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qMfy. 1.1.6
Map to the semi-analytic cohomology
To (1.9) is associated a long exact sequence, of which we use the following part: (1.11)
0—>()-->*£—»££—»ff1(T,'V?).
(The space (*V^°) is zero.) This implies that there is an injective map
Crql^-^CW0).
In [4] it is shown that there is an injective map rs : Mf^ —> H1 (T, 'Vf'™). The image is the parabolic cohomology subgroup 77par (T, 1^'°°). (For the modular group T - PSL2(Z), the parabolic cohomology group H^w (T,A) can be defined as the subgroup of classes in Hx (r, A) that can be represented by a cocycle c that satisfies cj = 0 for T = L j e Y.) It turns out that rs is equal to the composition cs o qs. So the quantum Maass forms constitute a space through which we can factorize the map from Maass cusp forms to cohomology. The injectivity of r^ implies that qs is injective.
7
Quantum Maass Forms 1.2
Eisenstein series, extended quantum Maass forms
The Eisenstein series Es(z) = M2s)ys + A(2s - l)yl~s
(1.12)
+ 2 ^ |nrV2,_!(W) A(M) = w - * r ( ^ £ ( i O ,
^Ks-l/2(2n\n\y)e2ninx,
crB(n) =
J V
gives a meromorphic family of elements of Mfs, complementary to the cusp forms. It is holomorphic in the region 0 < R e 5 < l , s ^ j . A modification of the theory of Lewis and Zagier works for the Eisenstein series; see §IV.l, [3]. The holomorphic periodic function associated to Es can be chosen as (1.13)
/,(r) =
±
VSftXl- 'M2»
±
, T ( 1 _ s) £ cr2s^n)e^
,
with the convention ± 1 = sign Im T. The period function ^ ( T ) = fs(j) T~2sfs(-l/r) is holomorphic on C . However, it has no smooth extension through 0. The function ifrs defined by (/^(JC) = ifss(x) for x > 0, and iffs(x) = |jc|_2'y^(—1/JC) is real analytic on R \ {0}, but it is not an element ofV?' 00 . The expression (1.13) is explicit, and implies that at the rational point 2, a, c € Z, c > 0, (a, c) - 1, we do not have a limit value, but an expansion: (1.14) Mr)
=
l
-c2s-2K(2s-\)-
± y-^Til
l
- s)T {s + I ) c-2*M2s) ( * (r - ^jf2S
-^-T(, + i)|((f))^,,/c ) + 0(l)
T-*f.
where ±1 = sinlmr, and where £(•, •) denotes the Hurwitz zeta function. The computations are similar to those for cusp forms. We use the fact that
8
R. Bruggeman
the twisted L-series associated to (1.13) have expressions in terms of the Hurwitz zeta function. The example of the Eisenstein series shows that there are two difficulties: (1) There are no limit values, but expansions. (2) The cocycle has values in a larger space than rV^,0°. If we do not want to end the study of quantum Maass forms here, generalizations are needed. 1.2.1
Type of quantum Maass forms
'Vf'00 consists of the semi-analytic vectors in qMff °° ^"qMfr
1.2.2
Extended quantum Maass forms
Let Rs be the space of systems p that associate to almost all £ e Q an expansion /?(£, T) holomorphic in r near £. Two such systems of expansions
9
Quantum Maass Forms
are equal if p i ( £ r ) = pifor) + o(l) (T ^ £) for almost all £. A_constant is a holomorphic function, so %s c ft*. The group T acts in ^ by (pb*y)(£T) = jy(T)-sp(y%,yT), where (1.16)
7 y ( T r = ((cr + d) 2 )"'
forr
erf
is a holomorphic extension of x i-» |c* + rf| . For any type t we define ££ = RspV's. This is a huge space. We refrain from giving a name to the elements of (Q'sf /Ws. To the Eisenstein series Es, we can associate an element of \Q"'sj /fcs, represented by the system of expansions in (1.14). It turns out that the middle term in (1.14), with ± ((T/(T - a/c))~2s, determines a system of expansions in *R%. So this term may be considered as trivial and should be divided out. It seems sensible to define 4ls c *RS as the systems of expansions of the form (1.17)
p(t,r)
= Jt^
+ ct + oQ.)
(T-Z+&.
It is a T-invariant subspace. We define Q?s = 0. See Proposition 2.1 in [1].
n^a±ne^,
Quantum Maass Forms
11
A representative g of au is a holomorphic function on U \ P^ for some neighborhood U of P^ in P£. Two representatives differ by a holomorphic function on a full neighborhood of P^. That difference corresponds to an element of "Vf. If y € T, then f^y is an other representative of au. So 7 H r r = g - g\^sy is a cocycle with values in fVf. This induces a map r^ : ££ —> 7/1 (T, 0 implies the absolute convergence in (2.4). The support of B is contained in (oo). The Poisson map is given by an integral around oo: yQ+T 2 ) *J|TM?
