40 68 96 111 123 127 128

25

32 66 88 105 115 120

40 69 96 112 123 126 128

26

31 66 86 105 113 118 120

39 63 94 110 121 127 128

27

31 64 86 103 113 120

39 65 94 112 121 125 128

28

31 63 86 105 113 119 120

39 66 94 110 121 126 128

29

30 56 74 93 107 114 119 119 120

38 56 82 96 115 120 127 128

Table 8 The integers di (j; k) for the adjoint module of E8 equipped with a Z2 -gradation


k

d0 (j; k); j = 1; 2; : : :

d1 (j; k); j = 1; 2; : : :

30

30 54 74 92 107 114 119 120

38 58 82 97 115 120 127 127 128

31

29 64 84 99 112 120

37 62 92 112 120 123 128

32

29 62 84 103 112 118 120

37 64 92 108 120 125 128

33

28 52 64 79 91 106 115 119 119 120

36 55 72 84 99 111 123 126 127 127 128

34

28 64 84 92 112 120

36 56 92 112 120 120 128

35

28 60 84 100 112 116 120

36 60 92 104 120 124 128

36

28 59 84 99 112 117 120

36 61 92 105 120 123 128

37

26 56 73 93 104 114 118 119 120

34 55 81 96 112 120 126 128

38

26 54 73 92 104 114 118 120

34 57 81 97 112 120 126 127 128

39

26 54 74 96 106 114 118 120

34 56 82 100 114 120 126 128

40

25 50 64 79 91 106 113 118 119 120

33 51 72 84 99 110 121 126 127 127 128

41

25 49 64 79 91 105 113 119 119 120

33 52 72 84 99 111 121 125 127 127 128

42

24 56 72 93 103 112 117 119 120

32 52 80 94 111 120 125 128

43

24 54 72 91 103 114 117 119 120

32 54 80 96 111 118 125 128

44

24 53 72 91 103 113 117 120

32 55 80 96 111 119 125 127 128

45

23 54 71 84 101 114 117 117 120

31 48 79 96 109 112 125 128

46

23 50 71 88 101 110 117 119 120

31 52 79 92 109 116 125 126 128

47

22 42 57 76 90 104 108 114 118 120

30 45 65 80 98 109 116 120 126 127 128

48

22 50 63 79 91 106 112 116 119 120

30 48 71 84 99 108 120 126 127 127 128

49

22 48 63 79 91 104 112 118 119 120

30 50 71 84 99 110 120 124 127 127 128


627


628

k

d0 (j; k); j = 1; 2; : : :

d1 (j; k); j = 1; 2; : : :

50

22 50 68 88 100 110 115 119 120

30 51 76 92 108 116 123 126 128

51

21 50 63 78 91 106 112 113 119 120

29 42 71 84 99 105 120 126 127 127 128

52

21 46 63 78 91 102 112 117 119 120

29 46 71 84 99 109 120 122 127 127 128

53

21 45 63 79 91 103 112 116 119 120

29 47 71 83 99 108 120 123 127 127 128

54

21 42 57 76 90 101 108 114 118 119 120

29 44 65 79 98 105 116 120 126 127 128

55

21 42 57 75 90 100 108 114 118 120

29 44 65 80 98 106 116 120 126 126 128

56

20 36 47 62 73 84 91 101 108 114 118 119 119 120

28 39 55 66 81 88 99 106 116 120 126 126 127 127 128

57

20 46 63 83 94 106 112 117 119 120

28 47 71 86 102 111 120 124 127 128

58

19 42 57 76 89 102 108 114 117 120

27 44 65 80 97 107 116 120 125 127 128

59

19 44 60 78 90 102 109 116 118 120

27 45 68 82 98 108 117 122 126 127 128

60

18 41 56 77 87 100 107 113 116 119 120

26 41 64 78 95 104 115 120 124 127 128

61

18 40 56 75 87 100 107 113 116 120

26 42 64 80 95 104 115 120 124 126 128

62

18 42 57 76 88 101 107 114 117 119 120

26 43 65 79 96 105 115 120 125 127 128

63

18 42 57 75 88 100 107 114 117 120

26 43 65 80 96 106 115 120 125 126 128

64

17 40 56 73 86 100 106 112 116 120

25 41 64 80 94 103 114 120 124 125 128

65

17 40 56 75 86 99 106 113 116 119 120

25 41 64 78 94 104 114 119 124 126 128

66

16 36 47 62 72 84 91 101 107 114 117 119 119 120

24 38 55 66 80 88 99 106 115 120 125 126 127 127 128

67

16 40 56 74 86 98 106 112 116 118 120

24 40 64 76 94 102 114 118 124 126 128



k

d0 (j; k); j = 1; 2; : : :

d1 (j; k); j = 1; 2; : : :

68

16 40 56 70 86 100 106 110 116 120

24 40 64 80 94 100 114 120 124 124 128

69

16 40 56 73 86 97 106 112 116 119 120

24 40 64 77 94 103 114 118 124 125 128

70

15 38 47 65 74 92 98 103 109 114 117 117 120

23 32 55 64 82 88 106 112 117 120 125 128

71

15 34 47 63 74 88 98 105 109 114 117 119 120

23 36 55 66 82 92 106 110 117 120 125 126 128

72

15 35 47 62 71 84 91 101 107 113 116 119 119 120

23 35 55 66 79 88 99 105 115 120 124 126 127 127 128

73

15 34 47 62 71 84 91 100 107 114 116 119 119 120

23 36 55 66 79 88 99 106 115 119 124 126 127 127 128

74

14 34 46 63 74 88 96 105 109 114 116 119 120

22 35 54 66 82 92 104 110 117 120 124 126 128

75

14 34 46 62 71 83 91 100 106 113 116 119 119 120

22 35 54 65 79 88 99 106 114 119 124 125 127 127 128

76

14 34 46 61 71 84 91 100 106 114 116 118 119 120

22 35 54 66 79 87 99 106 114 118 124 126 127 127 128

77

13 30 39 55 64 76 82 93 99 106 109 114 117 118 119 119 120

21 30 47 56 72 78 90 96 107 110 117 120 125 126 127 128

78

13 29 39 54 64 75 82 92 99 105 109 114 117 119 119 120

21 31 47 57 72 79 90 97 107 111 117 120 125 125 127 127 128

79

13 34 46 61 71 82 91 99 106 112 116 119 119 120

21 33 54 64 79 88 99 106 114 118 124 124 127 127 128

80

13 34 46 59 71 84 91 99 106 114 116 117 119 120

21 33 54 66 79 86 99 106 114 116 124 126 127 127 128

81

13 33 46 61 71 83 91 100 106 112 116 118 119 120

21 34 54 64 79 87 99 105 114 118 124 125 127 127 128

82

12 24 30 41 47 58 64 75 81 88 91 97 100 106 109 114 117 118 118 119 119 120

20 26 38 44 55 61 72 78 89 92 99 102 108 111 117 120 125 125 126 126 127 127 128

83

12 26 34 47 55 68 74 84 90 100 104 108 110 114 116 119 119 120

20 28 42 50 63 71 82 88 98 104 112 114 118 120 124 126 127 127 128


629


630

k

d0 (j; k); j = 1; 2; : : :

d1 (j; k); j = 1; 2; : : :

84

12 32 42 59 68 82 89 99 104 110 113 116 118 120

20 31 50 60 76 84 97 104 112 116 121 124 126 127 128

85

12 31 42 58 68 81 89 99 104 110 113 117 118 120

20 32 50 61 76 85 97 104 112 116 121 123 126 127 128

86

11 28 38 54 63 76 83 94 99 106 109 114 116 118 119 120

19 29 46 56 71 79 91 98 107 111 117 120 124 125 127 128

87

11 30 39 55 63 76 82 93 98 106 109 114 116 118 119 119 120

19 29 47 56 71 78 90 96 106 110 117 120 124 126 127 128

88

11 29 39 54 63 75 82 92 98 105 109 114 116 119 119 120

19 30 47 57 71 79 90 97 106 111 117 120 124 125 127 127 128

89

10 26 34 47 55 67 74 84 90 99 103 108 110 114 116 119 119 120

18 27 42 50 63 71 82 88 98 104 111 114 118 120 124 125 127 127 128

90

10 26 34 47 55 68 74 84 90 100 103 108 110 114 116 118 119 120

18 27 42 50 63 70 82 88 98 103 111 114 118 120 124 126 127 127 128

91

10 28 38 53 62 74 82 91 97 104 109 113 115 118 119 120

18 28 46 55 70 78 90 96 105 110 117 119 123 124 127 127 128

92

10 28 38 52 62 76 82 91 97 106 109 112 115 118 119 119 120

18 28 46 56 70 76 90 96 105 108 117 120 123 124 127 128

93

9 26 34 47 55 66 74 84 90 98 103 107 110 114 116 119 119 120

17 25 42 50 63 71 82 88 98 103 111 114 118 120 124 124 127 127 128

94

9 26 34 47 55 68 74 84 90 100 103 107 110 114 116 117 119 120

17 25 42 50 63 69 82 88 98 101 111 114 118 120 124 126 127 127 128



631

i; j

k

4,5; 4,6; 4,8; 5,6; 5,7; 5,9; 5,12; 5,14

3

7,8; 8,9; 8,12; 8,14

6

9,10; 9,11; 10,12; 10,14; 11,12; 11,14

7

12,13; 12,16; 12,17; 12,18; 12,19; 12,20; 12,21; 13,14; 14,16; 14,17;

9

14,18; 14,19; 14,20; 14,21 11,13; 11,15; 11,16; 11,17; 11,22

10

14,15; 14,22; 14,23; 14,24; 14,25; 14,27; 14,28; 14,30; 14,31; 14,32;

12

14,33; 14,34; 14,36; 14,38; 14,41 15,16; 15,17; 15,18; 15,19; 15,20; 15,21

13

18,22; 19,22; 20,22; 21,22

17

19,20; 19,23; 19,25; 19,27; 19,31; 19,34; 20,23; 20,24; 20,26; 21,23

18

21,24; 21,26

19

21,28; 21,29; 21,30; 21,32; 21,35; 21,36; 21,37; 21,38; 21,39; 21,42;

19 20

21,43; 21,44; 21,45; 21,46; 21,47; 21,50; 21,54; 21,57; 21,58; 21,62 21,25; 21,27; 21,31; 21,34

20

24,25; 24,27; 24,31; 24,34; 25,26; 26,27; 26,31; 26,34

23

26,28; 26,30; 26,32; 26,33; 26,36; 26,38; 26,41

24

27,28; 27,29; 28,31; 28,34; 29,31; 29,34

25

29,35; 29,39; 29,44

26 28

30,31; 30,34; 31,32; 31,33; 31,35; 31,37; 31,40; 31,42; 31,48; 31,51;

27

32,34; 33,34; 34,35; 34,37; 34,40; 34,42; 34,48; 34,51 30,32; 30,33; 30,35; 30,36; 30,37; 30,39; 30,40; 30,42; 30,43; 30,45;

27 28

30,48; 30,51 29,30; 29,32; 29,33; 29,36; 29,38; 29,41

28

34,36; 34,38; 34,39; 34,41; 34,43; 34,45

31

33,35; 33,36; 33,37; 33,38; 33,39; 33,42; 33,43; 33,44; 33,45; 33,46;

32

33,47; 33,50; 33,54; 33,57; 33,58; 33,62; 35,36; 35,38; 35,41; 36,37; 36,40; 36,42; 36,48; 36,51; 37,38; 37,41; 38,40; 38,42; 38,48; 38,51; 41,42 40,41; 41,48; 41,51

33 j

Table 9 Irreducible components O1k of O1i \ O1j


632

i; j

k

37,39; 37,44; 39,40; 39,42; 39,48; 39,51; 40,44; 42,44; 44,48; 44,51

35

38,39; 38,43; 38,45; 39,41; 41,43; 41,45

36

40,42; 40,43; 40,45; 40,46; 40,47; 40,50; 40,54; 40,57; 40,58; 40,62;

37

42,43; 42,45; 42,46; 42,47; 42,49; 42,54; 42,55; 42,56; 43,48; 43,51; 45,48; 45,51; 46,48; 46,51; 47,48; 47,51; 48,54; 51,54 41,44; 41,46; 41,47; 41,50; 41,54; 41,57; 41,58; 41,62

38

43,44; 44,45

39

48,49; 48,55; 48,56; 49,51; 51,55; 51,56

40

48,52; 48,59; 48,63; 51,52; 51,59; 51,63

40 42

48,50; 48,57; 48,58; 48,62; 50,51; 51,57; 51,58; 51,62

42

45,46; 45,47; 45,49; 45,50; 45,52; 45,53; 45,55

43

54,57; 54,59; 54,60; 56,57

45 46

54,58; 54,61; 54,63; 54,64; 54,68; 56,58

45 47

56,61

45 47 49

56,59; 56,60

45 49

56,63; 56,64; 56,68

45 55

47,49; 47,50; 47,52; 47,53; 47,57; 47,59; 47,60; 49,50; 49,54; 49,57;

46

49,58; 49,62; 50,54; 50,55; 50,56; 52,54; 53,54; 55,57 54,55; 55,58; 55,62

47

55,61; 55,65; 55,67; 55,70

47 49

51,53; 51,61; 51,64; 51,65; 51,66; 51,68; 51,69; 51,73; 51,75; 51,79

48

52,55; 52,56; 53,55; 53,56; 55,59; 55,60

49

52,53; 53,59; 53,63

49 50

56,65; 56,67; 56,70

49 54

52,57; 52,58; 52,62; 53,57; 53,58; 53,62

50

59,63; 60,63

52 57

61,63; 63,65; 63,67; 63,70

52 58

60,61; 60,64; 60,65; 60,66; 60,68; 60,69; 60,73; 60,75; 60,79

53 59

56,62

54

56,69; 56,71; 56,74

54 55 j



633

i; j

k

58,59; 58,60; 59,62; 60,62

57

61,62; 62,63; 62,64; 62,68

58

64,65; 64,67; 64,70; 65,68; 67,68; 68,70

61

66,68; 68,69; 68,72

64

66,69; 66,71; 66,74

64 65

66,67; 66,70; 67,69; 67,73; 67,75; 67,79; 69,70; 70,73; 70,75; 70,79

65

72,76; 72,77; 72,78; 72,80; 72,82

66 67 69

72,73; 72,75; 72,79

66 69

70,71; 70,72; 70,74; 70,76; 70,78; 70,80; 70,81; 70,83; 70,85; 70,88;

67

70,89 71,72; 72,74

67 69

71,73; 71,75; 71,79; 73,74; 74,75; 74,79

68 69

82,84; 82,86

70 75 76

77,84; 77,86; 77,91; 77,93

70 76

82,91; 82,93

70 78

74,76; 74,77; 74,78; 74,82

71

75,76; 75,77; 75,80; 76,79; 77,79; 79,80

73

80,81; 80,84

74 76

78,79; 79,81; 79,82; 79,83; 79,84; 79,87; 79,90; 79,94

75

78,81; 78,83; 78,84; 78,85; 78,86; 78,87; 78,90; 78,92; 78,94; 81,82;

75 76

82,83; 82,85 82,87; 82,90; 82,92; 82,94

75 77

77,78; 77,80; 77,81; 77,83; 77,85; 77,88; 77,89; 78,80; 80,82

76

82,95; 82,98; 82,99; 82,100; 82,104

77 78

82,88; 82,89

78

83,85; 83,86; 83,87; 83,88; 83,91; 83,92; 83,96; 85,87; 85,90; 85,94;

80 81

87,88; 87,89; 88,90; 88,94 86,87; 86,90; 86,94; 87,91; 87,93; 90,91; 91,94

80 84

83,84; 84,85; 84,88; 84,89

81

89,90; 89,94

83 j



634

i; j

k

90,93; 93,94

83 84

86,88; 86,89; 88,92; 89,92

85

91,92; 92,93

86

92,95; 92,98

86 87

90,92; 90,96; 92,94; 94,96

87

95,96; 96,98

87 91

89,91; 89,96

88

93,95; 93,98; 93,99; 93,100; 93,104

89 91

94,95; 94,97; 94,98; 94,102

90

93,96

91

96,99; 96,100; 96,104

91 92

99,102; 97,99

92 95

97,100; 97,104

92 93 95

100,102; 102,104

92 98

97,98; 98,101

93 95

101,102

97

98,99

95

100,103; 103,104

98 99

100,101; 101,104

93 99

104,105; 104,107

100

103,105

101 102

106,107; 107,110

103 105

108,110; 109,110; 110,111

106

108,109; 109,112

106 107

111,112

108 j



®1

®3

®4

¯7 ®2

®5

®6

®7

®8

®0 = ®¡120

¯6 ¯5

¯4

¯3

¯2

¯1

¯8

Fig. 1 The extended base of E8 and a base of R0

635


636

176 172 D4 21 166 D4 (a1 ) 20 19 A + 164 3 A1 18 168

D4 (a1 ) + A1 25 24 A3 + 2A1 23 2A2 + 2A1 22

2A2 + A1 17 14 2A2 15 16 12 A2 + 3A1 13

162 156 154 148

A3 11

9 A + 2A1 10 2 7 A 2 + A1 8 4A1 6

136 128 A2

4

115 E8

238

114 E8 (a1 )

236

113 E8 (a2 )

234

111 E8 (a3 ) 112 E7 E8 (a4 ) 109 110 108

232 230 228 226

146

114

240

5

224 222 220 218

112

3 3A1

216

92

2 2A1

214

58

1 A1

212

0

0

210

106 E8 (b4 ) 107 E7 (a1 ) 104 E8 (a5 ) 103 105 E8 (b5 ) 102 100 D 7 101 98 E8 (a6 ) E7 (a2 ) 99 97 93 E6 + A 1 D7 (a1 ) 94 95 96 92 E8 (b6 ) 89 E7 (a3 ) 90 91 A E6 (a1 ) + A1 88 86 7 87 E6 85 D6 D7 (a2 ) 84 83 82 E6 (a1 ) 79 81 77 78 80 D5 + A2 E7 (a4 ) 74 A6 + A1 75 76 D6 (a1 ) 72 73

Fig. 2 The closure diagram of E8 (bottom and top)

A6 70 71


210 208

72 73 D6 (a1 ) D5 + A1 66

206 204 202 200 198 196 194 192 190 188 184 182 180 178 176

E6 (a3 ) + D5 (a1 ) + D5 56 E6 (a3 ) 54,55 D5 (a1 ) + A1 48,49

70 71 A6 67 E8 (a7 ) 68 69 E7 (a5 ) 64 65 63 60 D6 (a2 ) 62 61 A1 A + A1 A2 59 585 A 4 + A3 57 53 52 D4 + A2 51 A 4 + A2 + A 1 47 A5 50 45 A4 + A2 46 42 43 44A4 + 2A1

D5 (a1 ) 41 40 2A3 37 A4 + A1 39 38 D4 + A1 34 D4 (a1 ) + A2 33 35 36 31 A3 + A2 + A1 32 29 A4 30 A + A2 26 283 27 24 25 D4 (a1 ) + A1 Fig. 3 The closure diagram of E8 (middle)

637

638


®¡67 ®¡72

®99

®¡53 ®¡63

Fig. 4 Type D4 (a1 ) + A1


639

k nodes . . .. .. .. .. .. .. ..

...

:::

::: Dn (ak ): n nodes, 1 µ k µ

n 2

¡

1 (n ¶ 4)

:::

Dn (a1 )0 : n nodes (n ¶ 5)

::: ::: Dn (ak )0 : n nodes, k =

n 2

¡

1 (n even ¶ 6)

Fig. 5 Nonstandard D types

... . .

.. .. .. . . . . .. ..

