Arithmetic of Hyperbolic Three-Manifolds (Graduate Texts in Mathematics)

Graduate Texts in Mathematics 1

TAKBUTIIZARING. Introduction to

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ORAVBRIWATKINS. Combinatorics with

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MANIN. A Course in Mathematical Logic.

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Emphasis on the Theory of Graphs.

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Analysis.

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Dirichlet Series in Number Theo ry. APosTOL. Modular Functions and

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and Representation Theory.

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Topological Spaces.

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J. ..p. SERRE. A Course in Arithmetic. HuoHBSIPIPBR. Projective Planes.

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ALEXANDBRIWERMER. Several Complex

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SPITZER. Principles of Random Walk.

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Theory I: Elements of Functional Analysis.

BROWNIPBARCY. Introduction to Operator MASSEY. Algebraic Topology: An Introduction. CRoWBulFox. Introduction to Knot KOBLITZ. p adic Numbers, p..adic

Theory.

..

Analysis, and Zeta-Functions. 2nd ed. LANG. Cyclotomic Fields. Classical Mechanics. 2nd ed.

ARNoLD. Mathematical Methods in WmTBHBAD. Elements of Homotopy

Theory. I
Preface

Vll

..

utrticulary via quadratic forms. These are discussed in Chapter 10, where also show how these arithmetic Kleinian groups fit into the wider realm , general discrete arithmetic subgroups of Lie groups. Some readers may wish to use this book as an introduction to arithmetic 1\ leinian groups. A short course covering the general theory of quaternion lp;ebras over number fields, suitable for such an introduction to either arithmetic Kleinian groups or arithmetic Fuchsian groups, is essentially self­ C'ontained in Chapters 2, 6 and 7. The construction of arithmetic Kleinian p,nn1ps from quaternion algebras is given in the first part of Chapter 8 and t. I ntain consequences of this construction appear in Chapter 11. However, i r t.he reader wishes to investigate the role played by arithmetic Kleinian J'.n Hip s in the general framework of all Kleinian groups, then he or she must further assimiliate the material in Chapter 3, such examples in Chapter 4 interest them, the remainder of Chapter 8, Chapter 9 and as much of < �l1apter 12 as they wish. For those in the field of hyperbolic 3-manifolds and 3-orbifolds, we have u leavoured to make the exposition here as self-contained as possible, given l.lu' eonstraints on some familiarity with basic aspects of algebraic number llu 'ory, as mentioned earlier. There are, however, certain specific exceptions In this, which, we believe, were unavoidable in the interests of keeping the :;iz(' of this treatise within reasonable bounds. Two of these are involved in :d.('PH which are critical to the general development of ideas. First, we state \vit.ltout proof in Chapter 0, the Hasse-Minkowski Theorem on quadratic lc H'lrtH and use that in Chapter 2 to prove part of the classification theorem r. •r quaternion algebras over a number field. Second, we do not give the full l•nu>f in Chapter 7 that the Tamagawa number of the quotient A�/Al is I. ;t.lt.hough we do develop all of the surrounding theory. This Tamagawa 1111111her is used in Chapter 11 to obtain volume formulas for arithmetic 1\ l • i tian groups and arithmetic Fuchsian groups. We should also mention I hat. the important theorem of Margulis, whereby the arithmeticity and 111 ,1 1-arithmeticity in Kleinian groups can be detected by the denseness or •I iscn�teness of the commensurator, is discussed, but not proved, in Chapter I 0. llowever, this result is not used critically in the sequel. Also , on a :an all number of occasions in later chapters, specialised results on algebraic 111111tl >er theory are employed to obtain specific applications. Mauy of the arithmetic methods discussed in this book are now available i11 t.lu� computer program Snap. Once readers have come to terms with :;••JIH' of these methods, we strongly encourage them to experiment with this \\'(HI< l 2, this field has no real places and c/>(n)/2 complex places, where ¢ is Euler ' s function. 5.

In a similar way, the real subfield Q (cos 21rjn) of the cyclotomic field is totally real.

If a

E

k, then the norm and trace of a are defined by

In the case where a = t and k = Q ( t ) , these are the product and sum, respectively, of the conjugates of t. As such, they are, up to a sign, the constant and leading coefficients of the minimum polynomial and so lie in Q . If K denotes a Galois closure of the extension k I Q , then K can be taken to be the compositum of the fields ai (k) , = 1, 2, For each a in the Galois group, Gal(K I Q), the set {am} is a permutation of the set {ai}· Thus for each a E k, NkiQ (a), TrkiQ (a) are fixed by each such a and so lie in the fixed field of Gal(K I Q ) [i.e. , .1\iiQ (a) and TrkiQ(a) lie in Q ] . If [k : Q] = let { Oi, a2, · · · , ad} be any set of elements in k. If

i

... , d.

d,

with Xi

E

Q , then for each monomorphism ot,

rI

,bus one readily deduces that the set { a1, a2, . . . , ad } is a basis of k I Q i (' and only if det [ai ( ai )] =f 0. The element 8 = det[ai ( aj ) ] lies inK and for a E Gal(K I Q), a( 1. The primes ·pi are said to lie over or above p, or pZ. Note that fi is the degree of the (�xtension of finite fields [Rk /Pi Z/pZJ . If [k : Q] = d, then N (pRk ) = pd and so :

d=

g

e f i i · L il =

(0. 16)

Oefinition 0.3. 7 The prime number p is said to be ramified in the exten­ .� ion k I Q if, in the decomposition at {0. 1 5}, some � > 1. Otherwise, p is

nn'f'fLrnified. r l 'l u �

followi ng t.heorent

• Ti t u i uaut..

of Dcdckind connects ramification with the dis­

14

0. Number-Theoretic Menagerie

Theorem 0.3.8 A prime number p is ramified in the extension k I Q if and only if p I �k · There are thus only finitely many rational primes which ramify in the extension k I Q .

If P is a prime ideal in Rk with IRk /P I = q (= pm ) , and l l k is a finite extension, then a similar analysis to that given above holds. Thus in R�,, (0. 17)

where, for each i , Rt / Qi is a field of order qfi . The ei , fi then satisfy (0. 16) where [l : k] = d. Dedekind's Theorem 0.3.8 also still holds when �k is replaced by the relative discriminant, and, of course, in this case, the ideal P must divide the ideal 8t l k · Now consider the cases of quadratic extensions Q (v' d) I Q in some de­ tail. Denote the ring of integers in Q (V d) by od . Note that from (0. 16) , there are exactly three possibilities and it is convenient to use some special terminology to describe these. 1 . pOd = P 2 (i.e., g = 1 , e1 = 2 and so /1 = 1 ) . Thus p is ramified in Q (v'd) I Q and this will occur if p I d when d = 1(mod 4) and if p l 4d when d ¢ 1(mod 4) . Note also in this case that Od/P rv 1Fp , so that N(P) = p.

2.

pOd = PI P2 (i.e., g = 2, e 1 = e2 = !I = /2 == 1 ) . In this case, we say that p decomposes in Q (J d) I Q . In this case N('A ) = N(P2) = p.