\l~s
\(T-Z)(T-Z)J
dr 1+T2'
with R > 1, R > \z\. After a change of variables: -2s
usBiz) = ^ L £ i „ r v V - f (i + If (i + * V dr • iFi(l; 2 - 2*; 2mn{r + x)) — T
for sufficiently large R. Insert the power series expansion for iFi and the
(
2\i-l
/
\l-2s
1 + ^2 J and (1 + 7) • Some computations lead to the power series expansion of h/2-s, and give the desired terms in (2.2). It turns out that B\%ST(T) = B(r) + o(l) as r ^ 00. To check this, we use again the power series expansion of 1F1 and the binomial formula for powers of 1 + i. So the contribution of B(T) to p(<x>, r) can be transported to all £ € Q to give an element of 9^. Now we have a system of expansions p\ e *R.S given by pi(£, T) = g(r) - m(£, T) with m e f\^, and p(oo, T) = CIQQT 1 Ceo "T" o(l) (T ^> 00). Let
y = [c2] 6 r PI(£T)
such that
^ = r°° e Q- As T -^ ^: = M r ' V f c i © (00, r_1r) =
^"'^^"'^-(ffll^Kr-'T))
13
Quantum Maass Forms = M r M ( g ( r lr) + ry(y
1
r)-m(oo,y ! T ) )
= iy{y~lr)s (ry{y-lT) + pi(oo, y" 1 ^)
rf
1_
= (Anal) + (Anal) (d^ (— -
_ ^ ) + Coo + o(\)] ,
C2(T-0)
c
where (Anal) means a function that is holomorphic in T on a neighborhood of P^. This shows that p\ e r ) ( T ) = AIT + A0 + o(l)
(r -** oo).
For £ = 700, the subgroup ^ c T fixing £ is generated by n% = yTy~x. The one-sided average Av+ is obtained from Av£ by conjugation. Let ceZ1 (T, TJ 0 ). We define pe GLd(T\) for primes I of the coefficient field T c Q . We suppose that p is associated to a pure (absolute Hodge) motive in the sense of Deligne (see [D]). We assume that p does not contain the trivial representation as a subquotient. We write S for the finite set of ramification of p and p\ is unramified outside S U {oo, (}, where t is the residual characteristic of I. We write p = y e Or||£|p < l} and often write W := OT,V, where Oj is the integer ring of T. Often we just write p for p p which acts on V = T$. For simplicity, we assume that p £ S. Assume that q \ I, and let E((X) = det(l -pq(Frobe)\vi X) € T[X\. We always assume thatp p is ordinary in the following sense: p restricted to Gal(Qp/Qp) is upper triangular with diagonal characters N"' on the inertia Ip for the p-adic cyclotomic character N ordered from top to bottom as a\ > 02 > • • • > 0 > • • • > adThus (N"\ 0
* N"i
P\i„ = V 0
0
••• N"d I
In other words, we have a decreasing filtration T'+lp c T'p stable under Gal(Qp/Qp) such that gr>(p)(-i) := (T'plTMp)(-i), the Tate twists, is unramified. Define d
HP(X) = Y\ det(l - Frobp^wJX)
= [~[(1 - ajX). j=\
i
Then it is believed to be EP(X) = HP(X) if p £ S and EP(X)\HP(X) otherwise. In any case, ord p (a 7 ) e Z. Let us define (07 J
ifordp(or;)> 1,
[pa]
1
if ordp(or;) < 0
and put e = \{j\pj = p)\.
&(p) = f](l -fijp-1) and 8+(p) = f ] (1 -fijp-1).