640


®98

®93

®¡77 ®¡71

®¡73

®¡75

®89

®96

®96

®¡66 ®¡77

E6 (a3 )0 : orbit 55

E6 (a3 ): orbit 55 ®¡93 ®¡89

®100 ®¡79

®99

®¡63

®¡78 ®¡89

®¡81

®96

®¡73

®94

E6 (a1 )0 : orbit 77

E6 (a1 ): orbit 77

®¡80

®¡79 ®99

®¡81 ®¡82 ®108

®105

0

®¡83

E8 (a7 ) : orbit 67 ®¡79 ®¡81

®105

®¡77

®¡83 ®¡85 ®106 ®99 00

®¡70

E8 (a7 ) : orbit 67 Fig. 6 Some nonstandard types on 6 or 8 nodes


641

®102 ®99

®¡84

®¡78

®¡67

®103

®¡69 ®90

®¡87 ®¡80

E7 (a5 ): orbit 64 ®¡81

®99

®¡70

®¡84

®96

®¡70

®98

®¡87

®¡87

®¡90 ®¡81

®93

®¡80

®¡80 ®93

®¡70

®¡90 ®¡85

®¡76

®93

®¡84 ®¡88

®103

®95

®93

®¡85

E7 (a2 ): orbit 97

®99

E7 (a4 )00 : orbit 76 ®100 ®¡84

®96

®¡90

®98

®¡76

®93

0

E7 (a3 ): orbit 90

®¡81

®93

E7 (a4 ): orbit 76

E7 (a4 )0 : orbit 76 ®¡87

®99

E7 (a3 ) : orbit 90

®93

®¡88 ®103

®¡86

®¡85

®¡81

®95

E7 (a2 )0 : orbit 97

Fig. 7 Some nonstandard types on 7 nodes

®¡81


642

®99

®¡87 ®96

®95

®¡88 ®¡85

®95

®93

®93

®¡84

®¡85 ®¡88

®96

®¡87

E7 (a1 )0 : orbit 103 ®98 ®¡83

E7 (a1 ): orbit 103

®90 ®95

®¡85

®¡76 ®102

®¡87 ®96

®¡84 ®¡89

®93

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E8 (a7 ): orbit 68

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E8 (a6 ): orbit 98

®93

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E8 (a5 ): orbit 105

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®96

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E8 (b4 ): orbit 107

®100

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E8 (b4 )0 : orbit 107

Fig. 8 Some nonstandard types on 7 or 8 nodes


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E8 (a4 ): orbit 110 ®¡86

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E8 (a2 )00 : orbit 113

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E8 (a1 )0 : orbit 114

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E8 (a1 ): orbit 114

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E8 (a2 )0 : orbit 113

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643

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Fig. 9 Some nonstandard types on 8 nodes

®96

CEJM 4 (2003) 644{649

On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces E. Ballico ¤1 , R. Notariy2, M.L. Sprea¯coz2 1

2

Dept. of Mathematics University of Trento 38050 Povo (TN), Italy Dipartimento di Matematica Politecnico di Torino 10129 Torino, Italy

Received 26 June 2003; accepted 5 September 2003 Abstract: Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of Pn with one point as support and with high regularity index. ® c Central European Science Journals. All rights reserved. Keywords: zero-dimensional scheme, regularity index, minimal free resolution MSC (1991): 14N05

1

Introduction

z

y

¤

For any zero-dimensional scheme Z » Pn let hZ be its Hilbert function, i.e. de¯ne hZ : N ! N by the formula hZ (t) := dim(Im(½Z;n;t )), where ½Z;n;t : H 0 (Pn ; OPn (t)) ! H 0 (Z; OZ (t)) is the restriction map. The Hilbert functions of zero-dimensional degree d subschemes of Pn are known since Macaulay: this are the so-called O-sequences. In [4] Geramita, Maroscia and Roberts characterized the Hilbert functions of ¯nite reduced subschemes of Pn : both the function and its ¯rst di®erence function must be O-functions. For some geometrically interesting classes of zero-dimensional subschemes of Pn there are very strong restrictions for the possible Hilbert functions (see [3], [5] and references therein for the case of fat points). E-mail: [email protected] E-mail: [email protected] E-mail: maria.sprea [email protected]

E. Ballico et al. / Central European Journal of Mathematics 4 (2003) 644{649

645

Here we study the problem of ¯nding restrictions on the Hilbert function for the case of curvilinear subschemes of Pn supported by a unique point of Pn . We will ¯nd very strong restrictions for the Hilbert function of such a curvilinear scheme Z if Z has large regularity index ¿ (Z) (see Theorem 2.5), but that, contrary to the case of fat points, all very high regularity indices may occur (see Proposition 2.8). We will also show that when ¿ (Z) > d=2 ¡ 1, then also the Betti numbers bi;j (Z), 1 µ i µ n ¡ 1, j ¶ 0, of the minimal free resolution of Z have some restrictions (see part (d) of Theorem 2.5). If Z » P2 we compute the minimal free resolution of IZ ; too.

2

Restrictions on the Hilbert function

We work over an algebraically closed base ¯eld K. Let R = K[x0 ; : : : ; xn ] and let Pn = Proj(R): If Z » Pn is a closed subscheme, we set IZ its saturated homogeneous ideal, and IZ its ideal sheaf. As usual, we set hi (Pn ; IZ (j)) = dimK H i (Pn ; IZ (j)): At ¯rst, we state the result for Z » P2 . Proposition 2.1. Fix d ¶ 4; and x such that d > x > d=2+1: Let P 2 P2 and let Z » Pn be a zero-dimensional curvilinear scheme such that Zred = fP g: If h1 (P2 ; IZ (x ¡ 2)) 6= 0 and h1 (P2 ; IZ (x ¡ 1)) = 0 then a) there is a line D such that length(D \ Z) = x and Z is contained in a double line; b) a minimal free resolution of IZ is 0 ! O(¡ d + x ¡

2) © O(¡ x ¡

1) ! O(¡ 2) © O(¡ d + x ¡

c) the Hilbert function hZ of Z is 8 > >0 > > > > > < 2t + 1 hZ (t) = > > t+d¡ x+1 > > > > > :d

1) © O(¡ x) ! IZ ! 0

if t < 0 if 0 µ t µ d ¡ if d ¡

x

xµtµx¡

if t ¶ x ¡

1

1:

Proof 2.2. Since Z is curvilinear, there is an open subset U containing P in P2 and a smooth curve C » U with Z » C. Notice that Z is exactly the e®ective Cartier divisor dP of C. For any integer y µ d there is a unique curvilinear subscheme Zy of Z with length(Zy ) = y: the e®ective Cartier divisor yP of C. First we will check the existence of a line D » P2 such that length(D \ Z) = x. By hypothesis, the regularity index ¿ (Z) of Z is equal to x ¡ 2: By [2],Corollaire 2, with s = 2 there is a line D » P2 such that length(D \ Z) = x. Let W be the residual scheme of Z with respect to D. We have length(W ) = d ¡ x < x and W ³ Z. Since W is the e®ective Cartier divisor (d ¡ x)P of C and d ¡ x µ x, W is contained in the Cartier divisor xP of C, i.e. W ³ Z \ D. In particular we have W » D. Thus Z is contained in the double line 2D.

646


Furthermore, we have the following exact sequence 0 ! IW (¡ 1) ! IZ ! IZ\DjD ! 0:

(1)

From the collinearity of W and Z \ D we deduce that the ideal IW is minimally generated by L; G where L is a linear form such that D = V (L); and G has degree d ¡ x; while the ideal ID\ZjD is generated by only one form F of degree x: The ¯rst map of the sequence is the product by L; while the second map is the restriction onto D: The ideal IZ\D » R is generated by L; F where F is a lifting of F in IZ : But IZ\D » IW and so we get that F 2 IW : Looking at the cohomology sequence associated to the sequence (1) we get that the ideal IZ is minimally generated by L2 ; LG; F; where F 2 IW is a lifting of F : From F 2 IW we deduce the existence of a; b 2 R such that F = aL + bG; and so aL2 + bLG ¡ LF = 0: Moreover, L; G is a regular sequence. Then, we get the resolution of IZ : 0 ! O(¡ d + x ¡

2) © O(¡ x ¡

1) ! O(¡ 2) © O(¡ d + x ¡

1) © O(¡ x) ! IZ ! 0

where O = OP 2 : From the free resolution of IZ we can compute the Hilbert function hZ of Z and so we have the claim. Remark 2.3. It is easy to write the last map of the computed resolution of IZ and it is 0

1

B G a C B C B¡ L b C: B C @ A 0 ¡ L Remark 2.4. Any smooth curve containing Z has degree at least x as we can see from the generators of the ideal IZ : Now, we can state the analogous result if Z » Pn ; n ¶ 3: Theorem 2.5. Fix positive integers d ¶ 4, and x such that d > x > d=2 + 1: Let P 2 Pn , n ¶ 3, and let Z » Pn be a length d zero-dimensional curvilinear scheme such that Zred = fP g. If h1 (Pn ; IZ (x ¡ 2)) 6= 0 and h1 (Pn ; IZ (x ¡ 1)) = 0 then: (a) there is a line D » Pn such that length(D \ Z) = x and Z » D (1) , where D (1) denotes the ¯rst in¯nitesimal neighborhood of D in Pn , i.e. the closed subscheme of Pn with (ID )2 as ideal sheaf; (b) every quadric hypersurface containing D in its singular locus contains Z and in particular h0 (Pn ; IZ (2)) ¶ n(n ¡ 1)=2; (c) h1 (Pn ; IZ (t)) µ x ¡ 1 ¡ t for d ¡ x ¡ 1 µ t µ x ¡ 2; (d) bi;x+i¡2 (Z) 6= 0 and bi;j (Z) = 0 for 1 µ i µ n and j ¶ x + i ¡ 1.


647

The non-vanishing of the Betti number bi;x+i¡2 (Z) follows from part (a) because it is forced by the existence of a line D such that length(D \ Z) = x; for instance the existence of the collinear subscheme Z \ D of Z shows that the homogeneous ideal of Z requires at least a minimal generator of degree ¶ x. In the proof we shall use the following well-known and easy lemma, usually called Horace lemma, that we state without proof. Lemma 2.6. Let W be a projective variety, D an e®ective Cartier divisor of W , E a vector bundle on W and Z a closed subscheme of W . Let Y be the residual scheme of Z with respect to D, i.e. the closed subscheme of W with Hom(ID;W ; IZ;W ) as ideal sheaf. Then hi (W; E « IZ;W ) µ hi (D; (EjD) « IZ\D;D ) + hi (W; E(¡ D) « IY;W ) for i = 0; 1. Proof 2.7. As in the proof of Proposition 2.1 let U be an open subset of Pn containing P and let C » U be a smooth curve with Z » C: Z is the e®ective Cartier divisor dP on C: Let D ³ Pn be the Zariski tangent space of C at P . Since C is smooth at P , D is a line, and it is the only line of Pn with w := length(Z \ D) ¶ 2. Since h1 (Pn ; IZ (x ¡ 1)) = 0, and x ¡ 1 µ d ¡ 2 we have Z * D, i.e. w < d. Moreover, w µ x: Let E be the osculating plane of C at P , i.e. the only plane E 0 of Pn such that D » E 0 and the intersection multiplicity of C with E 0 at P is at least 1 + ¹P (C \ D), where ¹P (C \ D) is the multiplicity of the intersection between C and D in P: Since w < d, the plane E is the unique plane E 0 of Pn such that length(E 0 \ Z) > w. There is a unique subscheme Y of Z with length(Y ) = w + 1: the e®ective Cartier divisor (w + 1)P of C. The plane E is the linear span of Y . Claim We have w = x. Proof (of the Claim) Fix a general codimension 3 linear suspace A of Pn and let f : Pn nA ! Pn be the linear projection from A. Since A is general, A \ E = ; and in particular D \ A = ;. Hence by Nakayama’s lemma f induces an isomorphism of Z with f (Z). Hence, taking cones with A as vertex, we easily see that h1 (Pn ; IZ (t)) µ h1 (P2 ; If (Z)(t)). Hence we may apply Proposition 2.1 to f (Z) taking instead of x the maximal integer x0 such that h1 (P2 ; If (Z)(x0 ¡ 2)) 6= 0. The line f (D) is the tangent line to f (C) at f (P ). The condition A \ E = ; implies that f (Y ) spans P2 . Thus f (Y ) \ f (D) = f (Y \ D) = f (Z \ D), i.e. lenght(f (Y ) \ f (D)) = w + 1. Hence x0 = w. Since x0 ¶ x and w µ x, we obtain w = x, proving the Claim. During the proof of the Claim we also checked the inequalities in part (c). Let H » Pn be the hyperplane spanned by D and a general (n ¡ 3)-dimensional linear space and W the residual scheme of Z with respect to H. In the proof of the Claim we also obtained length(Z \ H) = x, i.e. length(W ) = d ¡ x. Since Z is curvilinear, W » Z and length(W ) = d ¡ x µ x = length(Z \ D), we saw at the beginning of the proof that W » D. Hence W is contained in any hyperplane H 0 with D » H 0 . Thus Z is contained in any reducible quadric hypersurface containing D in its singular locus (use for instance Horace Lemma). Taking linear combinations of such quadric hypersurfaces we see that every quadric hypersurface singular along D contains Z. Since D (1) is the intersection of all such

648


quadrics, we have Z » D (1) , concluding the proof of parts (a), (b) and (c). To prove part (d) it is su±cient to prove h1 (Pn ; iPn (i + x ¡ 1) « IZ )) 6= 0 and h1 (Pn ; iPn (j) « IZ )) = 0 ©(n¡1) © OD (¡ 1) and hence for for 1 µ i µ n ¡ 1 and j ¶ i + x. We have Pn (1)jD ¹= OD ¡ ¢ i a every integer i with 1 µ i µ n ¡ 1 we have Pn (i)jD ¹= O © OD (¡ 1)b , where a = n¡1 i ¡ ¢ and b = n¡1 . Hence the vanishing result in part (d) follows by applying twice Horace i¡1 Lemma with respect to a general hyperplane H containing D, while the non-vanishing part follows from the non-vanishing of the corresponding Betti numbers of the subscheme Z \ D of Z. At last, we want to give an existence result, proving that we can construct such curvilinear schemes. Proposition 2.8. Fix integers n ¶ 2, x > 0 and d ¶ x + n ¡ 1, P 2 Pn and a line D » Pn . Then there exist a curvilinear zero-dimensional scheme Z » Pn spanning Pn such that length(Z) = d and length(Z \ D) = x. If d ¶ 4 and x > d=2 + 1, then ¿ (Z) = x ¡ 2. Proof 2.9. Set X := Spec(K[t]=(td )). Use the homogeneous polynomials associated to the n + 1 polynomials 1; t; tx ; tx+1 : : : ; tx+n¡2 to obtain an embedding j : X ! Pn such that Z := j(X) is a solution of the ¯rst part of Proposition 2.8. The last part follows from the ¯rst one and Theorem 2.5. Remark 2.10. Assume char(K) = 0. Fix positive integers n, d and P 2 Pn . There is a zero-dimensional curvilinear scheme Z » Pn such that Zred = fP g, length(Z) = d and with maximal rank ([1]). The regularity index ¿ (Z) of such scheme Z is the ¯rst positive ¡ ¢ ¶ d. integer x such that n+x n

Acknowledgments The authors were partially supported by MURST and GNSAGA of INdAM (Italy).

References [1] C. Ciliberto and R. Miranda: \Interpolations on curvilinear schemes", J. Algebra, Vol. 203, (1998), pp. 677{678. [2] Ph. Ellia and Ch. Peskine: \Groupes de points de P2 : caractere et position uniform", In: Algebraic Geometry, Proceedings, L’Aquila 1988, pp. 111{116, Lect. Notes in Math. 1417, Springer-Verlag, 1990. [3] G. Fatabbi and A. Lorenzini: \On a sharp bound for the regularity index of any set of fat points", J. Pure Appl. Algebra, Vol. 161, (2001), pp. 91{111. [4] A.V. Geramita, P. Maroscia, L. Roberts: \The Hilbert function of a reduced kalgebra", J. London Math. Soc., Vol. 28, (1993), pp. 443{452.


649

[5] P.V. Thiên: \Segre bound for the regularity index of fat points in P3 ", J. Pure App. Alg., Vol. 151, (2000), pp. 197{214.

CEJM 4 (2003) 650{660

Stable vector bundles over cuspidal cubics Lesya Yu. Bodnarchuk1¤ , Yuriy A. Drozd2y 1

2

Fachbereich Mathematik, UniversitÄat Kaiserslautern, 67663 Kaiserslautern, Germany Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 01033 Kyiv, Ukraine

Received 31 July 2003; accepted 30 August 2003 Abstract: We give a complete classi cation of stable vector bundles over a cuspidal cubic and calculate their cohomologies. The technique of matrix problems is used, similar to [2, 3]. ® c Central European Science Journals. All rights reserved. Keywords: vector bundles, cohomologies, matrix problems MSC (2000): 14H60; 14H45, 15A21

1

Introduction

y

¤

Stable vector bundles over projective curves have been widely investigated. In particular, it has been proved that there are coarse moduli spaces of such bundles and they have good compacti¯cations; the dimensions of these spaces have been found, etc. (cf. [6]). Nevertheless, not so much is known about explicit structure of stable bundles, especially over singular curves. It seems that there is a unique general result in this direction, namely that of [1] describing stable vector bundles over a nodal cubic. This description has been derived from the description of all vector bundles over such a curve [3]. If we consider more complicated singularities, e.g. a cuspidal cubic, it follows from [3] that the description of all vector bundles is a wild problem, i.e. contains a classi¯cation of all representations of all ¯nitely generated algebras. Thus, if we are going to study stable vector bundles over such curves, we have to ¯nd another approach. Fortunately, there is one used before in quite di®erent situation, namely in study of representations of \mixed" E-mail: [email protected] E-mail: [email protected]

L. Bodnarchuk, Yu. Drozd / Central European Journal of Mathematics 4 (2003) 650{660

651

Lie groups (i.e. neither reductive nor solvable) [2]. It combines the technique of \matrix problems" used in [3] with the concept of \general position", allowing to restrict matrix considerations by rather simple cases, especially to avoid reductions that lead to wild fragments, thus making possible a recursive construction of all stable vector bundles. Using this method, we prove the following main result. Theorem 1.1. Let C » P2 be a cuspidal cubic over an algebraically closed ¯eld k. (1) The rank r and degree d of a stable vector bundle over C are always coprime. (2) For every pair (r; d) of coprime integers with r > 0 the moduli space VB(r; d) of stable vector bundles of rank r and degree d over C is isomorphic to the a±ne line A1 . Note that A1 ’ k+ is just the Picard group Pic± (C) [5, Example II.6.11.4]. Moreover, we explicitly construct a universal family F (r; d; ¹) of stable vector bundles of rank r and degree d depending on the parameter ¹ 2 A1 and calculate their cohomologies. Note that the matrix problem used in these calculations coincides with that arising in the description of representations of groups of echelon matrices [2]. Perhaps there would be an intrinsic reason for this coincidence, but at the moment we have no idea of what nature it can be. There is also evidence that an analogous result must be valid for other degenerate cubics. We hope to present it in the near future.

2

Matrix reduction

We use the technique of [3] to reduce the description of vector bundles to some matrix calculations. Let C be a cuspidal cubic with the singular point p, ¼ : C~ ! C be its normalization. (C is the compacti¯cation of the curve y 2 = x3 and p = (0; 0).) Then ~ = O ~ . We also C~ ’ P1 and ¼ ¡1 (p) = fqg, one point set. We denote O = OC and O C denote by VB(C) the category of vector bundles or, the same, locally free coherent sheaves ~ We over C. For every vector bundle F over C set F~ = ¼ ¤ F . It is a vector bundle over C. always identify F with ¼ ¡1 F » F~ . Note that C~ n fqg ’ C n fpg and the sections of F~ and F coincide on this common part, while F~q ¼ Fp ¼ t2 F~q , where t is the local parameter ~ Thus V = Fp =t2 F~q is a vector subspace in W = F~q =t2 F~q . The latter at the point q 2 C. ~ q =t2 O ~q ’ k[t]=(t2 ). Moreover, dim V = rk W = rk F~ is a free module over the algebra O and V generates W as k[t]=(t2 )-module. On the contrary, if E is a vector bundle over C~ of rank r and V is an r-dimensional subspace in W = Eq =t2 Eq generating W as k[t]=(t2 )module, its preimage in E is a vector bundle F over C of rank r. We consider the following category T : ° Its objects are pairs (E ; V ), where E is a vector bundle over C~ and V is a subspace in W = Eq =t2 Eq such that dim V = rk E and V generates W as k[t]=(t2 )-module. ° A morphism (E ; V ) ! (E 0 ; V 0 ) is a morphism of sheaves Á : E ! E 0 such that Á(V ) ³ V 0 , where Á is the induced morphism Eq =t2 Eq ! Eq0 =t2 Eq0 .