(i.e., g = 1 , e 1 = 1 , /1 = 2) . In this case, we say that p is inert in the extension. Note that N(P) = # . The deductions here are particularly simple since the degree of the exten­ sion is 2. To determine how the prime ideals of Rk lie over a given rational prime p can often be decided by the result below, which is particularly useful in computations. We refer to this result as Kummer's Theorem. (It is not clear to us that this is a correct designation, and in algebraic num­ ber theory, it is not a unique designation. However, i� this book, it will uniquely pick out this result.) 3.

pOd = P

Let Rk = Z[B] for some 8 E Rk with minimum polynomial h. Let p be a (rational} prime. Suppose, over 1Fp , that

Theorem 0.3.9

where hi E Z[x] is monic of degree fi and the overbar denotes the natural rnap Z[x] --+ 1Fp [x] . Then Pi = pRk + hi (O)Rk is (L ]J'rirne ideal, N (Pi) = pfi

(l,nd

,. . 1

'" ' · � ,n t · -.z

, .�

.

•

.

·rt·,. ,

.

•

0.3 Ideals in Rings of Integers

15

There is also a relative version of this theorem applying to an extension l I k with Rt = Rk [8] and P a prime ideal in Rk . As noted earlier, such extensions may not have integral bases. Even in the absolute case of k I Q, it is not always possible to find a 8 E Rk such that { 1, (}, (}2 , , 8d- l } is an integral basis. Thus the theorem as stated is not always applicable. There are further versions of this theorem which apply in a wider range of cases. •

•

•

Once again we consider quadratic extensions, which always have such a basis as required by Kummer's Theorem, with 8 = v'd if d ¢. 1(mod 4) and (} = (1 + v'd)/2 if d = l(mod 4) . In the first case, p is ramified if p I 4d . For other values of p, x2 - d E IFp [x] factorises if and only if there exists a E Z such that a2 = d(mod p) [i.e. if and only if ( ; ) = 1] . In the second case, if p is odd and p % d, then x2 - x + (1 - d)/4 E lFp [x] factorises if and only if (2x - 1 )2 - J E 1Fp [x] factorises [i.e. if and only if ( ; ) = 1] . If p = 2, then 1-d

{ x2x2 ++ xx + 1F1 2E[x]lF [x]

if d = 1(mod 8) if d = 5(mod 8). 4 2 Thus using Kummer's Theorem, we have the following complete picture of prime ideals in the ring of integers of a quadratic extension of Q. x

2

_

x+

=

E

In the quadratic extension, Q(v'd) I Q, where the integer d is square-free and p a prime, the following hold: 1 . Let p be odd. (a) If p I d, p is ramified. (b) If ( ; ) = 1 , p decomposes.

Lemma 0.3.10

{c) If ( ; ) = -1, p is inert.

2.

Let p = 2. (a) If d ¢ 1(mod 4), 2 is ramified. {b) If d = l(mod 8), 2 decomposes. (c) If d = 5(mod 8), 2 is inert.

lijxamples 0.3. 1 1 I

. 'rhe examples treated at the end of the preceding section will be con­ sidered further here. Thus let k = Q (t) where t satisfies Xi + x + 1. r rl l iH

SO

polynomial is irreducible mod 2, there is one prime ideal P2 i u /l�: lying over 2 and N(P ) = 2 3 • Modulo 3, the polynomial factor­ 2 iHPH as {:r - 1 ) (:r:2 + - 1), so that 3Rk = P�P!{ with N(P� ) = 3 and N CP:�' )

:r�

-- :fl . Mod u l o :� 1 , t.lu� polynot u i al factoris 1 only occurs for finitely many P. Thus for the unramified primes P, we have [i : k] = fg and (f, g) determines the splitting type of P. For the absolute quadratic case, the splitting types are (1, 2) and (2, 1) and there are infinitely many primes of each type. The density theorems are concerned with generalising this. Cou t. iunc to

and J(' f, �/ ( Q;. )

H.HHU IIlC

-·

{

rr

C

that p is nnranlified

fl I fT( Q; )

-

( Q;, ) } ,

ill t.hc�

t i lt '

0, does indeed do that. Now give a similar method for the cases where d = l(mod 4) . 5. Find fundamental units in Q (v'7) and in Q (v'13) . 6. Determine all the units in k = Q(a), where a satisfies :i - 2 = 0. 7. Order the Galois embeddings of k such that O"t , a2 , . . . , O"r1 are real and O"r1 + r2 + i = O"r1 +i for i = 1 , 2, . . , r2 . Let .\ be the mapping from Rk to lRr1 +r2 defined by .\( u ) = (In l at ( u ) l , . . . , In l ar 1 ( u ) l , 2 ln l ar1 +1 (u) l , . . . 2 ln l ar1 +r2 ( u ) l ) . Show that .\(Rk) is a lattice in the r(= r1 + r2 - I)-dimensional subspace V of Rr1 + r2 , where 1.

n

.

,

V

=

{ (XI , X2 ,

.

•

. , Xr1 + r2 )

L Xi = 0 }

·

The volume of a fundamental cell for this lattice times (n +r2 ) - 1 12 is called the Regulator of k. It is clearly an invariant of the isomorphism class of k. If { U t , u 2 , . . . , Ur } is a set of fundamental units for k, show that the Regulator of k is the determinant of any r x r minor of U, where U is the r + 1 x r matrix with entries .ei I ai ( Uj ) I , where .ei = 1 or 2 according to whether O"i is real or not. Calculate the regulator for Q (v'3) and for the field Q (a ) , where a satisfies x3 2 = 0. -

0.5

Class Groups

cl a "'H p;ronp i u t.( 'p;P n-i i n f.h al.

r rl u �

..

l llllllher field gi veH a, t l l e�U·n n·( � of how far t.I H � ring of 1 1 1 1 1 n i H ' f" f i(dd is frou a I H � i u g a pr i u < · i pal i d ( 'al dor nai n , as

of

a.

0.5 Class Groups

23

the "class" refers to classes of ideals modulo principal ideals (see below) . This group has a role to play when we come to consider the structure of arithmetic Kleinian groups. Additionally, further investigations into the structure of arithmetic Kleinian groups lead to the consideration of certain ray class groups. It is more natural to consider these after the introduction of valuations and so we will defer the consideration of these ray class groups until §0.6. Let us denote the group of fractional ideals of k by Ik and recall that it is a free abelian group on the set of prime ideals of Rk ( Theorem 0.3.4) We will call Ik the ideal group of k. The subset ./l; of non-zero principal fractional ideals ( i.e. , those of the form aRk for a E k* ) , is a subgroup of Ik .

Definition 0. 5 . 1 • •

The class group, Ck , of k is the quotient group Ik /Pk . The class number, always denoted h , of k is the order of the class group.

The second definition depends, of course, crucially on the following funda­ rnental and important result: Theorem 0.5.2

The order of the class group of a number field is finite.

The definition of class group can be formulated independently of frac­ tional ideals. If necessary, to distinguish ideals from fractional ideals, we nse the terminology integral ideals. Note that, by the definition of frac­ tional ideals, each coset of Pk in Ik can be represented by an integral ideal I of Rk . Define two integral ideals I, J to be equivalent if there exist ton-zero elements a, (3 E Rk such that ai = (3 J . Thus in the group Ik , / ,J- 1 = (a- 1(3) Rk E Pk so that I, J belong to the same coset of Pk in I�: . Defining multiplication of these equivalence classes of integral ideals by [ I] [J] [I J] is well-defined and gives the class group. Clearly, the class number is 1 if and only if Rk is a principal ideal domain. Again, Minkowski's Theorem can be used to prove the finiteness of the ('lass number. This is used to obtain relationships between the norms of i( leals and the discriminant. In particular, Theorem 0.5.2 will follow quite n'adily from Theorem 0.5.3 below. The inequality in this result is some­ l.inles referred to as Minkowski 's bound and it is used, in particular, in ( '( nnputations in specific fields. I

=

' l"l1eorem 0. 5.3 In every class of ideals of the number field k, where d t '!J'I 'f �(! [k : Q] n, there exists a non-zero (integral} ideal J such that =

N ( ,J)


0 for all x E K and v (x) 0 if and only if x {ii} v(xy) v(x)v (y) for all x, y E K . (iii) v (x + y) < v(x) + v ( y ) for all x, y E K . =

:

K =

---+

JR+ , such

0.