652


The preceding considerations make obvious the following result (cf. also [3]). ~ Fp =t2 F~q ) is an equivProposition 2.1. The functor T : VB(C) ! T mapping F to (F; alence of categories. ~ Recall that all indecomposable vector bundles over C~ ’ P1 are just twists O(d) of the L ~ ~ F ’ O(d) structure sheaf [4]. Especially, for some multiplicities md . d md Note that the arithmetic genus of C is 1 and that of C~ is 0. Therefore, the exact sequence 0 ! F ! F~ ! W=V ! 0 and the Riemann{Roch theorem give that degC F = Â(F ) = Â(F~ ) ¡

r = degC~ F~ ;

where Â(F ) denotes, as usual, the Euler{Poincar¶e characteristic: Â(F ) = dim H0 (F ) ¡ dim H1 (F ). Since W=V is zero outside the unique point q, Â(W=V ) = dim(W=V ) = r. Denote by sl(F ) = deg F = rk F , the slope of F . Recall [6] that a vector bundle F is said to be stable if sl F 0 < sl F for every proper subsheaf F 0 » F . As the arithmetic genus of C is 1, there is an easier criterion for F to be stable. Lemma 2.2. A vector bundle F over C is stable if and only if End F = k. This condition is always necessary. Indeed, if End F 6= k, there is an endomorphism f that is neither zero nor an isomorphism. Denote by F 0 = Im f ’ F = Ker f . If F is stable, sl F 0 < sl F . But it implies that sl(Ker f ) > sl F , so F is not stable. To prove that it is also su±cient, note the following easy result. Lemma 2.3. Let F ; G be coherent sheaves of O-modules. (1) If one of them is locally free, Hom(F ; G ) ’ F _ « G . (2) If F is locally free, (F _ « G )_ ’ G _ « F . Proof. (1) There is a natural morphism Á : F _ « G ! Hom(F ; G ), which is isomorphism if either F = O or G = O. Since Á is an isomorphism if and only if all induced morphisms of stalks are isomorphisms, it implies the claim. (2) Hom(F _ « G ; O) ’ Hom(G ; Hom(F _ ; O)) ’ G _ « F . Now the Riemann{Roch theorem implies that for any coherent sheaves F ; G one of which is locally free, dim Hom(F ; G ) ¡

over C,

dim Ext(F ; G ) = Â(F _ « G ) = deg(F _ « G ) = rk F deg G ¡

especially dim Hom(F ; G ) 6= 0 if sl G locally free,

rk G deg F ;

> sl F . Moreover, by the Serre’s duality, if F is

dim Ext(F ; G ) = dim H1 (Hom(F ; G )) = dim H1 (F _ « G ) = = dim H0 (G

_

« F ) = dim Hom(G ; F ):


653

Thus if sl G = sl F and Hom(G ; F ) 6= 0, also Hom(F ; G ) 6= 0. Suppose now that F is not stable and F 0 » F is such that sl F 0 ¶ sl F . Then, as we have seen, Hom(F ; F 0 ) 6= 0, so the composition of a non-zero homomorphism F ! F 0 with the embedding F 0 ! F gives a nontrivial endomorphism of F . Corollary 2.4. If F is a stable vector bundle, then fd j md 6= 0g = fc; c + 1g for some integer c. Proof. Otherwise there are integers a; b such that a µ b ¡ 2 and ma 6= 0; mb 6= 0. There ~ ~ ~ q ) ³ t2 O(b) ~ q . It gives rise ! O(b) is a nonzero homomorphism f : O(a) such that f (O(a) 2 ~ Fp =t F~q ) is a nontrivial to an endomorphism Á of F~ such that Á = 0, so (Á; 0) 2 End( F; endomorphism. By Proposition 2.1 it corresponds to a nontrivial endomorphism of F . Hence F is not stable. From now on we consider the full subcategories VBc (C) » VB(C) consisting of all ~ ~ O(c+1) vector bundles F such that F~ ’ aO(c)©b for some integers a; b. Note that in this case rk F = a + b and deg F = (a + b)c + b. Under the equivalence T it corresponds to the ~ ~ O(c+1). full subcategory Tc » T consisting of all pairs (E ; V ) with E = aO(c)+b The shift F 7! F (c) of the category of coherent sheaves induces equivalences VB 0 (C) ! VBc (C) ~ © bO(1), ~ and T0 ! Tc , so we only have to consider T0 . If E = aO then Eq =t2 Eq ’ aW0 © ~ ! O(1) ~ bW1 , where W0 = W1 = k[t]=(t2 ). Homomorphisms O induce homomorphisms 2 of k[t]=(t )-modules W0 ! W1 mapping 1 7! ¸ + ¹t and t 7! ¸t. Note that there are ~ ~ and all endomorphisms of both O ~ and O(1) ~ !O no non-zero homomorphisms O(1) are just multiplications by a scalar. Given a pair (E ; V ) from T0 , choose bases w10 ; w20 ; : : : ; wa0

of aW0 ;

w11 ; w21 ; : : : ; wb1

of bW1 ;

v1 ; v2 ; : : : ; vr and write vj =

a X

of V

((®0ij + ¯ij0 t)wi0 + (®1ij + ¯ij1 t)wi1 ):

i=1

Since v1 ; v2 ; : : : ; vr must generate W , the matrix 0 0 ®11 : : : B : : : ... B B 0 ::: B® L = B a1 1 B ®11 : : : B @ : : : ... 1 ®b1 :::

0 ®1r

1

::: C C C ®0ar C 1 C C ®1r C ::: A 1 ®br

must be invertible. Therefore, changing the basis v1 ; v2 ; : : : ; vr , we may suppose that it is the unit r £ r matrix. Then, using automorphisms of E of the form id + f , where f arises

654


~ ! O(1), ~ from a homomorphism O we can make ¯ij1 = 0 for j µ a. Thus the remaining part of coe±cients ¯ijk form a matrix µ ¶ M1 M2 M= (1) 0 M3 where M1 is of size a £ a, M2 is of size a £ b and M3 is of size b £ b. We shall always suppose that the bases v1 ; v2 ; : : : ; vr ; w10 ; w20 ; : : : ; wa0 and w11 ; w21 ; : : : ; wb1 are chosen this way and call M a de¯ning matrix of the pair (E ; V ). If M 0 is a de¯ning matrix for another pair (E 0 ; V 0 ) and Á : (E ; V ) ! (E 0 ; V 0 ) is a morphism of pairs, let © = ©0 + ©1 t be the matrix of the homomorphism Á with respect to the bases w10 ; w20 ; : : : ; wa0 ; w11 ; w21 ; : : : ; wb1 of W and w 0 01 ; w 0 02 ; : : : ; w 0 0a0 ; w 0 11 ; w 0 12 ; : : : ; w 0 1b0 of W 0 . Note that µ ¶ µ ¶ S1 0 0 0 ©0 = and ©1 = (2) S2 S3 S4 0 where S1 is of size a0 £ a, S2 and S4 of size b0 £ a, and S3 of size b0 £ b, and the matrices ©0 ; ©1 uniquely de¯ne the morphism Á. The condition Á(V ) ³ V 0 means that S1 M1 = M10 S1 + M20 S2 ; S1 M2 = M20 S3 ; S3 M3 + S2 M2 =

(3)

M30 S3 :

On the contrary, if S1 ; S2 ; S3 satisfy equations 3, there exists a unique matrix S4 such that the pair ©0 ; ©1 given by equations 2 arises from a uniquely de¯ned morphism Á : (E ; V ) ! (E 0 ; V 0 ). We consider the category of matrix triples M. Its objects are triples (M1 ; M2 ; M3 ) as above and morphisms (M1 ; M2 ; M3 ) ! (M10 ; M20 ; M30 ) are triples (S1 ; S2 ; S3 ) satisfying conditions 3. A triple from M is called stable if it only has scalar endomorphisms. We denote by Ms the full subcategory of stable triples. Then we get the following result. Theorem 2.5. The category M is equivalent to the category VBc (C). Especially, the category Ms is equivalent to the full subcategory VBsc (C) » VBc (C) of stable vector bundles. We call the pair (a; b) the size of the triple M = (M1 ; M2 ; M3 ), the sum r = a + b the rank of this triple and b the degree of this triple. Note that r coincides with the rank of the corresponding vector bundle from VBc , while the actual degree of this vector bundle is d = rc + b. Especially r; d are coprime if and only if so are a; b. We also call the matrix µ ¶ M1 M2 0 M3 the de¯ning matrix corresponding to the triple M . It is indeed a de¯ning triple of a pair (E ; V ) from the category Tc , and thus de¯nes a vector bundle F 2 VBc (C). Especially F is stable if and only if so is the triple (M1 ; M2 ; M3 ).


3

655

Proof of the main theorem

Theorem 2.5 shows that the main theorem 1.1 is equivalent to the following. Theorem 3.1. (1) There is a stable triple of size (a; b) if and only if a; b are coprime. (2) For every pair (a; b) of coprime integers the moduli space T (a; b) of stable triples of size (a; b) is isomorphic to the a±ne line A1 . Proof. Suppose that a triple M = (M1 ; M2 ; M3 ) is stable. If b = 0 (the same for a = 0), we only have one matrix M1 . If a > 1, there is a nonscalar matrix S1 commuting with M1 , hence de¯ning a non-trivial endomorphism of the triple. If a = 1, M1 = ¹ 2 k. Hence, if either a or b is zero, then r = 1 and the moduli space is A1 (it coincides with the Picard group Pic± (C)). We denote the corresponding vector bundle (actually line bundle) from VBc (C) by F (1; c; ¹) (it is of degree c). From now on we suppose that both a and b are non-zero. First of all we show that rk M2 = min(a; b). Indeed, if rk M2 < min(a; b), there are invertible matrices S1 ; S3 such that both the ¯rst row and the last column of the matrix M20 = S1 M2 S3¡1 are zero. Replacing M by an isomorphic triple, we may suppose that M2 = M20 . Then the triple (Ia ; S2 ; Ib ), where S2 has only one non{zero element in the lower left corner, de¯nes a nontrivial endomorphism, so M is not stable. If a = b = rk M2 , we can make M2 a unit matrix and M3 = 0. Then conditions 3 for M = M 0 become S1 = S3 ; S2 = 0 and S1 M1 = M1 S1 . If a > 1, one can easily ¯nd a non-scalar matrix S1 such that these conditions hold, so M is not stable. If a = 1, S1 is just an element from k, so M is stable. Moreover, one cannot change M3 (which is also an element ¹ 2 k) without changing M2 and M1 , so the stable triples of this shape are (0; 1; ¹). they are of size (1; 1) and their moduli space is A1 . Suppose that a < b; then M2 can be chosen in the form (Ia 0). Using transformations 3, we can make M1 zero and transform M3 to the form µ ¶ N1 N2 ; (4) 0 N3 where N1 ; N2 ; N3 are of sizes, respectively, a £ a; a £ (b ¡ a); (b ¡ a) £ (b ¡ a). Moreover, if a triple (S1 ; S2 ; S3 ) is a homomorphism of M to another triple M 0 of the same form, with Ni replaced by Ni0 , one can check that S3 is of the form µ ¶ T1 0 S3 = T2 T3 such that T1 N1 = N10 T1 + N20 T2 ; T1 N2 = N20 T3 ; T3 N3 + T2 N2 = N30 T3 ;

656


S1 = T1 and the part S2 is also uniquely de¯ned by S3 . Thus mapping a triple (N1 ; N2 ; N3 ) of size (a; b ¡ a) to the triple µ µ ¶¶ N1 N2 0; (Ia 0); 0 N3 we get a full embedding M ! M, which induces a one-to-one correspondence between stable triples of size (a; b ¡ a) and those of size (a; b). The same result can be obtained if a > b: then we map a triple (N1 ; N2 ; N3 ) of size (a ¡ b; b) to the triple µµ ¶ µ ¶ ¶ N1 N2 0 ; ; 0 : 0 N3 Ib Therefore we are able to use induction on max(a; b), which immediately implies the claim of the theorem. Remark 3.2. One can easily check that the matrix problem given by the category M is actually wild, hence so is also a description of all vector bundles from VBc (X ). Note that the proof above is e®ective, i.e. enables to get an explicit description of stable vector bundles of any prescribed rank r and degree d (which must be coprime). To do it, we have ¯rst to ¯nd a; b; c such that r = a + b; d = rc + b. It means that b is the residue of d modulo r, c = [d=r] and a = r ¡ b. Having (a; b), suppose that a < b. If a = 1, the canonical de¯ning matrix M(1; b; ¹) is 0

0 1 0 B 0 0 1 B B::: ::: ::: B @ 0 0 0 0 0 0

1 ::: 0 0 ::: 0 0 C C .. . ::: :::C C: ::: 0 1 A ::: 0 ¹

Let a > 1; b = qa + b0 with 0 < b0 < b. Subdivide the de¯ning matrix µ ¶ M1 M2 0 M3 into a £ a blocks starting from the left upper corner; the last horizontal and vertical stripes will be of width b0 . Set all blocks zero, except those immediately over diagonal and the last two diagonal blocks, and set all square blocks immediately over the diagonal, except the last one, equal Ia , obtaining 0 1 0 Ia 0 ::: 0 0 0 B 0 0 Ia : : : 0 0 0 C B C B .. C . ::: ::: :::C B::: ::: ::: B C B 0 C 0 0 : : : 0 I 0 a B C @ 0 0 0 : : : 0 N1 N2 A 0 0 0 ::: 0 0 N3


657

(there are q + 2 horizontal and vertical stripes). If a > b = 1, the canonical de¯ning matrix M(a; 1; ¹) is 0 1 ¹ 1 0 ::: 0 0 B 0 0 1 ::: 0 0 C B C . B : : : : : : : : : .. : : : : : : C : B C @ 0 0 0 ::: 0 1 A 0 0 0 ::: 0 0 If a > b > 1; a = qb + a0 with 0 b £ b blocks, obtaining 0 N1 B 0 B B B 0 B B::: B @ 0 0

< a0 < b, start from the lower right corner and make N2 0 N3 Ib 0 0 ::: ::: 0 0 0 0

1 ::: 0 0 ::: 0 0 C C C ::: 0 0 C C: .. . ::: :::C C ::: 0 Ib A ::: 0 0

Here the ¯rst horizontal and vertical stripes are of width a0 . Now pass to the triple (N1 ; N2 ; N3 ) applying the same procedure, etc. The resulting triple is called the canonical form of a stable triple of size (a; b), and its de¯ning matrix M = M(a; b; ¹) is called the canonical de¯ning matrix. To obtain the corresponding vector bundle, consider the vector ~ ~ + 1) over C~ and take the O-subsheaf F = F (r; d; ¹) which © bO(c bundle E = aO(c) coincides with E outside p and is generated by the preimages of columns of the matrix Ir + tM at the point p. Example 3.3. Let a = 3; b = 11. The ¯rst step of reduction gives the matrix 0 1 0 I3 0 0 0 B 0 0 I3 0 0 C B C B 0 0 0 I3 0 C B C; @0 0 0 N N A 1 2 0 0 0 0 N3 the triple (N1 ; N2 ; N3 ) being of size (3; 2). The second step replaces µ ¶ N1 N2 0 N3 by

0

L1 @ 0 0

L2 L3 0

1 0 I2 A ; 0

where the triple (L1 ; L2 ; L3 ) is of size (1; 2). So we have to set 0 1 µ ¶ 0 1 0 L1 L2 = @0 0 1A: 0 L3 0 0 ¹

658


Thus the canonical de¯ning matrix 2 0 0 60 0 6 60 0 6 60 0 6 6 60 0 6 · ¸ 6 60 0 M1 M2 60 0 =6 60 0 0 M3 6 60 0 6 60 0 6 60 0 6 60 0 6 40 0 0 0

M(3; 11; ¹) is 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 ¹ 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0

3

7 7 7 7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 7 7 5

Here we keep lines showing the subdivision of the ¯rst step; double lines denotes the original division of this matrix into M1 ; M2 ; M3 . In the same way, the canonical de¯ning matrix M(7; 4; ¹) is 2 3 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 1 0 0 0 0 0 07 6 7 60 0 0 0 0 1 0 0 0 0 07 6 7 60 0 0 ¹ 1 0 0 1 0 0 07 6 7 60 0 0 0 0 1 0 0 1 0 07 6 7 60 0 0 0 0 0 1 0 0 1 07 6 7: 6 7 60 0 0 0 0 0 0 0 0 0 17 6 7 60 0 0 0 0 0 0 0 0 0 07 6 7 60 0 0 0 0 0 0 0 0 0 07 6 7 40 0 0 0 0 0 0 0 0 0 05 0 0 0 0 0 0 0 0 0 0 0 We note also the following easy consequence of our reduction procedure, which is useful, for instance, in calculating cohomologies. Corollary 3.4. For every stable trip (M1 ; M2 ; M3 ) of size (a; b) µ ¶ M2 rk ( M1 M2 ) = a and rk = b: M3

4

Cohomologies

Having an explicit description, we can calculate cohomologies of stable vector bundles F (r; d; ¹).


659

Theorem 4.1. The sheaves F = F (r; d; ¹) have the following dimensions of cohomologies: 8 if d > 0; 1, hence a > 0; b > 0 and d 6= 0. The exact sequence 0 ! F ! F~ ! W=V ! 0 gives rise to the exact sequence of cohomologies ~ ¡ h! W=V ! H1 (F ) ! H1 (F) ~ ! 0; 0 ! H0 (F ) ! H0 (F) ~ © bO(c ~ + 1), then r = a + b; d = cr + b and and dim W=V = r. Let F~ = aO(c) ½ (c + 1)r + b if c ¶ ¡ 1, 0 ~ dim H (F ) = 0 if c < ¡ 1. Note that d > 0 if and only if c = [d=r] ¶ 0. Let H be the image of H0 (F~ ) in W = F~q =t2 F~q . If c ¶ 0, global sections generate F~ , so H = W and Im h = W=V , wherefrom dim H0 (F ) = cr + b = d. If c = ¡ 1, Corollary 3.4 easily implies that dim(H +V ) = a+ 2b, so dim(Im h) = b and H0 (F ) = 0. We have proven formula 5. Now the Riemann{Roch theorem implies formula 6.

Acknowledgments This paper has been prepared during the stay of the second author at the University of Kaiserslautern supported by the DFG Schwerpunkt \Globale Methoden in der komplexen Geometrie". We are grateful for this opportunity. We also thank Igor Burban and GertMartin Greuel for useful discussions.

References [1] I. Burban: \Stable vector bundles on a rational curve with one node", Ukrainian Math. J., Vol. 55, No. 5, (2000).

660


[2] Y. Drozd: \Matrix problems, small reduction and representations of a class of mixed Lie groups", In: Representations of Algebras and Related Topics, Cambridge Univ. Press, 1992, pp. 225{249. [3] Y. Drozd and G.-M. Greuel: \Tame and wild projective curves and classi¯cation of vector bundles", J. Algebra, Vol. 246, (2001), pp. 1{54. [4] A. Grothendieck: \Sur la classi¯cation des ¯br¶es holomorphes sur la sphµere de Riemann", Amer. J. Math., Vol. 79, (1956), pp. 121{138. [5] R. Hartshorn: Algebraic Geometry, Springer, New York, 1977. [6] C.S. Seshadri: \Fibr¶es vectoriels sur les courbes alg¶ebriques", Ast¶erisque, Vol. 96, (1982).

CEJM 4 (2003) 661{669

On Reprensentation Theory and the Cohomology Rings of Irreducible Compact Hyperkahler Manifolds of Complex Dimension Four Daniel Guan Department of Mathematics The University of California at Riverside Riverside, CA 92521 U.S.A.