=

There is always the trivial valuation where v(x) 1 for all x =f 0, so we nssume throughout that our valuations are non-trivial. A valuation v on K defines a metric on K via d(x , y ) v (x - y) for all x, y E K and, hence, ( lefines K as a topological space. =

=

Two valuations v, v' on K are equivalent if there exists E JR+ such that v' (x) [v (x ) ]a for x E K .

J )efinition 0.6.2

,

=

A n alternative formulation of this notion of equivalence is that the valu­ ;d.ions define the same topology on K. I >efinition •

•

0.6.3

If the valuation v satisfies in addition {iv} v(x + y ) < max{ v(x ) , v(y) } for all x, y E K, then v is called a non-Archimedean valuation. If the valuation v is not equivalent to one which satisfies (iv), then v is Archimedean.

Non-Archimedean valuations can be characterised among valuations as 1 h oHc for which {v (n. lK) : n E Z} is a bounded set (see Exercise 0.6, N o. 1 ) . I � P nt iila 0. 6 . 4 •

I i ('o )

-

:

{

Let v be a non-Archimedean valuation on K . Let

("

c

I\ l v ( ( Y )

< I

},

26

0. Number-Theoretic Menagerie •

P(v)

= { a E K l v( a ) < 1} .

Then R(v) is a local ring whose unique maximal ideal is P (v ) and whose field of fractions is K. Definition 0.6.5 respect to v).

The ring R( v) is called the valuation ring of K (with

Now let K k be a number field. All the valuations on k can be determ­ ined as we now indicate. Let a k -t C be any one of the Galois embeddings of k. Define Va by vu (x) = l a(x) l , where I · I is the usual absolute value. It is not difficult to see that these are all Archimedean valuations and that Vu and Vu' are equivalent if and only if (a, a') is a complex conjugate pair of embeddings. Out of an equivalence class of valuations, it is usual to select a normalised one. For real a, this is just Vu as defined above, but for a complex embedding a, choose vu (x) l a(x) l 2 • Now let P be any prime ideal in Rk and let c be a real number such that 0 < c < 1. For x E Rk \ { 0 } , define Vp ( and np ) by vp (x) = cn-p(x ) , where np ( x) is the largest integer m such that x E pm or, alternatively, such that pm I xRk · It is straightforward to show that Vp satisfies {i}, {ii} and (iv). Since k is the field of fractions of RT 1. If np2 (x) > 1, then np2 ( 1 - y ) and np2 (y) > 1 so that 1 E P2 . Thus ·np2 (:.r ) 0 so that Vp1 , Vp'J. cannot be eqniva.Jent. . =

=

J\ 1 1 eq uivalence class of valna.t.ions is ea.l l f .p r

2.

x

to ad (�l es and id(�]es, H is one-dimensional with q(u) a =/= 0. Thus U represents a locally and, hence, so does V. Thus by Corollary 0.9 . 9, V represents a. Hence U and V are isometric. Now let U have dimension > 2 , u E U with q(u) = a =/= 0. As above, V also represents a. Thus there exists v E V with Q(v) = a. Let U' = < u > ..L and V' =< v > ..L . By assumption, for each v there exits an isometry av : Uv --+ Vv . Furthermore, there exists T E O(Vv) such that Tav ( u) = v Theorem 0.9. 12

=

n

( see Exercise 0. 9, No. 7) . Thus TUv gives an isometry from u� to v:. Thus by induction, there is an isometry U' --+ V', which can be easily extended to an isometry between U and V. D .

For basic results and results over P-adic fields, see Lam ( 1973 ) , Chapters 1 and 6. For the Hasse-Minkowski theorem and Hilbert Reciprocity, also see Lam (1973 ) , Chapter 6, but for full proofs, using ideles as described in preceding section, see O'Meara { 1 9 63 ) , Chapters 6 and 7. Exercise 0.9

Let V be a quadratic space (not necessarily regular} and let W be a regular subspace. Prove that V = W ..L W..L . 2. Let (V, q ) be a two-dimensional regular quadratic space over K. Prove that the following are equivalent: (a) V is isotropic. {b) The determinant of V is -1K* 2 • (( ) q ha,c; tlu� diagonal Jo·rrn :q :r:� . 1.

;

(d)

lJ ha.� Ut. t '

.fot·rn

:r 1 ;J:'.l .

-

0.9 Quadratic Forms

45

[A space such as described by these equivalent conditions is called a hyper­ bolic plane.J 3. Let V = M2 (K) and define B and B' by B(X, Y) = tr (XY) and B' (X, Y) = tr (XYt ) , respectively. Show that (V, B) and (V, B) are regular quadratic spaces and find orthogonal bases for each. Show that they are isometric if and only if -1 E K* 2 • 4 . Let kp be non-dyadic. (a} Prove that every regular quadratic space of dimension > 5 over kp is isotropic. {b) Show that if 1, u, 1r and u1r are the coset representatives in Mp/kp 2 given in Exercise 0. 7, No. 6 with u E Rp, then the four-dimensional quadratic space with diagonal form if - 1rx� - ux� + 1rux� is anisotropic. [This result is also true for dyadic fields ; see §2.5.] 5. Show that the four-dimensional form

xi + 3x� + (2 + V!O) x� + (2 - V!O) x� is anisotropic over Q (J l O) . 6. Determine if the following quadratic forms represent 1 or not: (a) 15x2 - 21y2 in Q . {b) 2x2 + 5y 2 in � . (c) c l+FJ )x2 - c l - p)y 2 in Q (J-7) . {d) (t + l ) x2 + ty2 in Q (t) , where t satisfies af + x + 1 = 0. 7. Let (V, B) be a quadratic space over a field K of characteristic =f 2 and let a E K*. Let A = {v E V I q(v) = a}. Prove that if v, w E A, then at ll�ast one of v ± w is anisotropic. Deduce that O(V) acts transitively on A.

1 Kleinian Groups and Hyperbolic Manifolds

/\ s

indicated in the Preface, this book is written for those with a reasonable knowledge of Kleinian groups and hyperbolic 3-manifolds, with the aim of extending their repertoire in this area to include the applications and i •nplications of algebraic number theory to the study of these groups and r r tanifolds.This chapter includes the main ideas and results on Kleinian groups and hyperbolic 3-manifolds, which will be used subsequently. There : t.re no proofs in this chapter and we assume that the reader has at least passing knowledge of some of the ideas expounded here. In the Further B.eading at the the end of the chapter, references are given for all the n�sults that appear here so that deficiencies in the presentation here may I remedied from these sources. : t.