Received 13 August 2003; accepted 8 September 2003 Abstract: In this paper, we continue the study of the possible cohomology rings of compact complex four dimensional irreducible hyperkahler manifolds. In particular, we prove that in the case b2 = 7, b3 = 0 or 8. The latter was achieved by the Beauville construction. c Central European Science Journals. All rights reserved. ® Keywords: Hyperkahler, cohomology rings, Betti numbers, holomorphic symplectic, Lie group, Lefschetz triple MSC (2000): 14F25, 14M99, 53C26, 53D35, 32Q55

1

Introduction

The study of higher dimensional hyperkÄahler manifolds has received much attention. It is evident that there are only a few examples of these manifolds, especially in complex dimension four (see [11]). Can we classify them as in the case of K3 surfaces? In [11], we combine the results of the Riemann-Roch formula and the representation generated by the KÄahler classes to give a picture of the Hodge diamonds of irreducible compact hyperkÄahler manifolds of complex dimension four and obtained Proposition 1.1. If M is an irreducible compact hyperkÄahler manifold of complex dimension 4, then 3 µ b2 µ 23. Moreover, (1) if b2 = 23, then b3 = 0. The Hodge diamond of M is the same as that of Fujiki’s ¯rst example. (2) if b2 6= 23, then b2 µ 8, and when b2 = 8, we have b3 = 0. 2 )(8¡b2 ) (3) if 3 µ b2 µ 7, then b3 µ min( 4(23¡b ; 12 (b2 + 4)(23 ¡ b2 )): b2 +1

662

D. Guan / Central European Journal of Mathematics 4 (2003) 661{669

(4) in b2 (5) in (6) c2

the case of b2 = 3; 4; 5; 6, b3 = 4l with l µ 17 if b2 = 3, l µ 15 if b2 = 4, l µ 9 if = 5, and l µ 4 if b2 = 6. the case of b2 = 7, b3 = 0; 4 or 8. 2 ^2 H 2 if and only if (b2 ; b3 ) = (5; 36); (7; 8); (8; 0); (23; 0):

The last item is very important since all the known examples are in this category. Therefore, those Hodge diamonds are the best candidates for being the Hodge diamonds of irreducible compact hyperkÄ ahler manifolds. We also doubt the existence of the case of b2 = 3. Therefore, much more work should be done in these directions. We announced there that, in statement 5), the case of b3 = 4 can be removed. The proof was left out there since we need a deeper knowledge on representations on the cohomology. One of our main results here is: Theorem 1.2. If M is a complex four dimensional irreducible compact hyperkÄahler manifold with b2 = 7, then b3 = 0 or 8. This means that if b2 = 7, then we have the Hodge diamond of the Kummer variety or b3 = 0. Together with the result in the case of b2 = 23, we see that our results are quite sharp. To prove this we have to deal with the representation of a Lie algebra generated by the KÄahler classes. We obtain some results on the structure of the cohomology ring of compact irreducible hyperkÄahler manifolds. We ¯rst obtain in general: Theorem 1.3. Fix a KÄahler class x on an irreducible compact hyperkÄahler manifold. The multiplication by x can be regarded as the root vector of the root e1 if we regard g = so(4; b2 ¡ 2) as a special orthogonal group with a properly chosen metric diag(1; ¡ 1; 1; 1; ¡ 1; ¢ ¢ ¢ ; ¡ 1; 1): We shall explain the choice of the metric in the proof. Applying this result to the case of complex dimension four we obtain: Theorem 1.4. If M is a complex four dimensional irreducible compact hyperkÄahler b2 manifold, then b3 = k2[ 2 ] for some integer k. In particular, if b2 = 7, b3 = 8k. Combining Theorem 1.4 with the results from [11] we obtain the Theorem 1.2. In our argument we use some technique used in [7], [8]. It turns out that the theory of spinors makes the proofs more e±cient. We shall give the real structure of the cohomology rings in more detail and some examples for the possible Hodge algebras as Jordan-Lefschetz modules (see [14]) in Section 3.


2

663

The Multiplication of a Kahler Class on Irreducible Compact Hyperkahler Manifolds

By the results in [22], there is an so(4; b2 ¡ 2) action on the cohomology ring which is induced by the KÄahler classes. When b2 is odd this Lie algebra is a simple Lie algebra of type B with rank b22+1 . When b2 is even this Lie algebra is a simple Lie algebra of type D with rank b22+2 . Here we only deal with the case of odd b2 . The case of a even b2 is similar. In this section we only apply the complex representation theory. We shall study the real structure in the next section. To apply the representation theory of the complex orthogonal groups, we eventually adopt the spinor theory in this paper. Here we recall some results about Cli®ord algebras, see [6, Chapter II, III], [13, Chapter II, VI] and [17, Chapter 9] for the details. Let V be a complex vector space of dimension n with the standard norm, S be the spinor representation of the even Cli®ord algebra if n = 2r ¡ 1, S1 ; S2 be the even and odd half spinor representations of the even Cli®ord algebra if n = 2r. Then S1 ; S2 are the irreducible representations of so(2r; C) corresponding to the fundamental weights 12 (e1 +¢ ¢ ¢+er¡1 §er ) of dimension 2r¡1 . ^r V is a sum of two irreducible representations V1r and V2r of highest weights e1 + ¢ ¢ ¢ + er¡1 § er . All other ^k V = V k ; k < r are irreducible with V k ; 0 < k < r ¡ 1 being the fundamental representations. We have (see [Cv p.96]): X

« 2 S1 =

V k + V1r ;

k=r(mod2) 0·k·r¡1

X

2

« S2 =

V k + V2r ;


X

S1 « S2 =

V k:

k6=r(mod2) 0·k·r¡1

And for the symmetric products and the skew-symmetric products we have: X

S 2 S1 =

V k + V1r ;

k=r(mod4) 0·k·r¡1 2

X

2

^ S1 = ^ S2 =

V k;

k=r+2(mod4) 0·k·r¡1

X

2

S S2 =

V k + V2r :


The case of n = 2r ¡ 1 can be regarded as that of a codimension 1 subspace of a space with dimension 2r, as in [6, p.107]. Then S comes from S1 . From V k ; 0 < k < r (resp. V 0 or V1r ) in the even case we obtain V k¡1 + V k (resp. V 0 or V r¡1 ). V k are irreducible with V k ; 0 < k < r ¡ 1 fundamental representations. And we have (see also [6, p.107]): «2 S =

X

0·k·r¡1 2

S S=

V k; X

k=r+3; r(mod4) 0·k·r¡1

V k;

664

D. Guan / Central European Journal of Mathematics 4 (2003) 661{669 X

^2 S =

V k:

k=r+1; r+2(mod4) 0·k·r¡1

This discussion can be thought of as a partial extension of some arguments we used in [8]. Proof (of Theorem 1.3). By [22](see also [23], [14]), the complex cohomology ring is a representation of the Lie algebra g = so(4; b2 ¡ 2) which is generated by all Lefschetz sl(2) for all the KÄahler classes in H 2 for all deformations of the manifold M . As in [22], we have a decomposition of g = g¡2 © g0 © g2 as a graded Lie algebra, where g¡2 (resp. g2 ) consists of the linear combinations of ¤! (resp. L! ) with all the possible KÄahler classes ! in H 2 . We have g0 = so(3; b2 ¡ 3) © RH. Fix a KÄahler class x1 . We want to prove that the multiplication by x1 (L3 in [22]) can be regarded as a root vector with root e1 if we regard g = so(4; b2 ¡ 2) as a special orthogonal group with a properly chosen metric diag(1; ¡ 1; 1; 1; ¡ 1; ¢ ¢ ¢ ; ¡ 1; 1) (we call so(1; 1; 2; b2 ¡ 3; 1)). We can ¯rst choose the metric such that the so(3; b2 ¡ 0

B 02;2

g0 = B @

1

02;b2

0b2 ;2 so(2; b2 ¡

3) component of g0 is:

3; 1)

C C; A

where so(2; b2 ¡ 2; 1) is the special orthogonal Lie algebra with metric diag(1; 1; ¡ 1; ¢ ¢ ¢ ; ¡ 1; 1) and 0k;l is the k £ l zero matrix. We let our Cartan subalgebra be the set of the elements diag(A1 ; ¢ ¢ ¢ Ar ; 0) with 0

1

B 0

Ai = B @

ai C

si a i 0

s1 = sr = 1; si = ¡ 1 if 2 µ i µ r ¡ 1 and r =

C A

b2 +1 . 2

Then, the root vector with root e1 is

0

1

0 0 01;b2 ¡1 1 C B B C B C B 0 0 01;b2 ¡1 1 C B C B C B C B C B 0b2 ¡1;1 0b2 ¡1;1 0b2 ¡1;b2 ¡1 0b2 ¡1;1 C B C @ A ¡ 1

and is in g = so(1; 1; 2; b2 ¡ diag(2A; 0; ¢ ¢ ¢ ; 0) with

1

01;b2 ¡1

0

3; 1). If fx1 ; H; ¤x1 g is the standard KÄahler triple, then H is 0

B A=B @

1

0 1C 10

C: A


665

The subalgebra generated by all the Kij in [22] is a K1 = so(3) factor of a maximal compact Lie subalgebra in so(2; b2 ¡ 3; 1). And we can choose our metric such that K12 generates diag(0; 0; so(2); 0; ¢ ¢ ¢ ; 0) and the other factor of this compact Lie subalgebra is K2 = diag(0; 0; 0; 0; so(b2 ¡ 3); 0), which commutes with L3 . We obtain that x1 can be found by the condition [x1 ; K2 ] = 0; [x1 ; K12 ] = 0; [H; x1 ] = 2x1 and is in the form we expected. Q. E. D. Proof (of Theorem 1.4). Since the cohomology ring H ¤ is a representation of the graded Lie algebra g = so(4; b2 ¡ 2) with elements of even degrees, we see that B = H 3 © H 5 is a representation of g. And for any element in B, the sl(2) representation introduced by x1 has dimension 2. This is only possible if all the irreducible representations are with highest weight 12 (e1 + ¢ ¢ ¢ + er ) since all the possible highest weights are r¡1 X

k=1

ak (

k X

ei ) +

i=1

r ar X ei ; 2 i=1

where ak are positive integers, i.e. if any ak > 0 with k < r, then the coe±cient of e1 is at least 1. Therefore, the irreducible representations are all the even spinor representations b2 + 1 (see, e.g., [7, Appendix], [12, p.69], [13, p.378]). The dimension of B must be k2 2 for some integer k. We obtain our desired conclusion. Q. E. D.

3

Real Structures and Examples

To give a more precise picture of the possible cohomology ring we shall apply a delicate representation theory of the real semisimple Lie groups. By [20] and [13] we shall calculate the class of the Tits algebra with applying [13, p.378{379]. Lemma 3.1. The standard spinor representation of the real even Cli®ord algebra of so(3; 2k) has dimension 2k+2 for k = 0; 3(mod4) and 2k+1 for k = 1; 2(mod4). Proof. We apply [17, p.333 Remark 9.2.12] to our case. We only need to calculate the Cli®ord invariant c = c(3; 2k). If c is the quaternion (i.e., c = ¡ 1), then the dimension is 2 £ 2k+1 where 2 is the degree of the quaternion and 2k+1 is the degree of the even Cli®ord algebra. If c is the real ¯eld (i.e., c = 1), then the dimension is the degree of the even Cli®ord algebra, i.e., 2k+1. We apply the formula in [17, p.81]. Notice that (1; ¡ 1) = (1; 1) = 1 and (¡ 1; ¡ 1) = ¡ 1, we obtain: 2

c(3; 8l) = ¡ s(3; 8l) = ¡ (¡ 1)C8l = ¡ (¡ 1)4l(8l¡1) = ¡ 1;

666

D. Guan / Central European Journal of Mathematics 4 (2003) 661{669 2

c(3; 8l + 6) = s(3; 8l + 6) = (¡ 1)C8l+

6

= (¡ 1)(4l+3)(8l+5) = ¡ 1;

similarly for k = 4l + 1 and 4l + 2. Q. E. D. Lemma 3.2. The even Ci®ord algebra C0 (3; 2k ¡ 3) is a simple algebra over C for even k and is a sum of two simple algebras if k = 2r + 1. 2k (2k ¡ 1)

Proof. The discriminant of (3; 2k ¡ 3) is D = (¡ 1) 2 (¡ 1) = (¡ 1)¡k+1 (see [13, xix]). Hence D = ¡ 1 if k is even and D = 1 if k is odd. We obtain our lemma by [13, p. 88 Theorem 8.2]. Q. E. D. Lemma 3.3. The even Cli®ord algebra C0 (3; 4r ¡ 1) is a sum of two simple algebras of class R if r is odd and is a sum of two simple algebras over quaternions if r is even. Proof. Here we ¯rst prove that C0 (3; 2k + 1) = C(3; 2k) (see [6, p.47]). Let V = V0 + Rx2k+4 such that q(x2k+4 ) = ¡ 1 and V0 is orthogonal to x2k+4. We consider the map f : C(3; 2k) ! C0 (3; 2k + 1) = C1 (3; 2k)x2k+1 + C0 (3; 2k) generated by f (1) = 1 and f (y) = yx2k+4 for all y 2 V0 . Then, f (y 2 ) = f (q(y)) = q(y) = ¡ q(y)q(x2k+4) = q(yx2k+4): We obtain an isomorphism. Apply Lemma 3.1. Then we have Lemma 3.3. Q. E. D. Now we are ready to give some examples for the possible cohomology rings: (1) b2 = 3. g = so(4; 1) = sp(1; 1); g 0 = so(3) = su(2) = sp(1). K12 is in a Cartan subalgebra of su(2) and can have the form diag(i; ¡ i) (see [21, Lemma 2.2]). The spinor representation of so(3) is su(2), which comes from the quaternion representation sp(1). In the case l = 1, the real dimension of H 3 is the real dimension of the standard representation of su(2), i.e., 4. If the complex basis of the representation of su(2) is e1 ; e2 , then H 2;1 is determined by K12 and is generated by e1 ; e¹2 . If (z1 ; z2 ) is the coordinates, then H 6 contains the tracefree skew-Hermitian forms on (z1 ; z2 ). H 4;2 =< z1 ^ z¹2 >; H 3;3 =< iz1 ^ z¹1 ¡ iz2 ^ z¹2 >; H 2;4 =< z2 ^ z¹1 >. (2) b2 = 4. g = so(4; 2) = su(2; 2); g 0 = so(3; 1) = sl(2; C). K12 is in a Cartan subalgebra of a compact Lie subalgebra su(2) of sl(2; C) and can have the form diag(i; ¡ i). The spinor representation of so(3; 1) is sl(2; C), which is complex, with real dimension 4. If l = 1 and the complex basis of the representation of sl(2; C) is (e1 :e2 ), then H 2;1 is generated by e1 ; e¹2 . Let (z1 ; z2 ) be the coordinates, then H 6 contains all the skew-Hermitian forms on (z1 ; z2 ). H 4;2 =< z1 ^ z¹2 >; H 3;3 =< iz1 ^ z¹1 ; iz2 ^ z¹2 >; H 2;4 =< z2 ^ z¹1 >.


667

(3) b2 = 5. g = so(4; 3); g 0 = so(3; 2) = sp(2; R). A maximal compact Lie subalgebra of so(3; 2) is so(3) £ so(2) with center so(2). A maximal compact Lie subalgebra of sp(2; R) is sp(2; R) \ so(4) containing all the matrices 0

1

B A BC C A

(AjB) = B @

¡ B A

with A a 2 £ 2 skew-symmetric matrix and B a 2 £ 2 symmetric matrix. This Lie algebra is isomorphic to u(2) with the isomorphism: ® : A + iB ! (AjB): Therefore, the so(3) in so(3; 2) is corresponding to the su(2) in u(2) and K12 can have a form 0 1 B 0 0 1 0 C B C B C B C B 0 0 0 ¡ 1C B C: ®(diag(i; ¡ i)) = B C B¡ C B 1 0 0 0 C B C @ A

0 10 0

The spinor representation of so(3; 2) is sp(2; R), which is real, with real dimension 4. If l = 1 and (e1 ; e2 ; e3 ; e4 ) be the real basis of the spinor representation, then K12 acts as the complex structure and K12 e1 = ¡ e3 ; K12 e2 = e4 . H 2;1 =< e1 + ie3 ; e2 ¡ ie4 >. H 4;2 =< (e1 + ie3 ) ^ (e2 ¡ ie4 ) >; H 3;3 =< e1 ^ e3 ¡ e2 ^ e4 ; e1 ^ e4 ¡ e2 ^ e3 ; e1 ^ e2 ¡ e3 ^ e4 >; H 2;4 =< (e1 ¡ ie3 ) ^ (e2 + ie4 ) >. (4) b2 = 6. g = so(4; 4); g 0 = so(3; 3) = sl(4; R). The even half spinor representation of so(3; 3) is sl(4; R) with real dimension 4. The odd half spinor representation is the adjoint of the even representation. A maximal compact Lie subalgebra of so(3; 3) is so(3) £ so(3) and a maximal compact Lie subalgebra of sl(4; 0 R) 1is so(4) with B 0 1C C. Therefore, A

a root system generated by diag(J; ¡ J ); diag(J; J) with J = B @

¡ 10

K12 can have a form diag(J; J). If l = 1, we might assume that H 3 is the even half spinor representation and (x1 ; x2 ; x3 ; x4 ) is a real basis, then K12 = diag(J; J) acts as a complex structure on R4 . For our convenience, we let ek = x2k¡1 + ix2k for k = 1; 2. Then, H 2;1 =< e1 ; e2 >. H 6 contains all the skew-symmetric forms on R4 and H 4;2 =< e1 ^ e2 >; H 3;3 =< iek ^ e¹l ; k; l = 1; 2 >. (5) b2 = 7. If b3 = 8, this occurs as the Kummer manifold K2 which ¯rst appeared in [5]. g = so(4; 5); g 0 = so(3; 4). The spinor representation is real with dimension 8.

Acknowledgements I would like to thank the Department of Mathematics of Princeton University and the University of California at Riverside for their support. I also thank the NSF for the

668


fellowship which made this work possible. I thank Professors F. Bogomolov, M. Gromov, S. Kobayashi and Y.-T. Siu, for their constant support. Professor Bogomolov also pointed out a °aw in an earlier version. I thank Professor T. Springer for showing me [13] while he was reading [7], which eventually led me to apply spinor theory in this paper. Finally, I would like to take this chance to thank Professor Hitchin and Sawon, Salamon, and S.-T. Yau for their kindly help with [11]. Supported by NSF Postdoctoral Fellowship DMS-9627434.

References [1] F. Bogomolov: \KÄahler manifolds with trivial canonical class", Izv. Akad. Nauk. SSSR Ser. Mat., Vol. 38, (1974), pp. 11{21. [2] F. Bogomolov: \The decomposition of KÄ ahler manifolds with a trivial canonical class", Mat. Sb. (N.S.), Vol. 38, (1974), pp. 573{575. [3] F. Bogomolov: \Hamiltonian KÄahlerian manifolds", Dolk. Akad. Nauk. SSSR, Vol. 243, (1978), pp. 1101{1104. [4] F. Bogomolov: \On the cohomology ring of a simple hyperkÄ ahler manifold (on the results of Verbisky)", Geom. Funct. Anal., Vol. 6, (1996), pp. 612{618. [5] A. Beauville: \Vari¶et¶es KÄahleriennes dont la premiµre classe de Chern est nulle", J. Di®erential Geometry, Vol. 18, (1983), pp. 755{782. [6] C. Chevalley: The Algebraic Theory of Spinors, Columbia University Press, New York, 1954. [7] D. Guan: \Toward a classi¯cation of compact complex homogeneous spaces", preprint, 1998. [8] Z. Guan: \Toward a classi¯cation of almost homogeneous manifolds I|linearization of the singular extremal rays", International J. Math., Vol. 8, (1997), pp. 999{1014. [9] D. Guan: \Examples of compact holomorphic symplectic manifolds which are not KÄahlerian II", Invent Math., Vol. 121, (1995), pp. 135{145. [10] D. Guan: \Examples of compact holomorphic symplectic manifolds which are not KÄahlerian III", Intern. J. of Math., Vol. 6, (1995), pp. 709{718. [11] D. Guan: \On Riemann-Roch formula and bounds of the Betti numbers of irreducible compact hyperkÄahler manifold of complex dimension four", ¯rst version in 1999, a part of paper appeared in Math. Res. Letters, Vol. 8, (2001), pp. 663{669. [12] J. Humphreys: Introduction to Lie Algebras and Representation Theory, GTM 9, Springer-Verlag, New York, 1987. [13] M. Knus, A. Merkurjev, M. Rost, J. Tignol: The Book of Involutions, Colloquium Publications 44, AMS, Providence, Rhode Island, USA, 1998. [14] E. Looijenga and V. Lunts: \A Lie algebra attached to a projective variety", Invent. Math., Vol. 129, (1997), pp. 361{412. [15] K. O’Grady: \Desingularized moduli spaces of sheaves on a K3", J. Reine Angew. Math., Vol. 512, (1999), pp. 49{117.


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[16] K. O’Grady: \A new six dimensional irreducible symplectic variety", preprint, 2000. [17] W. Scharlau: \Quadratic and Hermitian Forms", Grundlehren der mathematischen Wissenschaften 270, Springer-Verlag, Berlin{Heidelberg{New York{Tokyo, 1985. [18] S. Salamon: \Riemannian Geometry and Holonomy Groups", Pitman Research Notes in Mathematics, Series 201, 1989. [19] S. Salamon: \On the cohomology of KÄahler and hyperkÄahler manifolds", Topology, Vol. 35, (1996), pp. 137-155. [20] J. Tits: \Repr¶esentations lin¶eaires irr¶educibles d’un groupe r¶eductif sur un corps quelconque", J. reine angew. Math., Vol. 247, (1971), pp. 196{220. [21] M. Verbitsky: \Action of the Lie algebra of SO(5) on the cohomology of a hyperkÄahler manifold", Functional Anal. appl., Vol. 24, (1991), pp. 229{230, [22] M. Verbitsky: \Cohomology of compact hyperkÄ ahler manifolds and its applications", Geom. Funct. Anal., Vol. 6, (1996), pp. 601{611. [23] M. Verbitsky and D. Kaledin: HyperkÄahler Manifolds, International Press, Somerville, Massachusetts, USA, 1999. [24] S. Yau: \On the Ricci curvature of a compact KÄahler manifold and the complex Monge-Ampµere equation I", Comm. Pure Appl. Math., Vol. 31, (1978), pp. 339{411.

CEJM 4 (2003) 670{689

On quasistatic inelastic models of gradient type with convex composite constitutive equations Krzysztof CheÃlmi¶ nski1 2¤ 1

Fachbereich Mathematik und Statistik, UniversitÄat Konstanz, UniversitÄatsstr. 10, 78457 Konstanz, Germany 2 Cardinal Stefan Wyszy¶ nski University, ul. Dewajtis 5, Warsaw, Poland

Received 14 April 2003; accepted 4 September 2003 Abstract: This article de nes and presents the mathematical analysis of a new class of models from the theory of inelastic deformations of metals. This new class, containing so called convex composite models, enlarges the class containing monotone models of gradient type de ned in [1]. This paper starts to establish the existence theory for models from this new class; we restrict our investigations to the coercive and linear self-controlling case. ® c Central European Science Journals. All rights reserved. Keywords: global existence, energy method, monotone operators, Yosida approximation, inelastic deformations MSC (2000): 35Q72, 73E60, 73F99

1

Introduction

In this paper we study a class of models from the theory of inelastic deformations which is not contained in the class of monotone models. This new family of models enlarges the set of \generalized standard materials" de¯ned by B. Halphen and Nguyen Qouc Son in [18] and enlarges the set containing models of monotone-gradient type de¯ned in [1].

¤

Let us start with the de¯nition of a very large class of models in the theory of inelastic deformations, provided that we are interested in the quasistatic setting of the problem. Let » R3 be a bounded domain with smooth boundary @ . We have to ¯nd the displacement ¯eld u : £ R+ ! R3 , the Cauchy stress tensor T : £ R+ ! S 3 = R3£3 sym E-mail: [email protected]

K. CheÃlmiński / Central European Journal of Mathematics 4 (2003) 670{689

671

and the vector of internal variables z : £ R+ ! RN satisfying the following system of equations divx T (x; t) = ¡ F (x; t) T (x; t) = D("(u(x; t)) ¡

Bz(x; t))

1 (rx u(x; t) + rTx u(x; t)) 2 ³ ´ zt (x; t) 2 g ¡ ½rz Ã("(x; t); z(x; t))

"(u(x; t)) =

(1)

where the function F : £ R+ ! R3 describes the external forces acting on the material, D : S 3 ! S 3 is the elasticity tensor which is assumed to be constant in time and space and symmetric and positive de¯nite, the linear operator B : RN ! S 3 is the orthogonal projection of the vector z = ("p; y) 2 S 3 £ RN¡6 = RN onto the "p -direction (Bz = "p ), the multifunction g : D(g) » RN ! P(RN ) = fthe set of all subsets of RN g is the constitutive multifunction depending of the considered model and Ã ¶ 0 is the so called free energy function which we assume to be quadratic in all variables 1 ½Ã("; z) = D(" ¡ 2

Bz) ¢ (" ¡

1 Bz) + Lz ¢ z 2

with a positive semi-de¯nite and symmetric matrix L. Following the de¯nition from monograph [1] we say that the considered model is of pre-monotone type if the multifunction g satis¯es 8 z 2 D(g) 8 z ¤ 2 g(z) z¤ ¢ z ¶ 0 (2) and the matrix L is so chosen that the symmetric matrix = BT D B + L

is positive de¯nite:

If we additionally assume that 0 2 g(0) which is always satis¯ed for a very large class of models used in practice, then (2) will immediately follow for all multifunctions which are monotone at the point 0. Requirement (2) yields that system (1) is thermodynamical admissible, this means z ¤ ¢ ½rz Ã("; z) µ 0

for all z ¤ 2 g(¡ ½rz Ã("; z)) :

Let us denote the class consisting of all pre-monotone models by PM. All models used in practice and known to the author, belong to this class. For example the model of BodnerPartom ([12]), the model of Prandtl-Reuss ([3], [19], [20],), the model of ArmstrongFrederick ([5]), the model of Norton-Ho® ([20]) and many others. Mathematical analysis of all models from PM is not done until today. Only for several subclasses of PM or for single models we can ¯nd existence results in the literature. Let us present some important subclasses of PM. If we assume that the constitutive multifunction g is monotone and 0 2 g(0) then such a model is called of monotone type (shortly belongs to M). In the dynamical setting of the problem these models are studied in several papers: for

672


example in [1], [6], [8], [9] and [13]. In the class M, the existence theory of global in time solutions for all monotone models used in practice is very well understood (see [1] or [8] and the references therein). In the quasi-static case the existence theory can be constructed in a similar way as in the dynamical case (see [2], [11]). For non-monotone models there exists in the literature a mathematical analysis of single models ([12], [17]) and of the class L M containing global Lipschitz perturbations of monotone models, i.e. models with constitutive multifunctions of the form g = gM + L where gM is monotone and L is a global Lipschitz map ([12]). Moreover, we should mention here that there is a further subclass of non-monotone models considered in monograph [1]. This subclass denoted by T M contains all models which can be transformed into the class M. For details we refer the reader to [1]. This article presents a new subclass of PM not contained in M (also not contained in L M nor in T M) for which the existence theory of global in time solutions can be proved using the techniques from the theory of monotone operators. We say that a model is of convex composite type (shortly belongs to C C ) if the constitutive multifunction has the form 8 z 2 D(g) g(z) = @c (M ¯ ©)(z) where M is a convex function, © is a global Lipschitz C1;1 di®eomorphism and @c denotes the Clarke generalized gradient (see [15]). By the properties of © we see that the righthand side of the last equality is equivalent to 8 z 2 D(g) g(z) = rT ©(z) ¢ @M (©(z))

(3)

where now the symbol @ denotes the usual subgradient of the convex function M . For the general theory of subdi®erential evolution inclusions in nonconvex analysis we refer the reader to [16] and the references presented therein. Let us summarize the contents of this article. In Section 2 we formulate our initialboundary value problem and prove the existence of global in time solutions to this problem assuming that the model is coercive, i.e. the quadratic form Ã is positive de¯nite (compare with [1]). Section 3 is devoted to the investigation of convex composite models possessing the self-controlling property (for a de¯nition of this property and results for monotone self-controlling problems see [6], [7]). We present in this section the existence and uniqueness result for all convex composite and self-controlling models, provided that the self-controlling property is linear. The general case can be done following Remark 3.7 given at the end of the article.

2

Coercive models with convex composite inelastic constitutive equation

In this section we are going to formulate our initial-boundary value problem with an inelastic constitutive equation in the convex composite form. Moreover, assuming the coerciveness of the model, we will prove the existence and uniqueness result of global in


673

time, strong L2 solutions. Let us consider a pre-monotone model with the multifunction g having the form (3) in the quasi-static setting of the problem, meaning we will study the following system of equations divx T (x; t) = ¡ F (x; t) T (x; t) = D("(u(x; t)) ¡

Bz(x; t))

1 (rx u(x; t) + rTx u(x; t)) 2 ³ ´ zt (x; t) 2 @c (M ¯ ©) ¡ ½rz Ã("(x; t); z(x; t))

"(u(x; t)) =

½Ã("; z) =

1 D(" ¡ 2

Bz) ¢ (" ¡

(4)

1 Bz) + Lz ¢ z 2

with a convex and lower semicontinuous function M : RN ! R+ [ f+1g satisfying M (0) = 0 and 0 2 @M (0), and with a global Lipschitz C1;1 di®eomorphism © : RN ! RN (i.e. ©; ©¡1 ; D©; D©¡1 are global Lipschitz maps) satisfying ©(0) = 0. System (4) is considered with the following boundary condition of mixed type: the Dirichlet boundary condition on ¡1 » @ u(x; t) = gD (x; t) for x 2 ¡1 and t ¶ 0

(5)

and the Neumann boundary condition on ¡2 » @ T (x; t) ¢ n(x) = gN (x; t) for x 2 ¡2 and t ¶ 0

(6)

where n(x) is the exterior unit normal vector to the boundary @ at the point x. ¡1 and ¡2 are open in @ , and are disjoint, \smooth enough" sets satisfying @ = ¡1 [ ¡2 . Moreover the functions gD ; gN are given boundary data. Finally, the initial condition is in the form z(x; 0) = z 0 (x) (7) with a given initial data z 0 : ! RN . In this whole section we assume that the matrix L from the de¯nition of the free energy function Ã is positive de¯nite (which is equivalent to the coercivity of the model considered). We should mention here that the dissipation inequality (2) will not be used in the method presented below. Therefore our existence theory works for all models whose evolution is described by a system of type (4). The main idea to prove an existence result for system (4) is based on the so called partial Yosida approximation. This means that we are going to use the Yosida approximation of the maximal monotone operator @M . For ¸ > 0 let us de¯ne the function M¸ (z) = infN w2R

1 fjz ¡ 2¸

wj2 + M (w)g :

Hence, M¸ is a subquadratic, nonnegative map for which the gradient rM¸ is a global Lipschitz map and is the Yosida approximation of the operator @M . Using the function

674


M¸ we de¯ne a sequence of approximate problems divx T ¸ (x; t) = ¡ F (x; t) T ¸ (x; t) = D("(u¸ (x; t)) ¡

Bz ¸ (x; t))

1 (rx u¸ (x; t) + rTx u¸ (x; t)) 2 ³ ´ zt¸ (x; t) = r(M¸ ¯ ©) ¡ ½rz Ã("¸ (x; t); z ¸ (x; t))

"(u¸ (x; t)) =

½Ã("¸ ; z ¸ ) =

1 D("¸ ¡ 2

Bz ¸ ) ¢ ("¸ ¡

(8)

1 Bz ¸ ) + Lz ¸ ¢ z ¸ 2

with boundary conditions (5), (6) and with initial condition (7). Theorem 2.1 (existence for each approximation step). Let us assume that the boundary data gD ; gN and the external force F have the following regularity F 2 C1 (R+ ; L2 ( ; R3 )) 1

gD 2 C1 (R+ ; H 2 (¡1 ; R3 )) ;

(9) 1

gN 2 C1 (R+ ; H¡ 2 (¡2 ; R3 )) :

(10)

Then for all initial data z 0 2 L2 ( ; RN ) and for all ¸ > 0, there exists a unique, global in time solution (u¸ ; T ¸ ; z ¸ ) of problem (8) with boundary conditions (5), (6) and with initial condition (7). Moreover the solution has the following regularity u¸ 2 C1 (R+ ; H1 ( ; R3 )) ; T ¸ 2 C1 (R+ ; L2 ( ; S 3 )) ; z ¸ 2 C1 (R+ ; L2 ( ; RN )) : Proof 2.2 (of theorem 2.1). It is easy to see that only global Lipschitz nonlinearities arise in problem (8). Therefore the theorem follows from the general existence theorem for global Lipschitz models (see Theorem 3.1 in [2]). Next, we are going to prove some a priori estimates for the sequence of approximate solutions f(u¸ ; T ¸ ; z ¸ )g¸>0 . These estimates should be \so good" that we will be able to pass to the limit ¸ ! 0+ . Let us start with a lemma which presents the fundamental properties of convex composite °ows (compare with Proposition 2.4 p. 366 in [16]). Lemma 2.3. Let M : RN ! R+ [ f+1g be a convex, lower semicontinuous function, and © : RN ! RN be a C1;1 di®eomorphism. Then the following statements hold a) 9 C > 0

8 z1¤ 2 @c M ¯ ©(z1 ) 8 z2¤ 2 @c M ¯ ©(z2 ) ³ ´ z2 ) ¶ ¡ C kz1¤ kL2 ( ) + kz2¤ kL2 ( ) kz1 ¡ z2 k2L2 ( )

8 z1 ; z2 2 D(@c M ¯ ©) (z1¤ ¡

z2¤ ; z1 ¡

b) 9 C > 0 8 z 2 D(@c M ¯ ©) 8 z ¤ 2 @c M ¯ ©(z) 8 w 2 L2 ( ; RN ) Z Z M (©(w)) dx ¶ M (©(z)) dx + (z ¤ ; w ¡ z) ¡ Ckz ¤ kL2 ( ) kw ¡ zk2L2 ( ) :

(11)

(12)


675

Moreover the constant C in part a) and b) depends on the Lipschitz constants of ©, ©¡1 and D© and is independent of the function M . Proof 2.4 (of lemma 2.3). a) Denote by »1¤ and by »2¤ the elements from @M (©(zi )) such that zi¤ = D T ©(zi )»i¤ for i = 1; 2. Then we have (z1¤ ¡ z2¤ ; z1 ¡ z2 ) = (»1¤ ¡ »2¤ ; D©(z1 )(z1 ¡ z2 ))+(»2¤ ; (D©(z1 )¡ D©(z2 ))(z1 ¡ z2 )) : (13) Moreover there exists µ 2 [0; 1] such that (»1¤ ¡

»2¤ ; ©(z1 ) ¡

©(z2 )) = (»1¤ ¡

»2¤ ; D©(z1 + µ(z1 ¡

z2 ))(z1 ¡

z2 )) :

(14)

Adding and subtracting the left-hand side of (14) to and from the right-hand side of (13) and using the monotonicity of @M we obtain (z1¤ ¡

z2¤ ; z1 ¡

z2 ) ¶ (»1¤ ¡

»2¤ ; (D©(z1 ) ¡

D©(z1 + µ(z1 ¡

z2 )))(z1 ¡

z2 ))

+(»2¤ ; (D©(z1 ) ¡ D©(z2 ))(z1 ¡ z2 )) ³ ¶ ¡ C kD©(z1 ) ¡ D©(z1 + µ(z1 ¡ z2 ))kL2 ( ) kz1 ¡ z2 kL2 ( ) ´ +kD©(z1 ) ¡ D©(z2 )kL2 ( ) kz1 ¡ z2 kL2 ( ) (k»1¤ kL2 ( ) + k»2¤ kL2 ( ) ) ¶ ¡ 2CL(k»1¤ kL2 ( ) + k»2¤ kL2 ( ) )kz1 ¡

z2 k2L2 ( )

(15)

where L is the Lipschitz constant of the derivative D©. Finally from the de¯nition of »i¤ we get k»i¤ kL2 ( ) = kD ¡T ©(zi )zi¤ kL2 ( ) µ Ckzi¤ kL2 ( ) for i = 1; 2

(16)

where the constant C does not depend on zi (note that by assumption ©¡1 is a Lipschitz map). Inserting (16) into (15) ends the proof of part a). b) Similarly as in a) we denote by » ¤ the element from @M (©(z)) such that D T ©(z)» ¤ = z ¤ . From the de¯nition of the set @M (©(z)) we conclude that Z Z M (©(w)) dx ¶ M (©(z)) dx + (» ¤ ; ©(w) ¡ ©(z)) :

Then by the expansion ©(w) = ©(z) + D©(z)(w ¡ z) + R(kw ¡ zkL2 ( ) ) where R is a bounded real function we obtain Z Z M (©(w)) dx ¶ M (©(z)) dx + (D T ©(z)» ¤ ; w ¡ z) ¡ Ck» ¤ kL2 ( ) kw ¡ zkL2 ( ) :

Finally, a similar inequality to (16) completes the proof.

676


Our next goal is to prove L1 (L2 ) estimates for the time derivatives fzt¸ g¸>0 which will imply the weak-¤ convergence of a subsequence of this family of functions. The ¯rst step in this investigation is showing the following theorem yielding L2 (L2 ) boundedness of this family. Note that the proof of the theorem below works for gradient °ows only. Theorem 2.5 (L2 (L2 ) estimates for time derivatives). Let us suppose that the data gD ; gN and the external force F possess the regularity (9)+(10) and the initial datum z 0 2 L2 ( ; RN ) satis¯es ©(B T T 0 ¡ Lz 0 ) 2 D(@M ) where the initial stress is the unique solution of the following linear system of equations divx T 0 (x) = ¡ F (x; 0) T 0 (x) = D("(u0 (x)) ¡ u0 (x)j¡1 = gD (x; 0) ;

Bz 0 (x))

(17)

T 0 (x) ¢ n(x)j¡2 = gN (x; 0) :

Then the family fTt¸ ; zt¸ g¸>0 is L2 (L2 ) bounded and the following inequality holds: there exists C > 0 such that for all t > 0 Z Z tZ Z tZ ¡1 ¸ ¸ ¸ ¸ D Tt ¢ Tt dx dt + M¸ (©(¡ ½rz Ã(" ; z ))) dx + Lzt¸ ¢ zt¸ dx dt

µC

nZ

0

M (©(B T T 0 ¡

Z t³ 0 kFt k2L2 ( ) + kgD;t k2 1 Lz )) dx +

0

H 2 (¡1 )

0

+ kgN;t k2 ¡ 12 H

(¡2 )

´

o dt :

Proof 2.6 (of theorem 2.5). The proof is based on the fundamental property of the gradient °ows which yields L2 (L2 ) boundedness of time derivatives (see for example [4]). Z d M¸ (©(¡ ½rz Ã("¸ ; z ¸ ))) dx dt Z = D T ©(¡ ½rz Ã("¸ ; z ¸ )) ¢ rM¸ (©(¡ ½rz Ã("¸ ; z ¸ ))) ¢ (¡ ½rz Ã("¸ ; z ¸ ))t dx =

Z

=¡

zt¸ Z

¢ (B D

¡1

T

Tt¸

Tt¸

¢

Lzt¸ ) dx ¡

Tt¸

dx ¡

Z

=

Z

Tt¸

Lzt¸

¢

zt¸

¢

Bzt¸

dx +

dx ¡ Z

Z

Lzt¸ ¢ zt¸ dx

(18)

Tt¸ ¢ "¸t dx :

Next we integrate (18) with respect to time and get Z Z tZ Z tZ ¡1 ¸ ¸ ¸ ¸ D Tt ¢ Tt dx dt + M¸ (©(¡ ½rz Ã(" ; z ))) dx + Lzt¸ ¢ zt¸ dx dt =

Z

0

M¸ (©(B T T 0 ¡

Lz 0 )) dx +

Z tZ 0

0

Tt¸ ¢ "¸t dx dt :

(19)

To end the proof we have to estimate the last integral from the right-hand side of (19) only. The equation of motion and the boundary conditions imply Z tZ Z tZ Z t Z t ¸ ¸ ¸ ¸ h gN;t ; ut i¡2 dt + h Tt¸ ¢ n; gD;t i¡1 dt Tt ¢ "t dx dt = Ft ¢ ut dx dt + 0

0

0

0


µ

Z

t

0

+

kFt kL2 ( ) ku¸t kL2 ( ) Z

dt +

t

0

Z

677

t

kgN;t kH¡ 12 (¡ ) ku¸t kH1 ( ) dt 2

0

kgD;t kH 12 (¡ ) (kTt¸ kL2 ( ) + kdiv Tt¸ kL2 ( ) ) dt

(20)

1

where in the last two terms we have used the continuity of the trace operator. More1 over, the brackets h ; i¡i denote the duality forms between the spaces H 2 (¡i ; R3 ) and 1 H¡ 2 (¡i ; R3 ) i = 1; 2. Finally, we estimate the H1 norm of the velocity using the ellipticity of the equation of motion ³ ´ ¸ ¸ kut kH1 ( ) µ C kFt kH¡ 1 ( ) + kBzt kL2 ( ) + kgD;t k 12 + kgN;t k ¡ 12 (21) H (¡1 )

H

(¡2 )

where the constant C > 0 does not depend on ¸. The elasticity tensor is positive de¯nite and the matrix L is positive de¯nite too, therefore we can absorb the terms kTt¸ kL2 ( ) and kBzt¸ kL2 ( ) arising in the right-hand side of (20) by the left-hand side of (19). Moreover, the assumption ©(B T T 0 ¡ Lz 0 ) 2 D(@M ) implies that Z Z T 0 0 M¸ (©(B T ¡ Lz )) dx µ M (©(B T T 0 ¡ Lz 0 )) dx Z µ rM (©(B T T 0 ¡ Lz 0 )) ¢ ©(B T T 0 ¡ Lz 0 ) dx :

Thus, inserting (21) into (20) and then (20) into (19) immediately ends the proof. According to the proved L2 (L2 ) estimates for the time derivatives of the sequence fT ¸ ; z ¸ g¸>0 and to property (11) we are going to improve the estimates and obtain the boundedness in the space L1 (L2 ). Theorem 2.7 (L1 (L2 ) estimate for time derivatives). Suppose that the external force F and the boundary data gD ; gN have the regularity: for all T > 0 F 2 W 2;1 ((0; T ); L2 ( ; R3 )) 1

gD 2 W 3;1 ((0; T ); H 2 (¡1 ; R3 )) ; 1

gN 2 W 2;1 ((0; T ); H¡ 2 (¡2 ; R3 )) : Further, assume that the initial datum z 0 2 L2 ( ; RN ) satis¯es ©(B T T 0 ¡

Lz 0 ) 2 D(@M )

where the initial stress T 0 is de¯ned as the solution of system (17). Then for all T > 0 the approximate sequence f"¸t ; zt¸ g¸>0 satis¯es Z Z 1 1 ¸ ¸ ¸ ¸ ¸ ¸ E ("t ; zt )(t) = D("t ¡ Bzt ) ¢ ("t ¡ Bzt ) dx + Lzt¸ ¢ zt¸ dx µ C(T ) 2 2 where the constant C(T ) does not depend on ¸.