)(�

1 .1

PSL (2 , CC ) and Hyperbolic 3-Space

' l 'lu� group PSL ( 2, C ) is the quotient of the group SL ( 2, C ) of all 2 x 2 1 1 1at.rices with complex entries and determinant 1 by its center {±I}. Ele­ I J u ' n tH of a subgroup r of PSL ( 2, C ) will usually be regarded as 2 x 2 l l l at. ri ces of determinant 1, so that the distinction between r and its pre­ l l l l a�c � iu SL ( 2, C ) is frequently blurred. In general, this is innocuous, but, if • u ·c ·• �sHa.ry, we usc the notation r = p- 1 (r) c SL ( 2, C ) to distinguish these

1 '. n H I pH.

V i a t. h e l i u ear

fra.etiona.l action

z �- t

az

+b '

I ,

48

1 . Kleinian Groups and Hyperbolic Manifolds

elements 'Y

=

(� �) of PSL (2, C ) are biholomorphic maps oft "'"

=

C U { oo }.

The action of each 1 E PSL ( 2, C ) onC extends to an action on the upper half 3-space H3 { (x, y, t) E JR3 I t > 0} via the Poincare extension. Each 1 is a product of an even number of inversions in circles and lines in C . RegardingC as lying on the boundary of H3 as t 0, ( i.e. , the sphere at infinity ) , each circle C and line l in C has a unique hemisphere C or plane i in H3 , orthogonal to C and meeting C in C or l. The Poincare extension of 'Y to H 3 is the corresponding product of inversions in C and reflections in l. When H3 is equipped with the hyperbolic metric induced from the line element ds defined by =

A

=

"

"

(1.1) then H3 becomes a model of hyperbolic 3-space, (i.e. , the unique three­ dimensional connected and simply connected Riemannian manifold with constant sectional curvature - 1 ) . The line element induces a distance met­ ric d on H3 and H3 is complete with respect to this in that every Cauchy sequence converges. The inversions in hemispheres and reflections in planes as described above are isometries of H 3 with the hyperbolic metric and generate the full group of isometries, Isom H3 . Thus, under the Poincare extension, the group PSL ( 2, C ) is identified with the subgroup Isont H3 of orientation-preserving isometries of H3 : PSL (2, C ) "' Isom + H3 . (1.2)

The whole book is concerned with subgroups r of PSL ( 2, C ) which satisfy various conditions, some of which are topological, but mainly these condi­ tions relate to the geometry of the action of r on H3 . The broad idea is to relate the geometry, on the one hand, to the algebra and arithmetic of the associated matrix entries, on the other. In this chapter, we collect together the main geometrical ideas which will form the basis of this association. Before embarking on a discussion of the conditions to be satisfied by r c PSL ( 2, C ) , we remark that, starting instead with PSL ( 2, JR) , the linear fractional action of its elements on C restricts to upper half 2-space If { (x, t) E JR2 I t > 0} so that PSL ( 2, JR) becomes identified with Isom+ H 2 for this upper half-space model of hyperbolic 2-space with metric induced 2 2 by ds 2 dx ttdt : =

=

(1.3) This

1netrie

ou

H2

iH the

the hyperhol ie ruetrie on Ha to PSL(2, C ) i sol l t < 'f.riPH o f It and

n�strietion of

p l at u ' y ---· 0 . N o tn f. h at Pl 0}.

' I 'he line element ds defined by ds2 dx r + dx� + dx� - dx � yields a I � i(�n1annian metric on the hyperboloid, making it a model of hyperbolic =

:�

space. Alternatively, consider the positive cone in V defined by I let

c+

=

{x

E

v I q(x)
0}

denote its projective image PC+ . There is an obvious bijection w i t. It the hyperboloid and using this, one obtains that A is another model of h,v pa('eH wi ll ha.ve a one-dimensional orthogonal complement in (V, q) and :11u

:

: :u

A

P VP ry hypt�r hol ic plane i n

A iH

t.he hnage

of

a space

v..l

where

q(v)

>

0.

1. Kleinian Groups and Hyperbolic Manifolds

50

The subgroup of the orthogonal group of (V, q) which preserves the cone c+ , o + (v, q) = {a E GL (V ) I q(a(v) ) = q (v ) for all v E v, and a(c+ ) = c+ }

(1.5)

induces an action on A, yielding the full group of isometries of this model of hyperbolic 3-space. A reflection in a space vl. where q(v) > 0 lies in o + (v, q) and has negative determinant. Thus for A, we have the description of the isometry groups as Isom A rv po + (V, q) , Isom+ A rv PSO+ (V, q) . (1.6) Exercise 1 . 1

Show that stereographic projection from a point on the unit sphere in R= is the restriction of an inversion in a sphere. Show that the upper half-space H3 can be mapped conformally to the unit ball 1.

B3 = { ( x1 , x2 , x 3 ) E � I xi + x� + x� < 1}

by a couple of stereographic projections. Use this to describe the hyperbolic metric on the ball and the geodesic lines and planes in the J13 model of hyperbolic space. 2. Show that inversions in hemispheres in IJ.3 with centres on the sphere at infinity are isometries of the hyperbolic metric in If3 . 3. Find a formula for the hyperbolic distance between two points (X}. , y 1 , t 1 ) and (x2 , Y2 , t2 ) in H3 . 4 - Show that, in the Lobachevski model, if two planes 11 and P2 in A are the images of v1 J.. and v2 l. , respectively, as described above, then (a) if P1 and P2 intersect in the dihedral angle 8, then cos fJ = -B (vt , v2) , where B is the symmetric bilinear form induced from q; {b) if P1 and P2 do not intersect and are not tangent, then the hyperbolic distance l between them is given by cosh l = -B(v1 , v2 ) . 5. Prove that PSL ( 2, C ) acts transitively on the set of circles and straight lines in C .

1.2

Subgroups of PSL (2, C )

Now let us consider various subgroups of PSL(2, C ) and the related geo­ metry. First, for individual elements 1 =I= Id, we have the following classi­ fication: •

'Y

is P l l i pt.ic i f

t.r 1

(

l�

au c l

j t .r 1 1

llowing presentation of the Bianchi group PSL ( 2, 01 ) :

1 .4.2


1. The Lens Space /,(p, q) is non-Haken. Examples of non-Haken 3-manifolds with infinite ft1ndamental group are discussed later in the examples. L

:

=

There are various equivalent statements of Mostow-Prasad rigidity, but l . he most succinct is that given a compact orientable 3-manifold whose interior supports a hyperbolic structure of finite volume, then this structure i H unique. This is the biggest distinction in the theory of surface groups and l typerbolic 3-manifold groups of finite volume. l•�xercise 1.6 I.

Let r be a subgroup of SL(2, CC ) . Let R(r) be the set of conjugacy classes o.f 'representations ¢ in Hom(r, SL(2, C )) where ¢ preserves parabolic ele­ nu�nts. Then V (r) , as described in this section is the component of this u.lqebraic set which contains the identity representation. (a} If r is a Schottky group which is free on generators, determine n

IiIrI

( v (r) ) . ( fJ) If r is such that JI2 ;r is a hyperbolic once-punctured torus, determine . 1 i 1 1 1 ( v (r)) . I

I s.