678


Proof 2.8 (of theorem 2.7). We start in the same manner as in the proof of Theorem 3.2 from [2]. Let us ¯x T > 0 and for h > 0 let us denote by ("¸h (t); zh¸ (t)) the shifted functions ("¸ (t + h); z ¸ (t + h)). Then for the di®erences ("¸h ¡ "¸ ; zh¸ ¡ z ¸ ) we obtain Z d n1 D("¸h ¡ "¸ ¡ Bzh¸ + z ¸ ) ¢ ("¸h ¡ "¸ ¡ Bzh¸ + z ¸ ) dx dt Z2 o 1 ¸ ¸ ¸ ¸ ¡ ¢ ¡ + L(zh z ) (zh z ) dx 2 Z ¸ D("¸h ¡ "¸ ¡ Bzh¸ + Bz ¸ ) ¢ ("¸h;t ¡ "¸t ¡ Bzh;t = + Bzt¸ ) dx

+ =

Z

Z

+

L(zh¸ ¡

(vh¸ ¡

¸ ¡ z ¸ ) ¢ (zh;t

v ¸ ) ¢ (Fh ¡

Z ³

D

T

©(wh¸ )

zt¸ ) dx = (using the notation v ¸ = u¸t )

F ) dx + h (Th¸ ¡ ¸

T ¸ ) ¢ n ; (vh¸ ¡ T

v ¸ ) i@ ¸

´

D ©(w ) ¢ rM¸ (©(w )) ¢ (wh¸ ¡

¢ rM¸ (©(w )h ) ¡

¸

w ¸ ) dx

where w ¸ = ¡ ½rz Ã("¸ ; z ¸ )). Using Lemma 2.3 and inserting the boundary data we arrive at the inequality d E ("¸h ¡ dt

"¸ ; zh¸ ¡

z ¸ )(t) µ h (Th¸ ¡

T ¸ ) ¢ n ; (@t gD;h ¡ @t gD ) i¡1 Z ¸ ¸ gN ) ; (vh ¡ v ) i¡2 + (vh¸ ¡ v ¸ ) ¢ (Fh ¡

+h (gN;h ¡

F ) dx

(22)

¸ kL2 ( ) + kzt¸ kL2 ( ) ) kB T (Th¸ ¡ +C(kzh;t

T ¸) ¡

L(zh¸ ¡

z ¸ )k2L2 ( )

where the constant C does not depend on ¸. Next, we integrate (22) with respect to t, divide the result by h2 and shift the di®erences operators in the ¯rst three terms of the right-hand side of (22) from the functions v ¸ and T ¸ onto the given data. Finally, we pass to the limit h ! 0+ and obtain E ("¸t ; zt¸ )(t) +2

Z

+2

0

+C

t

Z

t 0

kzt¸ (¿ )kL2 ( ) E ("¸t ; zt¸ )(¿ ) d¿

¸

kgN;tt (¿ )kH¡ 12 (¡ )kv (¿ )kH 12 (@ ) d¿ + 2 2

0

Z

µ

E ("¸t ; zt¸ )(0)

t ¸

kv (¿ )kL2 ( ) kFtt (¿ )kL2 ( ) d¿ + C + sup kgD;tt (t)kH 12 (¡ t2(0;T )

¸

+ sup kv (t)k t2(0;T )

L2 (

)

1)

n

Z

t

0

kT ¸ (¿ ) ¢ nkH¡ 12 (@ ) kgD;ttt (¿ )kH 21 (¡ ) d¿ 1

sup kgN;t (t)kH¡ 12 (¡

t2(0;T )

sup kT ¸ (t) ¢ nkH¡ 12 (@ )

t2(0;T )

sup kFt (t)k t2(0;T )

L2 (

)

o

2)

sup kv ¸ (t)kH 12 (@ )

t2(0;T )


679

where both constants C > 0 are independent of ¸. Estimating the boundary norms kv ¸ (t)k 21 and kT ¸ (t) ¢ nkH¡ 12 (@ ) as in (20) we arrive at the inequality H (@ ) E ("¸t ; zt¸ )(t)

µ

E ("¸t ; zt¸ )(0)

+C

Z

t

kzt¸ (¿ )kL2 ( ) E ("¸t ; zt¸ )(¿ ) d¿

0

+C(T ; ®) + ® sup kv ¸ (t)k2H1 ( ) + ® sup kT ¸ (t)k2L2 ( ) t2(0;T )

(23)

t2(0;T )

where ® > 0 is any positive number and the constant C(T; ®) does not depend on ¸. Applying the Gronwall inequality to (23) we get E ("¸t ; zt¸ )(t)

n Z t o³ µ exp C kzt¸ (¿ )kL2 ( ) d¿ E ("¸t ; zt¸ )(0) 0

+C(T ; ®) + ® sup kv

¸

t2(0;T )

(t)k2H1 ( )

+ ® sup kT t2(0;T )

¸

(t)k2L2 ( )

From Theorem 2.5 we conclude that Z t Z ´ 12 p³ t ¸ ¸ kzt (¿ )kL2 ( ) d¿ µ t kzt (¿ )k2L2 ( ) d¿ µ C(T ) : 0

´

:

(24)

(25)

0

Moreover, the H1 norm of the velocity can be estimated as in (21) and the L2 norm of the stress is estimated by kT

¸

(t)k2L2 ( )

µ 2kT

¸

(0)k2L2 ( )

+ 2t

Z

t 0

kTt¸ (¿ )k2L2 ( ) d¿ :

(26)

Finally, inserting (26) and (25) into (24) and choosing ® su±ciently small we obtain E ("¸t ; zt¸ )(t) µ C(T )(E ("¸t ; zt¸ )(0) + 1) : The assumption ©(B T T 0 ¡ Lz 0 ) 2 D(@M ) yields that the sequence E ("¸t ; zt¸ )(0) is bounded and the proof is complete. Theorem 2.7 allows us (in system (8)) to pass to the limit ¸ ! 0¤ in the weak-¤ topology of the space L1 (L2 ). Unfortunately, the inelastic constitutive equation (inclusion) is not linear and in general the weak convergence is not strong enough to ensure that the weak limit functions satisfy the original system (4). Therefore, we have to improve the weak convergence of the sequence of arguments of the nonlinear multifunction @c (M ¯ ©). Theorem 2.9 (strong convergence). Let us suppose that the boundary data gD ; gN , the external force F and the initial datum z 0 satisfy the assumptions from Theorem 2.7. Then for all T > 0 there exists a positive constant C(T ) such that 8 ¸; ¹ > 0

E ("¸ ¡

"¹ ; z ¸ ¡

z ¹ )(t) µ (¸ + ¹)C(T ) :

680


Proof 2.10 (of Theorem 2.9). By the energy method we arrive at the inequality d E ("¸ ¡ "¹ ; z ¸ ¡ z ¹ )(t) = dt Z n ¡ D T ©(w ¸ )rM¸ (©(w ¸ )) ¡

o D T ©(w ¹ )rM¹ (©(w ¹ )) ¢ (w ¸ ¡

w ¹ ) dx

where w i = ¡ ½rz Ã("i ; z i ) for i = ¸; ¹. Let us denote by J¸ the resolvent of the maximal monotone operator @M . Thus we conclude that

=¡

d E ("¸ ¡ dtZ ³

"¹ ; z ¸ ¡

z ¹ )(t) ¸

rM¸ (©(w )) ¡

¡

Z

¹

rM¸ (©(w ¸ )) ¡

¡

Z ³

¡

Z

´

rM¹ (©(w )) ¢ D©(w ¸ )(w ¸ ¡ ¸

rM¹ (©(w )) ¢ D©(w ) ¡

Z ³

=¡

³

¹

´ D©(w ) (w ¸ ¡ ¹

w ¹ ) dx

´ rM¹ (©(w ¹ )) ¢ (©(w ¸ ) ¡

©(w ¹ ))

´ rM¸ (©(w ¸ )) ¡ rM¹ (©(w ¹ )) ¢ ³ ´ D©(w ¸ ) ¡ D©(w ¸ + µ(w ¸ ¡ w ¹ ) (w ¸ ¡

³ rM¹ (©(w ¹ )) ¢ D©(w ¸ ) ¡

w ¹ ) dx

(27)

w ¹ ) dx

´ D©(w ¹ ) (w ¸ ¡

w ¹ ) dx

where µ 2 [0; 1]. The ¯rst integral from the right-hand side of (27) can be estimated as follows Z ³ ´ ¡ rM¸ (©(w ¸ )) ¡ rM¹ (©(w ¹ )) ¢ (©(w ¸ ) ¡ ©(w ¹ ))

=¡

Z ³

µ¡

rM¸ (©(w )) ¡

³

Z ³

¸

¹

rM¹ (©(w )) ¢

¸rM¸ (©(w ¸ )) + J¸ (©(w ¸ )) ¡

rM¸ (©(w ¸ )) ¡

´

¹rM¹ (©(w ¹ )) ¡

´ J¹ (©(w ¹ )) dx

´ ³ rM¹ (©(w ¹ )) ¢ ¸rM¸ (©(w ¸ )) ¡

1 (¸ + ¹)(krM¸ (©(w ¸ ))k2L2 ( ) + krM¹ (©(w ¹ ))k2L2 ( ) ) 4

(28)

´ ¹rM¹ (©(w ¹ )) dx

Inserting (28) into (27) and using the global Lipschitz property of D(©) we get d E ("¸ ¡ "¹ ; z ¸ ¡ z ¹ )(t) dt 1 µ (¸ + ¹)(krM¸ (©(w ¸ ))k2L2 ( ) + krM¹ (©(w ¹ ))k2L2 ( ) ) 4 ³ ´ +C krM¸ (©(w ¸ ))kL2 ( ) + krM¹ (©(w ¹ ))kL2 ( ) kw ¸ ¡

w ¹ k2L2 ( ) :

(29)


681

From the equality Mi (©(w i )) = D ¡T ©(w i )(zti ) for i = 1; 2 we see that the L2 norms krMi (©(w i ))kL2 ( ) are controlled in time. Moreover, there exists a constant C independent on ¸ and ¹ such that kw ¸ (t) ¡

w ¹ (t)k2L2 ( ) µ C E ("¸ ¡

"¹ ; z ¸ ¡

z ¹ )(t) :

(30)

Thus, inserting (30) into (29) yields d E ("¸ ¡ dt

"¹ ; z ¸ ¡

³ z ¹ )(t) µ C(T ) (¸ + ¹) + E ("¸ ¡

"¹ ; z ¸ ¡

Applying the Gronwall inequality immediately ends the proof.

´ z ¹ )(t) :

Theorem 2.11 (global existence for coercive convex composite models). Let us suppose that the boundary data gD ; gN , the external force F and the initial datum z 0 2 L2 ( ; RN ) satisfy all assumptions from Theorem 2.7. If the considered model is of convex composite type and is coercive then problem (4) with boundary conditions (5), (6) and initial condition (7) possesses a unique solution that is global in time (u; T ; z) 2 W1;1 ((0; T ); H1 ( ; R3 ) £ L2 ( ; S 3 £ RN )) for all T > 0 : Proof 2.12 (of Theorem 2.11). Let us denote by (u¸ ; T ¸ ; z ¸ ) the solution of problem (8) to the boundary conditions (5), (6) and to the initial condition (7). Then from Theorem (2.7) we have that: for all T > 0 there exists C(T ) > 0 such that sup fk"¸t kL2 ( ) ; kzt¸ kL2 ( ) g µ C(T )

t2(0;T )

and consequently sup fk"¸ kL2 ( ) ; kz ¸ kL2 ( ) g µ C(T ) :

t2(0;T )

From the elastic constitutive equation we conclude that the same property is shared by the sequence of plastic strains f"p;¸ g¸>0 and the sequence of stresses fT ¸ g¸>0 . Hence, we pass in system (8) to the weak-¤ limit in the space L1 (L2 ) and obtain that the limit functions satisfy the following system of equations divx T (x; t) = ¡ F (x; t) T (x; t) = D("(x; t) ¡

Bz(x; t))

1 (rx u(x; t) + rTx u(x; t)) 2 zt (x; t) = Â(x; t) "(x; t) =

where Â(x; t) = w ¡ lim¸!0+ r(M¸ ¯ ©)(¡ ½rz Ã("¸ ; z ¸ )). Moreover, the limit functions satisfy the boundary and initial conditions. To end the proof of the existence of a global in time solution it remains to verify that Â(x; t) 2 @c (M ¯ ©)(¡ ½rz Ã("(x; t); z(x; t))) for a:e: (x; t) 2 £ (0; T ) :

(31)

682


By the boundedness in L1 of the sequence fD T ©(¡ ½rz Ã("¸ ; z ¸ ))g¸>0 we can assume that the weak limit w ¡ lim¸!0+ rM¸ (©(¡ ½rz Ã("¸ ; z ¸ ))) = Â¤ exists. From Theorem 2.11 the sequence f¡ ½rz Ã("¸ ; z ¸ )g¸>0 = fB T T ¸ ¡ Lz ¸ g¸>0 converges strongly in L1 (L2 ) to ¡ ½rz Ã("; z). Consequently, the sequence f©(¡ ½rz Ã("¸ ; z ¸ ))g¸>0 converges strongly in L1 (L2 ) to ©(¡ ½rz Ã("; z)) (note that © is global Lipschitz). Thus, according to the weak-strong closedness of the graph of @M we conclude that ³ ´ ¤ 2 Â (x; t) @ M ©(¡ ½rz Ã("(x; t); z(x; t))) for a:e: (x; t) 2 £ (0; T ) : Finally, we are going to show that Â = D T ©(¡ ½rz Ã("; z))Â¤ which will immediately imply (31). D T ©(¡ ½rz Ã("¸ ; z ¸ )) ¢ rM¸ (©(¡ ½rz Ã("¸ ; z ¸ ))) = D T ©(¡ ½rz Ã("; z)) ¢ rM¸ (©(¡ ½rz Ã("¸ ; z ¸ ))) (32) ³ ´ + D T ©(¡ ½rz Ã("¸ ; z ¸ )) ¡ D T ©(¡ ½rz Ã("; z)) ¢ rM¸ (©(¡ ½rz Ã("¸ ; z ¸ ))) :

Passing to the weak-¤ limit in the space L1 (L2 ) we see that the ¯rst part in the righthand side of (32) converges to D T ©(¡ ½rz Ã("; z))Â¤ and the second part converges to 0. This completes the proof of the existence part. Let us assume that (u1 ; T 1 ; z 1 ) and (u2 ; T 2 ; z 2 ) are two solutions of problem (4) to the same external forces, to the same boundary data and to the same initial data. Then for the di®erences we get Z d 1 2 1 2 E (" ¡ " ; z ¡ z )(t) = ¡ (zt1 ¡ zt2 ) ¢ (¡ ½rz Ã("1 ; z 1 ) + ½rz Ã("2 ; z 2 )) dx dt µ (by Lemma 2:3 part a)) µ C(kzt1 kL2 ( ) + kzt2 kL2 ( ) ) kB T (T 1 ¡ µ C(kzt1 kL2 ( ) + kzt2 kL2 ( ) ) E ("1 ¡

T 2) ¡

"2 ; z 1 ¡

L(z 1 ¡

z 2 )k2L2 ( )

z 2 )(t)

Thus from the L2 (L2 ) boundedness of the time derivatives zt1 ; zt2 we get E ("1 ¡

"2 ; z 1 ¡

z 2 )(t) = 0 :

Using the coerciveness of the model we easily complete the proof.

3

Self-controlling convex composite models

In this entire section we assume that the convex function M does not take an in¯nite value and is di®erentiable. Thus, in this situation the operator @M is single-valued and @M (z) = frM (z)g. The goal of this section is a generalization of Theorem 2.11 in the case L ¶ 0 only. Thus we do not assume that our model is coercive. The main idea to prove existence of global solutions in this case is based on the so-called coercive approximation. This means that we approximate a noncoercive model by a sequence


683

of coercive problems and try to pass to the limit. This procedure does not work good for all noncoercive models. Nevertheless, there exists a subclass of models for which the coercive approximation works. Hence, let us suppose that the gradient rM possesses the self-controlling property, this means that 9 F : R+ £ R+ ! R+ continuous

8 z 2 D(r M )

kBr M (z)kL2 ( ) µ F (kLr M (z)kL2 ( ) ; kzkL2 ( ) ) : If the function F can be chosen in a±ne form, this means if kBr M (z)kL2 ( ) µ C(kLr M (z)kL2 ( ) + kzkL2 ( ) + 1) : then we say that r M possesses the linear self-controlling property. (This property possesses for example the Melan-Prager model. A mathematical analysis of this model can be found in [9]). These two de¯nitions are introduced in [13]. Moreover, for existence results in the case of self-controlling models of monotone type we refer to [6], [7], [9], [10]. Hence, following the idea of coercive approximation, for a ¯xed natural number k we de¯ne the sequence of approximate problems divx T k (x; t) = ¡ F (x; t) ³ T k (x; t) = D "(uk (x; t)) ¡

´ 1 Bz k (x; t) + "(uk (x; t)) k

1 (rx uk (x; t) + rTx uk (x; t)) 2 ³ ´ k k k k zt (x; t) = r(M ¯ ©) ¡ ½rz Ã (" (x; t); z (x; t))

"(uk (x; t)) =

½Ã k ("k ; z k ) =

1 D("k ¡ 2

Bz k ) ¢ ("k ¡

(33)

1 1 D"k ¢ "k Bz k ) + Lz k ¢ z k + 2 2k

with the boundary conditions uk (x; t) = gD (x; t) for x 2 ¡1 and t ¶ 0

(34)

T k (x; t) ¢ n(x) = gN (x; t) for x 2 ¡2 and t ¶ 0

(35)

and with the initial condition z(x; 0) = z 0 (x) :

(36)

As for problem (8) from the previous section we need here initial values for the stress and the displacement. Let us denote the unique solution of the linear problem by T k;0 and uk;0 divx T k;0 (x) = ¡ F (x; 0) ³ T k;0 (x) = D "(uk;0 (x)) ¡ uk;0 (x)j¡1 = gD (x; 0) ;

´ 1 Bz 0 (x) + "(uk;0 (x)) k

T k;0 (x) ¢ n(x)j¡2 = gN (x; 0) :

(37)

684


Thus the initial values T k;0 ; uk;0 are not constant in the approximation procedure. Using the standard elliptic estimates for the di®erences uk;0 ¡ ul;0 we conclude that kuk;0 ¡

³1

+

Bz 0 (x)

´

ul;0 kH1 ( ) µ C

k

1´ : l

Consequently, if k ! 1 then the sequence fuk;0 g converges in the space H1 ( ; R3 ) to u0 and T k;0 ! T 0 in the space L2 ( ; S 3 ) where (u0 ; T 0 ) is the unique solution of the problem divx T 0 (x) = ¡ F (x; 0) ³ T 0 (x) = D "(u0 (x)) ¡ u0 (x)j¡1 = gD (x; 0) ;

(38)

T 0 (x) ¢ n(x)j¡2 = gN (x; 0) :

To prove the existence of global in time solutions for system (33) for each approximation step we repeat the idea from Section 2 and study the partial Yosida approximation of problem (33) divx T k;¸ (x; t) = ¡ F (x; t) ³ T k;¸ (x; t) = D "(uk;¸ (x; t)) ¡

´ 1 Bz k;¸ (x; t) + "(uk;¸ (x; t)) k

1 (rx uk;¸ (x; t) + rTx uk;¸ (x; t)) 2 ³ ´ ztk;¸ (x; t) = r(M¸ ¯ ©) ¡ ½rz Ã k ("k;¸ (x; t); z k;¸ (x; t))

"(uk;¸ (x; t)) =

½Ã k ("k;¸ ; z k;¸ ) =

1 D("k;¸ ¡ 2

Bz k;¸ ) ¢ ("k;¸ ¡

(39)

1 1 D"k;¸ ¢ "k;¸ Bz k;¸ ) + Lz k;¸ ¢ z k;¸ + 2 2k

with boundary conditions (34), (35) and with initial condition (36). Assuming that the boundary data gD ; gN and the external force F have the regularity required in (9) and in (10) the solvability and uniqueness of solutions for system (39) follows in the same manner as for problem (8) studied in Section 2. Next, we are going to prove a priori estimates for the sequence f(uk;¸ ; T k;¸ ; z k;¸ g¸>0;k2N . The idea is to improve Theorems 2.5, 2.7 and 2.9 in the noncoercive but linear self-controlling case. Theorem 3.1 (a priori estimates). Let us suppose that the boundary data gD ; gN and the external force F possess the regularity required in Theorem 2.11. Moreover, assume that the initial data z 0 2 L2 ( ; RN ) satis¯es 8k2N fr M (©(B T T^ k;0 ¡

©(B T T^ k;0 ¡

Lz 0 ) 2 D(r M )

Lz 0 ))g is bounded in L2 ( ; RN )

where the initial value T^ k;0 is de¯ned by D("(uk;0 )¡ Bz 0 ) and the initial displacement uk;0 in the k-th approximation step is de¯ned as the unique solution of system (37). Then for all single-valued, convex composite models possessing the linear self-controlling property


685

it holds that: for allZ T Z> 0 there exists a positive constant C(T ) such that for all t µ T ³ ´ t 1 k;¸ D ¡1 Ttk;¸ ¢ Ttk;¸ + Lztk;¸ ¢ ztk;¸ + D"k;¸ ¢ i) " dx dt µ C(T ) t k t Z0 ³ 1 k;¸ k;¸ ´ k;¸ k;¸ k;¸ k;¸ k;¸ D("k;¸ ¡ ¢ ¡ ¢ D" ¢ "t ii) Bz ) (" Bz ) + Lz z + dx µ C(T ) t t t t t t k t iii) kT k;¸ ¡ T k;¹ k2L2 ( ) + kLz k;¸ ¡ Lz k;¹ k2L2 ( ) µ C(T )(¸ + ¹) : Proof 3.2 (of Theorem 3.1). First, we have to mention that the initial function T^ k;0 arising in the theorem is not equal to the initial stress T k;0 obtained from (37). Nevertheless, we see that for k ! 1 we obtain kT^ k;0 ¡ T k;0 kL2 ( ) ! 0. i) Starting in the same manner as in Theorem 2.5 we arrive at the equality Z Z tZ k k;¸ k;¸ D ¡1 T^tk;¸ ¢ T^tk;¸ dx dt M¸ (©(¡ ½rz Ã (" ; z ))) dx + 0 Z tZ + Lztk;¸ ¢ ztk;¸ dx dt =

Z

0

T

^ k;0

M¸ (©(B T

¡

0

Lz )) dx +

Z tZ 0

T^tk;¸ ¢ "k;¸ t dx dt

(40)

where T^ k;¸ = D("k;¸ ¡ Bz k;¸ ). Next, we have to estimate the last integral in the right-hand side of (40) Z tZ Z tZ Z Z 1 t k;¸ k;¸ k;¸ k;¸ k;¸ ^ D"k;¸ Tt ¢ "t dx dt = Tt ¢ "t dx dt ¡ t ¢ "t dx dt k 0 0 0 Z tZ Z t h T k;¸ ¢ n; uk;¸ = Ft ¢ uk;¸ t dx dt + t i@ dt 0 0 Z Z 1 t k;¸ ¡ D"k;¸ t ¢ "t dx dt : k 0 Inserting the boundary values and using an estimate similar to (21) we conclude that Z tZ Z tZ 1 k;¸ k;¸ D"k;¸ ¢ "k;¸ T^t ¢ "t dx dt µ ¡ t t dx dt k 0 0 +C(T ; ®) + ®kBztk;¸ k2L2 ( ) + ®kTtk;¸ k2L2 ( )

(41)

where ® is any positive number and the constant C(T ; ®) does not depend on ¸ and k. Next, by the linear self-controlling property we have kBztk;¸ k2L2 ( ) µ C(kLztk;¸ k2L2 ( ) + kD("k;¸ ¡

Bz k;¸ )k2L2 ( ) + kLz k;¸ k2L2 ( ) + 1) : (42)

Then we estimate the norms kD("k;¸ ¡ kD("k;¸ ¡

Bz k;¸ )k2L2 ( )

Bz k;¸ )k2L2 ( ) and kLz k;¸ k2L2 ( ) as follows Z t k;0 2 ^ µ 2kT kL2 ( ) + 2t kD("k;¸ ¡ Bztk;¸ )k2L2 ( ) dt t

kLz k;¸ k2L2 ( ) µ 2kLz 0 k2L2 ( ) + 2t

Z

0

t

0

kLztk;¸ k2L2 ( ) dt :

686


Next, we insert the last two inequalities into (42) and the result into (41). Choosing ® small enough we end the proof in the same manner as in Theorem 2.5. ii) If we de¯ne the total energy E k ("; z) associated with the k-th approximation step by Z Z Z 1 1 1 k E ("; z)(t) = D(" ¡ Bz) ¢ (" ¡ Bz) dx + D" ¢ " dx Lz ¢ z dx + 2 2 2k then in the same manner as in the proof of Theorem 2.7 we arrive at an inequality similar to (24) n Z t o³ k;¸ k k;¸ k;¸ E ("t ; zt )(t) µ exp C kztk;¸ (¿ )kL2 ( ) d¿ E k ("k;¸ t ; zt )(0) + C(T; ®) 0 ´ +® sup kv k;¸ (t)k2H1 ( ) + ® sup kT k;¸ (t)k2L2 ( ) ; (43) t2(0;T )

t2(0;T )

where ® is any positive number and the constant C(T ; ®) does not depend on ¸ and k. Next, as in part i) we estimate the H1 norm of the velocity using the ellipticity of the equation of motion kv k;¸ k2H1 ( ) µ C(kBz k;¸ k2L2 ( ) + C(T )) ; where the constants C and C(T ) do not depend on k and ¸. Then taking into account (42), estimating the L2 norm of the stress as in (26) and choosing ® su±ciently small we end the proof of the part ii). iii) It is not di±cult to see that in the proof of Theorem 2.9 we have not used that the model is coercive. Therefore, for ¯xed k 2 N we can repeat the proof here and obtain the statement of iii). By Theorem 3.1 we can pass to the limit ¸ ! 0+ in system (39) and obtain the existence and uniqueness of global in time solutions for system (33). Next, we are going to pass to the limit k ! 1. According to the results i) and ii) from Theorem 3.1 we see that to prove existence of global solutions for system (4) with the inelastic constitutive equation possessing the linear self-controlling property it remains to obtain the strong L2 convergence of the sequence f¡ ½rz Ã k ("k ; z k )g (here ("k ; z k ) is the solution of problem (33)). Theorem 3.3 (strong convergence of f¡ ½rz Ã k ("k ; z k )g). Let us suppose that the given data gD ; gN ; F and z 0 satisfy all assumptions from Theorem 3.1. If we denote by f("k ; T k ; z k )g the sequence of solutions of problem (33) and by ("; T ; z) the weak-¤ limit of this sequence, then Tk ¡ ! T ;

Lz k ¡ ! Lz in the L1 (L2 ) topology:

Proof 3.4 (of Theorem 3.3). The idea is to prove that the sequences fT k g and fLz k g are L2 Cauchy sequences: Z d n1 D("k ¡ "l ¡ Bz k + Bz l ) ¢ ("k ¡ "l ¡ Bz k + Bz l ) dx dt 2


Z o 1 k l k l ¡ ¢ ¡ + L(z z ) (z z ) dx 2 Z D("k ¡ "l ¡ Bz k + Bz l ) ¢ ("kt ¡ = =

Z ¡

"lt

Bztk ¡

+

Bztl ) dx

+

D("k ¡

Z

"l ¡

1 ( D"k ¡ k

Z

687

L(z k ¡

z l ) ¢ (ztk ¡

ztl ) dx

1 Bz k + Bz l + "k ¡ k 1 l D" ) ¢ ("kt ¡ l

µ (by Lemma 2:3 part a))

1 l " ) ¢ ("kt ¡ "lt) dx l Z ³ l "t ) dx ¡ B T (T^ k ¡ T^l ) ¡

L(z k ¡

µ C(kztk kL2 ( ) + kztl kL2 ( ) )kB T (T^ k ¡

T^ l ) ¡

L(z k ¡

´ z l ) ¢ (ztk ¡

ztl ) dx

z l )k2L2 ( )

1 1 +C( + )(k"kt kL2 ( ) + k"lt kL2 ( ) )(k"k kL2 ( ) + k"l kL2 ( ) ) k l where Tî = D("i ¡ Bz i ) for i = k; l and the constants C arising in the right-hand side do not depend on k and l. Observing that kB T (T^ k ¡ T^ l ) ¡ L(z k ¡ z l )k2L2 ( ) nZ µC D("k ¡ "l ¡ Bz k + Bz l ) ¢ ("k ¡ Z o k l k l + L(z ¡ z ) ¢ (z ¡ z ) dx

"l ¡

Bz k + Bz l ) dx

using the L1 (L2 ) boundedness of the sequences fztk g, f"kt g, f"k g and applying the Gronwall inequality we conclude that: for t µ T it holds Z 1 D("k ¡ "l ¡ Bz k + Bz l ) ¢ ("k ¡ "l ¡ Bz k + Bz l ) dx 2 Z ³ 1 1´ 1 + L(z k ¡ z l ) ¢ (z k ¡ z l ) dx µ + C(T ) : 2 k l Hence, the last inequality implies that the sequences fT^ k g and fLz k g are L1 (L2 ) Cauchy sequences. Moreover, we see that k ¡1 D"k ! 0 strongly in L1 (L2 ) and the sequence of stresses fT k g is a L1 (L2 ) Cauchy sequence, too. Theorem 3.5 (existence for convex composite, linear self-controlling models). Let us suppose that the boundary data gD ; gN and the external force F possess the regularity required in Theorem 2.7 and the initial datum z 0 2 L2 ( ; RN ) satis¯es 8k2N fr M (©(B T T^ k;0 ¡

©(B T T^ k;0 ¡

Lz 0 ) 2 D(r M )

Lz 0 ))g is bounded in L2 ( ; RN )

where T^ k;0 = D("(uk;0 ) ¡ Bz 0 ) and the initial displacement uk;0 is the unique solution of system (37). Moreover, suppose that the considered model is single-valued, convex

688


composite and the gradient r M has the linear self-controlling property. Then problem (4) with boundary conditions (34), (35) and with initial condition (36) possesses a unique, global in time solution (u; T ; z) 2 W1;1 ((0; T ); H1 ( ; R3 ) £ L2 ( ; S 3 £ RN )) for all T > 0 :

Proof 3.6 (of Theorem 3.5). The theorem can be proved in the same manner as Theorem 2.11. Remark 3.7. Theorem 3.5 yields existence of global in time solutions for linear selfcontrolling models, only. This can be improved for the entire class of self-controlling models, provided that the boundary data and the external force satisfy the so called safe-load condition. For the de¯nition of this condition and for results in the monotone case we refer to [13]. Remark 3.8. For monotone models of the gradient type the strong convergence of the arguments of the nonlinear constitutive function (multifunction) can be proved using the convexity of M . (For details we refer to [14], [9]). Unfortunately, this method does not work (in the general situation) for convex composite models. This can be observed from the inequality Z Z k k k k 2 (zt ; w ¡ w ) ¡ Ckzt kL2 ( ) kw ¡ w kL2 ( ) µ M (©(w)) dx ¡ M (©(w k )) dx

where w = ¡ ½rz Ã("; z) and w k = ¡ ½rz Ã k ("k ; z k ). The sequence w k converges only weakly to w and we cannot obtain (in the general situation) that Z Z M (©(w)) dx µ lim inf M (©(w k )) dx :

k!1

References [1] H.-D. Alber: Materials with memory, Lecture Notes in Math., Vol. 1682, Springer, Berlin Heidelberg New York, 1998. [2] H.-D. Alber and K. CheÃlmi¶ nski: \Quasistatic problems in viscoplasticity theory I: Models with linear hardening", A part of the monograph: I. Gohberg et al.: Operator theoretical methods and applications to mathematical physics. The Erhard Meister memorial volume, BirkhÄauser, Basel, in print. [3] G. Anzellotti and S. Luckhaus: \Dynamical evolution of elasto-perfectly plastic bodies", Appl. Math. Optim., Vol. 15, (1987), pp. 121{140. [4] J.P. Aubin and A. Cellina: Di®erential inclusions. Springer, Berlin Heidelberg New York, 1984. [5] M. Brokate: \Elastoplastic constitutive laws of nonlinear kinematic hardening type", Technical Report 97-14, Berichtsreihe des Mathematischen Seminars, Kiel, 1997.


689

[6] K. CheÃlmi¶ nski: \Coercive limits for a subclass of monotone constitutive equations in the theory of inelastic material behaviour of metals", Mat. Stos., Vol. 40, (1997), pp. 41{81. [7] K. CheÃlmi¶ nski: \On self-controlling models in the theory of inelastic material behaviour of metals", Contin. Mech. Thermodyn., Vol. 10, (1998), pp. 121{133. [8] K. CheÃlmi¶ nski: \Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations", Math. Meth. Appl. Sci., Vol. 25, (2002), pp. 1195{1230. [9] K. CheÃlmi¶ nski: \Coercive approximation of viscoplasticity and plasticity", Asymptotic Analysis, Vol. 26, (2001), pp. 105{133. [10] K. CheÃlmi¶ nski: \On noncoercive models in the theory of inelastic deformations with internal variables", ZAMM, Vol. 81, Suppl. 3, (2001), pp. 595{596. [11] K. CheÃlmi¶ nski: \On quasistatic models in the theory of inelastic deformations", Proceedings in Applied Mathematics, Vol. 1, (2002), pp. 401{402. [12] K. CheÃlmi¶ nski and P. Gwiazda: \Monotonicity of operators of viscoplastic response: application to the model of Bodner-Partom", Bull. Polish Acad. Sci.: Tech. Sci., Vol. 47, (1999), pp. 191{208. [13] K. CheÃlmi¶ nski and P. Gwiazda: \Nonhomogeneous initial{boundary value problems for coercive and self{controlling models of monotone type", Contin. Mech. Thermodyn., Vol. 12, (2000), pp. 217{234. [14] K. CheÃlmi¶ nski and Z. Naniewicz: \Coercive limits for constitutive equations of monotone-gradient type", Nonlinear Anal., Vol. 48, (2002), pp. 1197{1214. [15] F.H. Clarke: Optimization and nonsmooth analysis, CMR Universit¶e de Montr¶eal, Montr¶eal, 1989. [16] S. Guillaume: \Subdi®erential evolution inclusion in nonconvex analysis", Positivity, Vol. 4, (2000), pp. 357{395. [17] P. Gwiazda: \Non-homogeneous boundary value problem for the Chan-BodnerLinholm model", Math. Methods Appl. Sci., Vol. 23:11, (2000), pp. 1011{1022. [18] B. Halphen and Nguyen Quoc Son: J. M¶ec., Vol. 14, (1975), pp. 39{63.

\Sur les matµeriaux standards g¶en¶eralis¶es",

[19] P.-M. Suquet: \Evolution problems for a class of dissipative materials", Quart. Appl. Math., Vol. 38, (1980), pp. 391{414. [20] R. Temam: \A generalized Norton-Ho® model and the Prandtl-Reuss law of plasticity", Arch. Rational Mech. Anal., Vol. 95, (1986), pp. 137{183.

CEJM 4 (2003) 690{705

Absolutely continuous functions of several variables and di® eomorphisms Stanislav Hencl¤ , Jan Mal¶yy Department KMA of the Faculty of Mathematics and Physics Charles University Sokolovsk¶ a 83, CZ-18675 Praha 8, Czech Republic

Received 25 June 2003; accepted 12 September 2003 Abstract: In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under di¬eomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls. ® c Central European Science Journals. All rights reserved. Keywords: absolute continuity in several variables, change of variables MSC (2000): 26B30

1

Introduction

Absolutely continuous functions of one variable are admissible transformations for the change of variables in Lebesgue integration. Recently J. Mal¶ y [4] introduced a class of d-absolutely continuous functions giving a d-dimensional analog of the notion of absolute continuity from this point of view. A more general de¯nition of d-absolutely continuous functions suggested by L. Zaj¶³·cek has been introduced in [2]. Suppose that » Rd is an open set and 0 < ¸ µ 1. We say that a function f : ! Rm is d; ¸-absolutely continuous if for each " > 0 there is ± > 0 such that for each pairwise disjoint ¯nite family fB(xi ; ri )g of balls in we have X X L d (B(xi ; ri )) < ± =) (oscB(xi ;¸ri ) f )d < " : y

¤

i

E-mail: [email protected]¬.cuni.cz E-mail: [email protected]¬.cuni.cz

i

S. Hencl, J. Malý / Central European Journal of Mathematics 4 (2003) 690{705

691

Absolute continuity from [4] coincides with d; 1-absolute continuity. It has been proved in [2] and [4] that d; ¸-absolute continuity implies continuity, weak di®erentiability with gradient in Ld , di®erentiability almost everywhere and a formula on change of variables. It has been shown in [2] that d; ¸1 -absolutely continuous functions are the same as d; ¸2 absolutely continuous functions for 0 < ¸1 < ¸2 < 1, and, moreover, that we can pass from balls to cubes in the de¯nition of d; ¸-absolutely continuous functions (0 < ¸ < 1) without a®ecting the resulting class of functions. Also, for 0 < ¸ < 1, the d; ¸-absolutely continuous functions are stable with respect to bilipschitz, or even quasiconformal, change of variables (see [2] and [3] for details). This °exibility is broken in the limiting case of ¸ = 1. M. CsÄornyei [1] invented a method of how to construct complicated examples of functions relevant to questions of d-absolute continuity. She proved that there exists a 2; 1-absolutely continuous function with respect to balls, which is not 2; 1-absolutely continuous with respect to cubes. In this paper, we develop another branch of this method to produce further examples. In Sections 3 and 4 we construct a 2; 1-absolutely continuous function with respect to cubes, which is not 2; 1-absolutely continuous with respect to balls. In Sections 3, 4, and 5 we prove that there exists a function f and a di®eomorphism F : R2 ! R2 such that f~ = f ¯ F ¡1 is 2; 1-absolutely continuous but f itself is not 2; 1-absolutely continuous. Section 6 contains a proof that AC1d functions are stable under a C 1;1 bilipschitz change of variables. This implies that our example in Section 5 is quite sharp.