.7

Volumes and Ideal Tetrahedra

f'ar

we have stressed the requirement that hyperbolic manifolds and or­ l • i foldH have finite volume. In that case, the hyperbolic volume itself is a I upological invariant, even a homotopy invariant as follows from Mostow ' s I ( i gi I i ty Theorem given in the preceding section. In this section, we indicate l 1 • • w vohuncH and numerical approximations to volumes may be computed. l 1 1 l at.< �r applications , the results of such calculations serve, in particular, to d i st.i ugnish t nanifolds and as a guide to possible degrees of commensurab­ d i l .y I ' t wP< ' U I( l < ' i niau group:-; or t h n as:-;oeiatc�d ifol cls or orhifol ds. ,

c

u

t n au

1. Kleinian Groups and Hyperbolic Manifolds

70

t

y

A'

A

C'

B'

X

FIGURE

1 .6 .

Finite-volume fundamental polyhedra whose combinatorial structure is not too complicated can be decomposed into a union of tetrahedra (see Exercise 1 .7, No 1). In turn, these tetrahedra can be expressed setwise as a sum and difference of tetrahedra with at least one ideal vertex. Finally, locating such a tetrahedron so that the ideal vertex in the upper half-space model of H3 is at oo and the remaining vertices lie on the unit hemisphere centred at the origin, that tetrahedron can be decomposed into a sum and difference of tetrahedra of the standard form we now describe: Let Ta.,1 denote the tetrahedron in H3 with one vertex at oo and the other vertices on the unit hemisphere such that they project vertically onto C to form the Euclidean triangle T as shown in Figure 1 .6 with acute angle B' A'C' = a and the dihedral angle along BC is I · Note that the length of A' B' is cos I · These acute dihedral angles a , 'Y determine the isometry class of the tetrahedron Ta.,1. The volume of this tetrahedron is thus given by the convergent integral

which, after a little manipulation reduces to

j

�: j

7r1 2 2 sin - a ! ln Vol(Ta 7 ) = ' 2 sin - a 4 1

dO .

(1. 16)

This integral can be conveniently expressed in terms of the Lobachevski function whose definition we now recall. For (} =/= n1r, define £ ( fJ )

=-

·

· f. , fl

0

lu

12 si n u I du .

( I . J 7)

1.7

Volumes and Ideal Tetrahedra

71

is not difficult to see that this integral converges for (J E (0, 1r ) and admits a continuous extension to 0 and 1r with .C (O) == .C (1r) = 0. By further extension, the above definition allows one to extend .C in such a way that it admits a continuous extension to the whole of JR. On (0, 1r ) , £((} + n1r ) - .C ( fJ) is differentiable with derivative 0 and so is constant on 10, 1r] . Thus we obtain the following: It

Lemma 1 . 7. 1

The Lobachevski function .C is periodic of period

1r

and is

td,c;o odd. ' I ' he function .C has a uniformly convergent Fourier series expansion which ( · ; \,n be obtained via its connection with the complex dilogarithm function oo

n

1/l (z) = L :2 ' n=l

lzl < 1.

l z l < 1 , z 'lf; '(z) = - ln ( 1 - z), where we take the principal branch of log, ; ln(�w) dw . Indeed this definition can also be extended so that 1/; (z) = - J e2i8 ) - 1/;( 1 ) for the two l.o l z l = 1. Then comparing imaginary parts of 'lf; ( l •'o r

• 'x

1

>ansions of the dilogarithm function yields this next result.

l ..emma 1.7.2 .C (fJ)

has the uniformly convergent Fourier series expansion C( O) =

� f: sin� O) . n=l

n

H( ' t.urning to the calculation of volumes, it readily follows from ( 1 . 1 6 ) and ( I . 1 7) that Vol (T ;y ) cr

=

! [.c (a + -y) + .C (a - -y) + 2c(; - a) J .

( 1 . 18)

The Coxeter tetrahedron with Coxeter symbol given in l •' i gure 1 . 7 is the difference of two tetrahedra each with one ideal vertex as : d 1 own in Figure 1.8. The number n labelling an edge indicates a dihedral n n g J ( � of 1r / n. The tetrahedron with vertices A, B, C and oo is of the type 1 : . . described earlier, where a = 7r / 3 and, via some hyperbolic geometry, i s Hnch that sin 1 = 3 /(4 cos 7r / 5 ) . Thus by ( 1 . 1 8) , the volume of Ta.,1 1 : ; approximately 0.072165 . . . . The tetrahedron with vertices D, B, C and is not of the type Ta, 7 but can be shown to be the union of two such l c · t. rahedra minus one such tetrahedron and so the volume of the compact 1 • -l. ra.h cclron in Figure 1 .8 can be determined ( see Exercise 1.7, No. 4 ) . l·�x ample 1 . 7.3

1

�

' .

C)

· - ··-· -----

o ===== o

--

o

72

1. Kleinian Groups and Hyperbolic Manifolds

D

3 c

A

FIGURE 1 .8.

If in the tetrahedron described by Figure 1 .6, the vertex C was also ideal, then it would lie on the unit circle in C and a = I · In that case, Vol(Ta,a )

(See Exercise 1.7, No. 5.)

=

1 .C a ) .

2 (

( 1 . 1 9)

An ideal tetrahedron in H3 is a hyperbolic tetrahedron all of whose vertices lie on the sphere at infinity.

Definition 1 . 7.4

For such a tetrahedron, the dihedral angles meeting at each vertex form a Euclidean triangle from which it is easy to deduce that opposite angles of an ideal tetrahedron must be equal. Locating the ideal tetrahedron with dihedral angles {3 and 'Y (with sum 1r ) such that one vertex is at oo and the others on the unit circle centred at the origin, if the angles are all acute, then the ideal tetrahedron can be decomposed into the union of six tetrahedra, two each of the forms Ta,a , Tf3,{3 and T1 ,1 as described above. Thus the volume of an ideal tetrahedron with angles (3 and 1 is

a,

a,

.C (a) + .C (f3) + .C (I) ·

( 1 . 2 0)

This formula still holds if not all angles are acute. These ideal tetrahedra can be used to build non-compact manifolds obtained by suitable gluing, as discussed in §1.4.4, the particular example of the figure 8 knot complement appearing there. Furthermore, the procedure of hyperbolic Dehn surgery on a cusped manifold built from such tetrahedra can be described by de­ forming the initial tetrahedra to further ideal tetrahedra before completing the structure. The key to calculations involving these is an appropriate pararnetrisation ror

sneh OI'l(�H ted tetrahedra. NorT n a.l i He HO that th ree

0 , I an d

r'X.J

a.ud

t. l u �

fo u rth is

a

< �o l u p l < � x l l l l l l l i H 'r

�-=

of t.h

0.

lin at Such

1 .7 Volumes and Ideal Tetrahedra

73

FIGURE 1 .9. a

z =

is not uniquely determined by the oriented tetrahedron because i t depends on the choice of normalisation, but the other possibilities are �yclically related to it as z2 = 1 and Z3 = 1 1 , as shown in Figure 1. 9. l•'rom (1.20) , the volume of such a parametrised ideal tetrahedron is z1

z


0} bounded by the hyperbolic planes given by x ±1, y ±1, x2+y 2 +t2 4 and x 2 +y 2 +t2 8. Decompose P into tetrahedra and hence determine the volume of P. 2. Deduce {1. 16}. 3. Complete the proof of Lemma 1. 7. 2. 4 . The tetrahedron BC Doo in Figure 1 . 10 is that described in Figure 1. 8. Let Ta1 ,11 be the tetrahedron BCEoo , Ta2 , 12 CEFoo and Ta3 ,'"'f3 DEFoe, so that setwise =

=

=

=

=

=

=

Use the relationships between dihedral angles and face angles to determine the ai and li · Hence, using {1. 18}, show that the volume of the compact tetrahedron in Figure 1. 8 is approximately 0. 03905. . . .

FIGURE 1. 10.

Prove that .C(2a) 2.C(a) - 2.C(7r/2 - a). (See Exercise 11. 1, No. 5 for a generalisation of this.)

5.

=

1 . 8 Further Reading Most of the material in this chapter has appeared in a variety of styles in a number of books. Since we are taking the viewpoint that our readership will have some familiarity with the main concepts here, in this reference section, we will content ourselves with listing references section by section which cover the relevant material. These liHtH arc not intended to be proHeriptivc au d , i l ld( !< �d , t. I H� rnad < n· J u ay

wnl l

pn�f< �r

of.hc�r

HO I I IT( �H . W< ' t r ust. ,

however,

1.8 Further Reading

75

.hat any obscurities which do arise in this brief introductory chapter can )e clarified by consulting at least one of the sources given here. §1.1 See Anderson (1999) , Beardon (1983) , Thurston (1979) , Thurston (1997) , Ratcliffe (1994) , Vinberg (1993a) and Matsuzaki and Tanigu­ chi (1998) . s1.2 See Th.urston (1997) , Thurston (1979 ) , Vinberg (1993b) , Ratcliffe (1994) , Maskit (1988), Harvey (1977)and Matsuzaki and Taniguchi (1998) . s1.3 See Thurston (1997) , Thurston (1979) , Matsuzaki and Taniguchi (1998) , Ratcliffe (1994) and Elstrodt et al. (1998) . �j 1.4 See Elstrodt et al. (1998) , Vinberg (1993b ), Ratcliffe (1994) , Thurston (1997) , Thurston (1979) , Maskit (1988) . and Epstein and Petronio (1994) . �{1.5 See Gromov (1981) , Hempel (1976) , Jaco (1980) , Morgan and Bass (1984) , Thurston {1979) , Benedetti and Petronio (1992) , Ratcliffe (1994) , Dunbar and Meyerhoff (1994) , Neumann and Reid (1992a) and Matsuzaki and Taniguchi (1998). �� 1 .6 See Thurston (1979) , Ratcliffe (1994) , Benedetti and Petronio (1992) , Matsuzaki and Taniguchi (1998) , Weil (1960) , Garland (1966) , Mum­ ford (1976) , Mostow (1973) and Prasad (1973) . �i 1 . 7

See Vinberg (1993a) , Ratcliffe (1994) and Thurston (1979) .

2

Quaternion Algebras I

' l 'ltroughout this book, the main algebraic structure which plays a major ,(e in all investigations is that of a quaternion algebra over a number l i('ld. In this chapter, the basic theory of quaternion algebras over a field of ··l lctracteristic =f 2 will be developed. This will suffice for applications in the r. •I lowing three chapters, but a more detailed analysis of quaternion algebras w i l l need to be developed in order to appreciate the number-theoretic input i 1 1 the cases of arithmetic Kleinian groups. This will be carried out in a later · ·l tapter. For the moment, fundamental elementary notions for quaternion l�ebras are developed. In this development, use is made of two key results c • l l ecntral simple algebras and these are proved independently in the later � i•'erties

arc

2.2 Orders in Quaternion Algebras

85

For the converse, choose a basis {x1 , x2 , Xg , x4} of A such that each Xi E 0. Now the reduced trace defines a non-singular symmetric bilinear form on A (see Exercise 2.3, No. 1). Thus d = det(tr (xixj)) =/= 0. Let L = {E aixi I ai E R}. Thus L C 0. Now suppose a E 0 so that a = E bixi with bi E k. For each j, axi E 0 and so tr (axj) = L: bitr (xixi) E R. Thus bi E ( 1 /d)R and 0 c (1 /d ) L. Thus 0 is finitely generated and the result follows. Using a Zorn's Lemma argument, the above characterisation shows that every order is contained in a maximal order. D Let us consider the special cases where A = M2 ( k) . If V is a two­ dimensional space over k, then A can be identified with End(V). If L is a eomplete R-lattice in V, define End(L) = {a E End(V) I a (L) C L}. I f V has basis { e 1 , e 2} giving the identification of M (k) with End(V), then 2 Do Re 1 + Re2 is a complete R-lattice and End(Lo) is identified with the 1naximal order M2(R). For any complete R-lattice L, there exists a E R 2 :-;uch2 that aL0 c L C a - 1 L0• It follows that a End(Lo) C End(L) c (t, - End(L0). Thus each End(L) is an order. =

Let 0 be an order in End(V) . Then 0 C End(L) for some t ·omplete R-lattice L in V. Proof: Let L = {l E Lo I 0 .e C Lo}. Then L is an R-submodule of 1 /,0 . Also, if a End(Lo) c 0 c a - End(£0) , then a Lo c L. Thus L is a Lemma 2.2.8

('omplete R-lattice and 0 c End(L).

D

1\

simple description of these orders End( L) can be given by obtaining a sirnple description of the complete R-lattices in V. 'rheorem 2.2.9 Let L be a complete R-lattice in V. Then there exists a hasis {x, y } of V and a fractional ideal J such that L = Rx + Jy. J >roof: For any non-zero element y E V, L n ky Iy y, where Iy = {a E 1.- I ay E L }. Since L is a complete R-lattice, there exists f3 E R such that I 111 C R so that Iy is a fractional ideal. We first show that there is a basis { x, y } of V such that L = Ix + Iy y for some fractional ideal I. Let { e 1 , e2} be a basis of V and define I = {a E k I ae1 E L + ke2} . 1\ �ai n it is easy to see that I is a fractional ideal. Since I I-1 = R, there c · x ist E I and f3i E I - 1 such that 1 = E aif3i· Now aiel = .ei + l'i e 2 E f3il'i· Let w l u � re f.;. E L, {i E k. Thus e 1 = E f3ili + ry e 2, where I' "'(C2 , y = I claim that L = Ix + Iy y. First note that =

.1

f�·i

.r

(

'

1 -

l :t:

1 111 u .:__

=

r2 .

/ (t ' J

--

'YI':.d -+

/, n ky

=

r

(L !V·i ) +

Ln

k y c L.

2. Quaternion Algebras I

86

z

Conversely, suppose that = a( e1 + f3e 2 ) E L for some a, f3 E k. Now ae 1 = af3e 2 E L + ke2 so that E I. Thus a({3e2 + 1e 2 ) = a( e 1 + {3e2 ) - a( e 1 1' e2 ) E L. Hence = a( e 1 - 1'e2 ) + a(f3e 2 + 1 e2 ) E Ix + Iy y . It remains to show that we can choose y such that Iy = R.1 Suppose L = Ix+Iy y, above. Then there exist 81 , 82 E k such that 81 I - +�2 I;- 1 R. Let y' 81x+82 Y · Then Iy' = (I81 1 )n{Iy82 1 ) = ( 81I- 1 +82 I;- 1 ) - 1 R. D z -

z

-

=

a

as

=

=

Thus if End(V) is identified with M2 (k) , then for any complete R-lattice L, End(L) is a conjugate of M2 ( R; J) : =

{ (� :) I a, d E R, b E J- 1 , c E J }

(2.5)

for some fractional ideal J. Note that if R is a PID and so J = xR for some x E k, then M2 (R; J) is conjugate to M2 (R) and from Lemma 2.2.8 and Theorem 2.2.9, we obtain the following : Corollary 2.2.10

ate.