2

Preliminaries

We denote by B(x; r) the d-dimensional open ball with the center x and diameter r and by B(x; r) the corresponding closed ball. Q(x; r) stands for the open cube with sides parallel to coordinate axes and with the center x and side of length 2r. Throughout the paper we use the letter B only for balls and the letter Q only for cubes. Throughout the paper we consider an open set » Rd ; d > 1. We denote by oscA f the oscillation of f : ! Rm over the set A » , which is the diameter of f (A). We denote by L d the d-dimensional Lebesgue measure. We use the symbol j : : : j for the Euclidean distance. We denote by ei the i-th base vector in Rd . A function F : Rd ! Rd is said to be a di®eomorphism if there exists an inverse F ¡1 and both F and F ¡1 are continuously di®erentiable. Let K > 0. We say that a function F : Rd ! Rd is K-Lipschitz if jF (x) ¡ F (y)j µ Kjx ¡ yj for every x; y 2 Rd . The function F is Lipschitz if there is K > 0 such that F is K-Lipschitz. We say that a function F : Rd ! Rd is bilipschitz (K-bilipschitz) if both F and F ¡1 are Lipschitz (K-Lipschitz). We denote by C 1 the set of all continuously di®erentiable functions and by C 1;1 the set of all functions whose ¯rst order derivatives are Lipschitz. Given an open set » Rd and a function f : ! Rm we de¯ne the d; ¸-variation of f on by

692


V¸d (f; ) = sup

nX

(oscB(xi ;¸ri ) f )d : fB(xi ; ri )g is a pairwise

i

disjoint family of balls in

o

:

We denote by BV¸d ( ) the class of all functions such that V¸d (f; ) < 1. We de¯ne the space AÇd ( ) as the family of all d; ¸-absolutely continuous functions in BV¸d ( ). Similarly we de¯ne classes Q-AÇd ( ) and Q-BV¸d ( ) of \absolutely continuous functions or BV functions with respect to cubes". For example, we say that a function f : ! Rm is Q; d; ¸-absolutely continuous if for each " > 0 there is ± > 0 such that for each pairwise disjoint ¯nite family fQ(xi ; ri )g of cubes in we have X X L d (Q(xi ; ri )) < ± =) (oscQ(xi ;¸ri ) f )d < " : i

i

We use the convention that C denotes a generic positive constant which may change from expression to expression. In the following lemma and in the rest of the paper we will denote B u (z; ½) := B(z + ½2 e2 ; ½2 ): Lemma 2.1. Let ® 2 R and ¯; r > 0. Suppose that 2¯ r µ ® 2 :

(1)

Let z 2 R2 and u = z + ®e1 + ¯e2 . Then B(u; ¯) \ B u (z; r) = ;:

(2)

Proof. We may assume that z = 0. Let z u = 12 re2 be the center of B u (z; r). Then by (1) ju ¡

z u j2 = ® 2 + (r=2 ¡

¯)2 ¶ (r=2 + ¯)2 :

Hence the balls B(u; ¯) and B u (z; r) = B(z u ; r=2) are disjoint. Lemma 2.2. Let ®; ¯ > 0. Suppose 3 ¯ < ®: 2

(3)

Let z 2 R2 and u = z + ® e1 + ¯ e2 . Let Q = Q(c; a) be a cube such that c1 ¶ z1 , c2 ¶ z2 , Q \ B(z; ¯2 ) 6= ; and Q n B(z; 8®) 6= ;. Then ³ ¯´ » Q: B u; 2

(4)

Proof. We may assume that z = 0. Since Q \ B(z; ¯) 6= ; and Q n B(z; 8®) 6= ;, we have 4a > diam Q ¶ 8® ¡

¯:

(5)


693

Also Q \ B(z; ¯2 ) 6= ; implies that c1 ¡

aµ

¯ ; 2

c2 ¡

aµ

¯ : 2

(6)

By (3), (5) and (6), c2 ¡ c1 ¡ Clearly

aµ

aµ

¯ ; 2

¯ µ®¡ 2

³

¯´ » B u; 2

and from (7) it follows that µ ®¡

3 8® ¡ ¯ ¯µ < a µ c2 + a; 2 4 ¯ ¯ 8® ¡ ¯ ; ®+ µ < a µ c1 + a: 2 2 4 µ

®¡

¯ ¯ ;® + 2 2

¶

¶

¯ ¯ ;® + 2 2

£

µ

¯ 3 ; ¯ 2 2

£

¶

µ

¯ 3 ; ¯ 2 2

(7)

¶

» Q(c; a) :

Lemma 2.3. Let ®; ¯; °; t; r; R > 0 and c; z; w 2 R2 . Suppose that 16¯ µ ° µ ®;

15®2 µ °R;

and

® µ t µ 2®:

(8)

Let c1 ¶ z 1 ;

c2 ¶ z 2

and

w = z + te1 + °e2 :

(9)

Suppose that B(c; r) \ B(z; ¯) 6= ; and B(c; r) n B(z; R) 6= ; :

(10)

B(w; ¯) » B(c; r):

(11)

Then

Proof. We may assume that z = 0. From (10), (9) and (8) we obtain jcj < r + ¯;

jwj2 = t2 + ° 2 µ 5® 2

(12)

and R µ 2r + ¯:

(13)

15®2 µ °R µ °(2r + ¯) µ ®(2r + ¯);

(14)

Since by (8) we have 7® < 12 (15® ¡

¯) µ r

(15)

and 16¯ µ ® < r:

(16)

694


Now, we distinguish two cases. If 2c1 µ r and 2c2 µ r, then jcj2 µ r2 =2. Therefore, since w ¢ c ¶ 0, (12), (15) and (16) yield cj2 ¡

jw ¡

(r ¡

¯)2 = jwj2 + jcj2 ¡ 2w ¢ c ¡ r 2 ¡ ¯ 2 + 2r¯ r2 µ 5® 2 + ¡ r 2 ¡ ¯ 2 + 2r¯ 2 ¡ ¢ r2 < 5®2 ¡ + 2r¯ < 19 ¡ 12 + 18 r2 < 0: 2

If 2c1 ¶ r or 2c2 ¶ r, then in view of (9) and (8) we obtain 2w ¢ c = 2tc1 + 2°c2 ¶ °r:

(17)

Therefore (12), (17), (14), (16) and (8) give us jw ¡

cj2 ¡

(r ¡

¯)2 = jwj2 + jcj2 ¡

2w ¢ c ¡

µ 5® 2 + (r + ¯)2 ¡ µ 5® 2 + 4r¯ ¡ µ

(r ¡

°r ¡

¯)2

(r ¡

¯)2

°r

1 3

°(2r + ¯) + 4r¯ ¡ °r ´ ¡2 1 1 µ 3 + 48 + 4 ¡ 1 °r < 0: In both cases we obtain jw ¡

cj + ¯ µ r

which implies (11). Lemma 2.4. Let 0 < ½ µ r µ r~ µ 1=6. Denote Q = [¡ 1; 1] £ [¡ 2; 2], B = B([0; r]; ½) and B~ = B([0; r~]; ½). Then there exists a (1 + 6~ r)-bilipschitz C 1 mapping H : R2 ! R2 such that ( x for x 2 R2 n Q H (x) = (18) x + (~ r ¡ r)e2 for x 2 B : ~ Hence H (B) = B. Proof. Clearly there exists a C 1 function ’ : R ! [0; 1] such that ( 1 for t 2 [¡ r; r] ’(t) = 0 for t 2 = [¡ 1; 1] : and

4 j’0 (t)j µ : 3

Set G(x1 ; x2 ) = (~ r¡

r)’(x1 )’

H(x) = x + G(x)

³x ´ 2

2

e2 ;


695

It is easily seen that H is C 1 , and (18) is satis¯ed. We have ³4 4´ jrGj µ r~ + = 2~ r: 3 6

Now we apply an elementary observation that any "-Lipschitz perturbation of the identity with 0 < " < 1 is (1 ¡ ")¡1 bilipschitz. Taking into account that r~ µ 16 we obtain that the bilipschitz constant of H is estimated by (1 ¡ 2~ r)¡1 µ 1 + 6~ r as desired.

3

Construction

In this section we will construct two functions f; f~ : R2 ! R such that f 2 = AC12 (R2 ), f 2 Q-AC12 (R2 ) and f~ 2 AC12 . In the next section we will construct a bilipschitz di®eomorphism F : R2 ! R2 such that f~ = f ¯ F ¡1 . Thus we will obtain the following theorems. Theorem 3.1. There exists a function f : R2 ! R such that f 2 = AC12 (R2 ) but f 2 Q-AC12 (R2 ). Theorem 3.2. There exists a function f : R2 ! R and a bilipschitz di®eomorphism F : R2 ! R2 such that f 2 = AC12 but f~ = f ¯ F ¡1 2 AC12 . Our construction is performed in the Euclidean space R2 . For every n 2 N we write !n =

1 2n¡1 n!

and ¿n = 4n n! (n + 1)! :

For each n = 0; 1; 2; : : : we de¯ne inductively ¯nite sets Zn , Z~n of points, a radius rn > 0, continuous functions fn , f~n on R2 and some other objects. We set Z0 = Z~0 = f0g;

r0 = 1;

f0 = f~0 ² 0:

Let n ¶ 1 and assume that all objects assigned to 0; : : : ; n ¡ 1 are de¯ned. We construct ¯nite sequences f¯n;k : k = 0; 1 : : : ; ng, f°n;k : k = 1; : : : ; ng, and f®n;k : k = 1; : : : ; ng so that ¯n;0 = rn¡1 ; (19) 16¯n;k µ °n;k =

1 ®n;k ; 15®2n;k µ °n;k ¯n;k¡1 ; 36n3 2¯n;k rn¡1 µ ®2n;k :

Next we set

¯n;n ; 2 ½ ¾ ¡ i ¢ i z+p 1+ ®n;k e1 + (¡ 1) ¯n;k e2 ; 3n rn =

Zn =

[

z2Zn¡ 1 k=1;:::;n i=0;:::;2n+1 p2f¡1;1g

k = 1; : : : ; n;

(20) (21) (22) (23)

696


Z~n =

[

z2Zn¡ 1 k=1;:::;n i=0;:::;2n+1 p2f¡1;1g

½

¾ ¡ i ¢ i z+p 1+ ®n;k e1 + (¡ 1) °n;k e2 ; 3n

(24)

so that the sets Zn ; Z~n consist each of 4n(n + 1)¿n¡1 = ¿n points. We de¯ne a function f¹n : [0; 1) ! [0; 1) by

f¹n (t) =

8 > 0 > > > > > k > > > : n

t ¶ 12 ¯n;0 ; 16®n;k ¡t 1 ¡16®n;k ¯ 2 n;k¡ 1

16®n;k µ t µ 12 ¯n;k¡1 ; k = 1; : : : ; n; 1 ¯ 2 n;k

(25) 1;

t µ 16®n;n :

Using (20) we obtain 12 ¯n;k < 16®n;k < de¯ned and continuous. We set fn (x) = if z 2 Zn¡1 and jx ¡

µ t µ 16®n;k ; k = 1; : : : ; n ¡

1 ¯ 2 n;k¡1

and therefore the function f¹n is well-

!n ¹ fn (jx ¡ n

zj)

(26)

zj < rn¡1 . If no such z exists, we set fn (x) = 0. Similarly, we set !n ¹ f~n (x) = fn (jx ¡ n

zj)

(27)

if z 2 Z~n¡1 and jx ¡ zj < rn¡1 . If no such z exists, we set f~n (x) = 0. By (20) and (22){(24), the balls B(z; rn¡1 ), z 2 Zn¡1 , are pairwise disjoint, and, similarly, the balls B(z; rn¡1 ), z 2 Z~n¡1 , are pairwise disjoint, and thus the functions fn and f~n are well de¯ned. Plainly, f¹n is constant on [®n;k ¡ rn ; 3®n;k + rn ] thanks to (25), (20) and (22). Hence we obtain from the construction that for any z 2 Zn the function fn is constant on B(z; rn ): This concludes the n-th step. Finally, we set f=

1 X

fn ;

f~ =

n=1

1 X

f~n :

n=1

Since sup fn = sup f~n = !n R2

and

R2

X n

the functions f , f~ are continuous.

!n < 1;

(28)


4

697

Properties of f and f~

Proposition 4.1. The function f does not belong to BV12 (R2 ) (and thus it does not belong to AC12 (R2 )). Proof. Consider a ball B = B(z; rn¡1 ) where z 2 Zn¡1 . Thanks to (28) and the de¯nition of B u we have oscB u f = oscB fn = !n : Fix any z 2 Zn¡1 , k 2 f1; : : : ; ng, i 2 f0; : : : ; 2n + 1g and p 2 f¡ 1; 1g. If i is odd (i.e. (¡ 1)i ¯n;k < 0), then the balls µ

¡ i ¢ B (z; rn¡1 ) and B z + p 1 + ®n;k e1 + (¡ 1)i ¯n;k e2 ; rn 3n u

¶

(29)

are clearly disjoint. If i is even, then we can use Lemma 2.1 ((21) veri¯es (1), (22) implies rn < ¯n;k ) to obtain that the balls in (29) are disjoint. It follows from (29) and the de¯nition of Zn (23) that the balls B u (z; rn ) : are mutually disjoint. We have XX

(oscB u

z 2 Zn ;

2 (z;rn ) f ) ¶

n2N z2Zn

X

n = 1; 2; : : :

2 ¿n !n+1 =

n2N

X

n2N

1 = 1: n+1

It easily follows that f 2 = BV12 (R2 ). Since f has a compact support, this implies that f2 = AC12 (R2 ). Lemma 4.2. For every set K there exists an index n = n(K), for which oscK f µ 4 oscK fn : Proof. We may assume that K is compact. Since f is continuous, there exist x; y 2 K for which jf (x) ¡ f (y)j = oscK f . Let m be the smallest index for which fm (x) 6= fm (y) (if there is no index with this property, then the statement is trivial). It is easy to see from the de¯nition that spt f1 ¼ spt f2 ¼ spt f3 ¼ : : : : Clearly oscK f µ oscK fm + oscK fm+1 +

1 X

oscK fi :

(30)

i=m+2

We may assume that

1 X

i=m+2

oscK fi > oscK fm + oscK fm+1 ;

(31)

698


otherwise either m or m + 1 has the required property. However, we will show that (31) leads to a contradiction. It follows that at least one of x and y (say x) is in the support of fm+2 (and then of course in the support of fm and fm+1 , as well). We have 1 X

oscK fi µ

i=m+2

1 X

!i

i=m+2

´ 1 1 + + ::: 2(m + 2) 4(m + 2)(m + 3) !m+1 !m < < : m+1 m = !m+1

³

(32)

By (31) and (32), we obtain oscK fm+1
0:

Since both x and y belong to spt fm+1, jfm (y) ¡ fm (x)j is a constant multiple of !m =m and by (33) jfm (y) ¡ fm (x)j = 0. This contradicts the choice of m. Proposition 4.3. The function f belongs to Q-AC12 (R2 ). Proof. The previous Xlemma gives us Xthat for every pairwise disjoint system of cubes 2 Q1 ; Q2 : : : we have oscQi f µ 16 osc2Qi fn(Qi ) , thus it is enough to prove that for i

i

every " > 0 there exists a ± > 0 such that for every sequence of pairwise disjoint cubes P Q1 ; Q2 ; : : : with i L 2 (Qi ) < ± and for every sequence of positive integers n1 ; n2 ; : : : we have X osc2Qi fni < ": (34) i

The functions fn are Q; d; 1-absolutely continuous (because they are Lipschitz), and 1 X i=1

osc2Qi

fn i µ

1 N X X n=1 i=1

osc2Qi

fn +

1 X

X

osc2Qi fni ;

n=N +1 fi:n=ni g

thus it is enough to prove that if an N is large enough, then every sequence of pairwise disjoint cubes Q1 ; Q2 ; : : : and every sequence of integers ni > N satisfy (34). We ¯x a number N 2 N, and we ¯x a sequence of pairwise disjoint cubes Q1 ; Q2 ; : : : and a sequence of integers n1 ; n2 ; : : : larger than N . Put ½ ¾ !ni A1 = Qi : oscQi fni µ 10 ; ni


A2 =

½

Qi : oscQi fni

!n i > 10 ni

¾

699

:

First we consider the oscillations on the cubes in A1 . For a ¯xed cube Q = Qi 2 A1 and n = ni let Q? denotes (one of the) smallest sub-cubes Q? » Q, for which oscQ fn = oscQ? fn . Then there exists z 2 Zn¡1 for which Q? » B(z; rn¡1 ). Let Sk denote the set n o 1 Sk = x 2 B(z; rn¡1 ) : 16®n;k µ jx ¡ cj µ ¯n;k¡1 2 ~ » Q? and an integer for k 2 f1; : : : ; ng. Since oscQ? fn µ 10 !nn , we can ¯nd cube Q ~ » Sk and oscQ? fn µ 10 osc ~ fn . Thanks to (25) and (20), the 1 µ k µ n such that Q Q function f¹n is Lipschitz on [16®n;k ; 12 ¯n;k¡1 ] with Lipschitz constant 1 ¯ 2 n;k¡1

1 C C µ µ ; ¯n;k¡1 diam Sk ¡ 16®n;k

therefore (26) gives us osc2Q

fn µ C

osc2Q~

¡ ¢ ~ 2 ~ !n2 diam Q !n2 L 2 (Q) µ fn µ C 2 C : 2 n (diam Sk ) n2 L 2 (Sk )

~ i are pairwise disjoint, we have Thus, since the cubes Q X

osc2Qi fn µ C

~ i ½Sk i:Q

Hence

i:Qi ½Sk

X

osc2Qi

fn µ

i:Qi 2A1

=C

X !2 L n n2 L ~

1 X

~

2 (Q i ) 2 (Sk )

¿n¡ 1 n 1 X XX

n=N +1 m=1 k=1

¿n¡1 n

n=N+1

C

µC

!n2 : n2

!n2 = n2

1 X !n2 1 = C 10 : ni Similarly to the proof of Theorem 4.3 the inequality X

osc2Bi f~ni µ C

i:Bi 2A1

1 X

1 0 we choose ± > 0 from the de¯nition of absolute continuity of f on P F ( ). Further suppose that i L d (Bi ) < K± 2 . Then X X X ~i) µ L d (B L d (F (Bi )) µ K 2 L d (Bi ) < ± : i

Thus

i

X i

oscdBi f ¯ F µ C2d

i

X

oscdB~i f < C2d " :

i

Acknowledgments This research is supported in part by the Research Project MSM 113200007 from the Czech Ministry of Education and Grant No. 201/03/0931 from the Grant Agency of the · Czech republic (GA CR).

References [1] M. CsÄornyei: \Absolutely continuous functions of Rado, Reichelderfer and Mal¶y", J. Math. Anal. Appl., Vol. 252, (2000), pp. 147{166. [2] S. Hencl: \On the notions of absolute continuity for functions of several variables", Fund. Math., Vol. 173, (2002), pp. 175{189. [3] S. Hencl: \Absolutely continuous functions of several variables and quasiconformal mappings", preprint MATH-KMA-2002/89, Charles University, Prague. [4] J. Mal¶y: \Absolutely continuous function of several variables", J. Math. Anal. Appl., Vol. 231, (1999), pp. 492{508.

Central European Journal of Mathematics

Central European Journal of Mathematics

Central European Journal of Mathematics