If R is a PID, all maximal orders in M2 (k) are conjug­

Exercise 2.2

1. Complete the proof of Lemma 2. 2. 2. Establish the result in Examples 2. 2. 6, No. 4 ; that is, if I is an ideal of A, then the sets Ot(I) and Or(I) are, indeed, orders in A. 3. In the special case where R = Z, show that

2.

0

=

{ (� :) E M2 (Z) I a = d(mod 2), b = c(mod 2) }

is an order in M2 ( Q ) . 4- Show that

is a quaternion algebra A over Q and that A"' Z[l, i, j, ij] is not a maximal order in A. 5. Let A =

0

=

( -�·5 )

•

Show that the order

( - l ,g(�U2) and let R be the ring of integer8

R[l , ·i , j, 'i_j] +

·i. s an o·nlt'1' in It .

Rc� , 'tiJh f�'l 'f! ( V. ==

'in

Q (J G) .

( 1 + 't ) ( ( l - v'f,)/2 + .J ) /2 . b"ho·u1

Ld

thai. ('J

2.3 Quaternion Algebras and Quadratic Forms

2.3

87

Quaternion Algebras and Quadratic Forms

Let A be a quaternion algebra over F . From the norm map on the vector space A, define a symmetric bilinear form B on A by

B(x, y) so

=

1 [n(x + y) - n(x) - n(y)]

2

= 21 [xy + yx]

that A becomes a quadratic space (see §0.9) . If A has a standard basis { 1 , i, j, k}, then it is easy to see that these vectors constitute an orthogonal I 1asis of A. If A = , then the quadratic space has the quadratic form r r - ax� - bx� + abx� and so is obviously regular. The restriction of n, and hence B, to the pure quaternions Ao makes Ao into a regular three­ diinensional quadratic space. The forms n and B have particularly simple -x. Thus n( x) = -x2 and ' lescriptions on Ao since, for x E Ao, x I J ( x, y) = - � ( xy + yx) . The norm map will be referred to as the norm form ot t both A and Ao. Recall that a quadratic space V with a quadratic form tf V --+ F is said to be isotropic if there is a non-zero vector v E V such I. I tat q ( v) = 0. Otherwise, the space, or the form, is said to be anisotropic.

( 7}!-)

.

=

:

Theorem 2.3.1

For A =

( al ) , the following are equivalent:

(b) A is not a division algebra. (c) A is isotropic as a quadratic space with the norm form. (d) Ao is isotropic as a quadratic space with the norm form. (l) The quadratic form aa? + by 2

=

1 has a solution with (x, y)

(f) If E = F(Jb), then a E NEIF (E) .

l , r·oof:

·. � . I .

E

F x F.

The equivalence of (a) and (b) is just a restatement of Theorem

7. (h) =? ( c) . If A is not a division algebra, it contains a non-zero non­ I I I V(�rtible element x. Thus n(x) = 0 and A is isotropic. ( l·) =? (d) . Suppose x = ao + a 1 i + a 2j + a3 ij is such that n(x) = 0. If , - 0 , then x E Ao and Ao is isotropic. Thus assume that ao =/= 0 so that t l I ( 'H �t one of a 1 , a2 and a3 must be non-zero. Without loss, assume that I 0. Now from n(x) = 0, we obtain a6 - ba� = a(ai - ba�) . Let

, ,(

.

'' 1

..

y ==

b( aoa3 + a 1 a2 )i + a( ai - ba�)j + ( aoa 1 + ba2 a3 )ij.

_ \ st . raip;ht.forward eaJenlation gives that n(y) = 0 . Now suppose that Ao is . u l i so f. rop i < ' . , l 'h u � u - 0 a.ud , i n pa.rt.icu lar , - aa T + a.ha� = 0 . Thus r1,(z) == 0

2. Quaternion Algebras I

88

where z = a 1 i + a3 ij. Again, if Ao is anisotropic, this implies that a1 = 0. This is a contradiction showing that Ao is isotropic. (d) => (e) . An equation of the form -aai - ba� + aba� = 0 holds with at least two of a 1 , a2 and a3 non-zero. If a3 =/= 0, then the pair x, y, where x = a 2 jaa 3 , y = a 1 jba3 satisfy ax 2 + by 2 1. If a3 = 0, then x ( l +a) / 2 a and y = a2 ( l - a)/2aa l satisfy ax2 + by 2 = 1. (e) => (f) . Let ax5 + byfi = 1 . If xo = 0, then lb E F and E = F, in which case the result is obvious. Assuming then that xo =/= 0, a rearrangement shows that NE I F ( l / xo + /byo /xo) = a. (f) => (b). If lb = c E F, then t? = b = j 2 . So (c + j) (c - j) = 0 and A has zero divisors. Now suppose that lb fl. F. Then a E NEIF ( E) shows that there exist x1 , y1 E F, not both 0, such that a = xi - byr . Then n (x 1 + i + y 1 j) = 0 so that A has non-zero non-invertible elements. D ==

=

If the quaternion algebra A over F is such that A rv M2 (F), then A is said to split over F. Remark Recall that in §0.9, a Hilbert symbol (a, b) was defined for the quadratic form ax2 + by2 . This theorem relates the two definitions of Hilbert symbol. Thus ( al ) splits if and only if (a, b) = 1. Definition 2.3.2

Corollary 2.3.3

to M2 (F) .

The quaternion algebras ( lFa ) and ( a,;a ) are isomorphic

For (�) , the result follows immediately from ( e ) . For ( a,; a ) , the norm form on A0 is -ax2 + ay2 + a2 z 2 , which is clearly isotropic. D Proof:

The above results give several criteria to determine when a given qua­ ternion algebra is isomorphic to the fixed quaternion algebra M2 (F) . Now consider some examples which are not isomorphic to M2 (F). Let k be a number field and A = 7J! . By Lemma 2. 1.2, it can be assumed that a, b E Rk , the ring of integers in k. Now the form aa? + by2 = 1 has a solution in k if and only if the form aa? + by2 = z 2 has a solution in Rk with z =I= 0. If the form has a solution in Rk , then for any ideal I of Rk, there will be a solution in the finite ring Rk /I. This enables us to construct many examples which are not isomorphic to M2 (F). Take, for example, where p is a prime = - l (mod 4) . Then choosing I, as above, to be pZ, the congruence -x 2 + py 2 = z 2 (mod p) clearly has no solution. Thus by Theorem 2.3. l (e) , is not isomorphic to M2 (Q). On the other hand, noting that Pell ' s equation

( )

( -�,p) ,

( -�,p)

x2 ha.o.;

au i n t.egml sol n t. ion i f 71

=

-

p

y2 = -1

I ( mod

-1 ) ,

iu t.lwse ea.o.;es

( --�rl')

"'

M2

(Q ) .

2.3 Quaternion Algebras and Quadratic Forms

89

More generally, it is necessary to decide when two quaternion algeb­ ras over the same field are isomorphic by an isomorphism which acts like the identity on the centre. The important classification theorem, which precisely describes the isomorphism classes of quaternion algebras over a number field, will be given in §2.7 and Chapter 7, but for the moment we recast the problem in terms of quadratic forms.

Let A and A' be quaternion algebras over F. Then A a.nd A' are isomorphic if and only if the quadratic spaces .1\l and Ab are ·i.�. ometric.

Theorem 2.3.4

With norm forms n and n' , this last statement means that there ( �xists a linear isomorphism ¢ Ao --+ A� such that n' ( ¢( x ) ) = n ( x ) for all Proof: E

.r

:

A0 •

Thus suppose that 1/J A --+ A' is an algebra isomorphism. Then by the ('haracterisation of the pure quaternions in terms of the centre ( Lemma �. 1.4) , 1/J must map Ao to A� . Then for x E A0 , n ' ( 'lf; ( x ) ) = - 'l/J ( x ) 2 = ;, ( - x 2 ) = 1/J ( n ( x ) ) = n ( x ) . Thus Ao and A0 are isometric. Now suppose ¢ : Ao -t Ab is an isometry with {l, i, j, ij} a standard I )c\Sis of A . We will show that { ¢( i), ¢(j) , ¢( i)¢(j)} is a basis of .AfJ. Let A = First note that

·,

;� 'I

�

Examples 2. 7.6

'

�

1. Let A � ( -1Q-1 ) . Then �{A) = 2Z because A splits at all the odd , primes by Theorem 2.6.6. It is established in Exercise 2.6, No. 3 that A is ramified at the prime 2. Alternatively, A is ramified at the Archimedean place by Theorem 2.5.1 and so by Theorem 2.7.3 must also be ramified at the prime 2. 2. Let t = J (3 - 2 J5), k = Q(t) and A rv ( �,t ) . We want to determine �(A). Recall some information on k from §0.2. Thus k = Q (u) , where u = (1 + t) /2 and so u satisfies x4 - 2x3 + x - 1 = 0. Also Rk = Z (u] . Now k has two real places and since t is positive at one and negative at the other, A is ramified at just one of these Archimedean places. Further Nki Q (t) = -11 so that tRk = P is a prime ideal. The quadratic forn1 -x2 + ty 2 == 1 has no solution in kp since ( f ) = - 1 . Th i s follows front

• ,

�

'

·

'rl u �ore r n OJ) . f) .

Finally, I IH)( l u lo 2 ,

tl u '

1

poly uou t i al ;1:'1

2:r:1

I

:r

+

I

iH

,

'

2.8 Central Simple Algebras

101

irreducible, so, by Kummer's Theorem, there is just one prime in k lying over 2. Thus �(A) = t Rk by Theorem 2.7.3 . Exercise 2. 7 1.

Show that the following quaternion algebras split:

(- 1

+ V5, (1

- 3J5)/2

Q (V s )

)

.

:�.

Let A be a quaternion division algebra over Q . Show that there are in­ .finitely many quadratic fields k such that A ®Q k is still a division algebra f)'Oer k . :1. Let k = Q ( t ) where t satisfies � + x + = 0. Let A = (t' 1t2t ) . Show t.hat �(A) = 2Rk . 4 . {Norm Theorem for quadratic extensions) Let L I k be a quadratic l':r:tension and let a E k* . Prove that a E N(L) if and only if a E N( L ®k kv) for all places v. n . Let k be a number field and let L I k be an extension of degree 2. Let f> be an ideal in � which decomposes in the extension L I k and let P2 be ideal in Rk which is inert in the extension. Let Ql lie over P1 and let Q:l lie over P2 . Let A be a quaternion algebra over k and let B = L ®k A. I }Tove that A is ramified at P1 if and only if B is ramified at Ql . Prove I hat B cannot be ramified at Q2 .

1

,

1

an

�.8

Central Simple Algebras

l 11

our discussion of quaternion algebras so far in this chapter, two crucial rc�� ntlts on central simple algebras have been used. These are Wedderburn's Structure Theorem and the Skolem Noether Theorem. The latter result, in 1 ,;trticular, will play a critical role in the arithmetic applications to Kleinian J •. ronps later in the book. Thus, in this section and the next, an introduc­ l j, to central simple algebras will be given, sufficient to deduce these two t • ·s ults . These sections are independent of the results so far in this chapter. I J( �t F denote a field. Unless otherwise stated, all module and vector space ; u·f.ions will be on the right. n1

I

>(1)a = �( l ) (a) so ;, that A is surjective. D ��; Proof:

'f

I

,,,,

1 '

Definition 2.8.3 • •

r

An F-algebra A is central if Z(A) = F. An F-algebra A is simple if it has no proper two-sided ideals.

We now investigate properties of tensor products of simple and central algebras. For any two F-algebras, the tensor product A ®F B is defined and is also an F-algebra with dimp(A ® B) = (dimFA) (dimpB). Let A and B be F -algebras. 1. If A' and B' are subalgebras of A and B, respectively,

Proposition 2.8.4

T,

l '

In particular, if A and B are central, so is A ® B. 2. If A is central simple and B is simple, then A ® B is simple. In particular, if A and B are central simple, so is A ® B.

Let E = A ® B. 1. A routine calculation shows that Proof:

Choose a basis { bj } of B. Then, if e E C (A' � B') , has a unique PxprnHsiou = 2: (\'.:i C>¢ b.i with (\'.j E A . Let. E A ' . 'fh en fro1n ' l'(a' (··) I )

a-.'; (�

(a.' C · ) I ) f · a.1 1d

t.h( � u u i qu< �I U �HH ,

H

WP

a.

'

oht.ai u n:,; u.

£�

a1t 'r.J for Pach

·

2.8 Central Simple Algebras

103

and all a' E A'. Thus a-i E CA (A') and so e E CA (A') ®B. Now choose a basis { Cj} of CA (A') so that e has a unique expression e = L: Cj ® (3i with {3i E B. For b' E B', e ( 1 ® b') (1 ® b' ) e yields b' (3i f3i b' . Thus each {3i E CB (B') and the result follows. Let I =f 0 be an ideal of E. Let 0 =I= z E I so that z = L:� 1 ai ® bi with ai E A and bi E B and chosen among elements of I such that r is minimal. Thus all � and bi are non-zero and { a 1 , a2 , , ar } is linearly independent. Otherwise, after renumbering, a1 L:� 2 aiXi and z = L:� 2 ai ® ( b1xi + bi) , contradicting the minimality of r. In the same way, { b 1 , b2 , . . . , br } is linearly independent. We now "replace" {a 1 , a2 , . . . , ar } by a set {1 , a�, . . . , a�}. The set Aa1 A is a two-sided ideal and so Aa1 A = A. Thus 1 = L:i Cjatdj . So r z1 = :E(cj ® 1)z(dj ® 1) 1 ® b 1 + L a� ® bi E I.

j

=

�-

==

.

.

•

=

=

j

i=2

Now repeat for b1 to obtain an element r z ' 1 ® 1 + L a� ® b� E I. =

i =2

From the equality r z' (a ® 1) - (a ® 1)z' = L (a� a - aa�) ® b� E I, i= 2

the choice of r shows that a� a aa� for i = 2, 3, . . . r. Thus all of these a� E Z (A ) = F. However, { 1, a�, . . . , a�} is linearly independent. So 1 and 1 ® 1 E I. Thus I = E. D =

'r

=

I >efinition

For the F -algebra A, let A0 denote the opposite algebra nJ/t.f!re multiplication o is defined by 2.8.5

o

a b = ba \:Ia, b E A.