£-Invariant ofp-Adic L-Functions
19
Then the complex L-function is defined by L{s,p) = Yle E{(£~s)~l. We assume that the value at 1 is critical for L(s,p) (in the motivic sense of Deligne in [D]). We suppose to have an algebraicity result (conjectured by Deligne) that for a well defined period c+(p)( 1)) e C x such that §c+(p(l)) & S € ~~ for all finite order characters e : Z* —» pp°°(Q). Then we should have Conjecture 0.1. Suppose that s = 1 is critical for p. Then there exist a power series Oa"(X) € W[[X]] and a p-adic L-function L^n(s,p) = Q>T{yx~s - 1) interpolating L(l,p ® e)for p-power order character e such that ®pn(e(y) - 1) ~ 8(p ® £ ) § ^ f f with the modifying p-factor £(p) as above (putting £(p ® e) = 1 if s + I). The L-function L^(s,p) has zero of order e + ord i= i L{s,p)for a nonzero constant J?n(p) e C* (called the analytic JL-invariant), we have c + (p(i»
*->i (s - ly
where "lim^i " is the p-adic limit, c + (p(l)) is the transcendental factor of the critical complex L-value L(l,p), and £ + (p) is the product ofnonvanishing modifying p-factors. When e > 0, we call that L™{s,p) has an exceptional zero at s = 1. Here is an example. Start with a Dirichlet character x '• (Z/AfZ) —» Q with^(-l) = - 1 . Then c(p + (l)) = (2ni). If we suppose ;f = (—j for a square free positive integer D, the modifying Euler factor vanishes at s = 1 if the Legendre symbol {—) = 1 d - 1. However it is easy to check that the system Ad(p) is not critical if d > 2. Thus we assume that d = 2. Now require that pp : Gal(Q/F) -» GLa(W) be a Galois representation associated to a p-ordinary Hilbert Hecke eigenform (belonging to a discrete series at oo) over a totally real field F. We make Ad{pF) and consider the induced representation Indp Ad(pp) whose eigenvalues of Frobp has 1 with multiplicity e for the number e of prime factors of p in F. The system Ind^ Ad(pF) is critical at s=l. Arithmetic X-invariant. Returning to a general ordinary representation p = p p , we describe an arithmetic way of constructing p-adic L-function due to Iwasawa and others. We define Galois cohomologically the Selmer group _ Sel(p) c
tf1(Gal(Q/Q00),p®Qp/zi»)
for the Zp-extension Q«,/Q inside Q0v°) by the subgroup of cohomology classes unramified outside p whose image vanishes in Hl{lp, (fi/'F+p) ®z Qp/Zp). Here T+p is the middle filtration Txp and Ip is each inertia group at p. The Galois group T = Gal(Qoo/Q) acts on //1(Gal(Q/Q0O),p®Qp/Z/7) and hence on Sel(p), making it as a discrete module over the group algebra W[[T]] = lim W[r/rP"]. Identifying T with 1 + pZ p by the cyclotomic character, we may regard y e Y. Then W[[r]] = A by y t~* 1 + X. By the classification theory of compact A-modules of finite type, the Pontryagin dual Sel*(p) has a A-linear map into Yifeci A//A with finite kernel and cokernel for a finite set Q c A. The power series $ p = rj/en f(X)
22
H. Hida
is uniquely determined up to unit multiple. We then define Lp(s,p) = ®P(yl~s - 1)- Greenberg gave a recipe of defining £(p) for this Lp(s,p) and verified in 1994 the conjecture for this Lp(s,p) except for the nonvanishing of X(p) (under some restrictive conditions). For the adjoint square Ad(pf) for pF associated to a Hilbert modular form, the conjecture (except for the nonvanishing of -£(p)) was again proven in my paper [HOO] in the Israel journal (in 2000) under the condition that pF = (pF mod mw) is absolutely irreducible over Gal(Q/F[pip\) and the p-distinguishedness condition for pF\Gal(F IF ) f° r a ^ Pl/> (which we recall later). If there exists an analytic p-adic L-function Lapn(s,p) - O^Xy1-* - 1) interpolating complex L-values, the main conjecture of Iwasawa's theory confirms <J>p = 0£" up to unit multiple. Suppose now that pF is associated to a Hilbert modular Hecke eigenform of weight k > 2 over a totally real field F. Following Greenberg's recipe, we try to compute £{Ad[pF)) = £(Indvp Ad(pF)). By ordinarity, we have PF\GMQ IF) - ( o a ) w ' m t w o distinct diagonal characters av and /3P factoring through 7P -> Gal(Fp[jUpoo]/Fp) for the inertia group / p for all p|p. We consider the universal nearly ordinary deformation p : Gal(Q/Q) —> GLQ,{R) over K with the pro-Artinian local universal Kalgebra R. This means that for any Artinian local AT-algebra A with maximal ideal mA and any Galois representation pA : Gal(Q/F) -> GL2{A) such that 1. unramified outside ramified primes for pF; Z
P^lcaKQ /Fp) - (*0aA» ) W i t h ffA,p = a P
m
°
d
™A S U C h
that the diag
°"
nal characters factor through Ip -» Gal(F p [/y»]/F p ) for all p|p; 3. det(pA) = detpf; 4. PA = PF mod mA, there exists a unique ^-algebra homomorphism (p : R -» A such that (pops pA. Write Tp s Z p for the p-profinite part of Gal(Fp[^p°°]/Fp). Choose a generator y„ of r p and identify W[[TP]] with W[[XV]] by y p YX ap([yp>, Fv\))p^p)
J.
The above value is independent of the choice of the basis {cp}p. When F = Q, by a result of Kisin [Ki] 9.10, [Kil] and [Ki2] 3.4 (generalizing those of Wiles [W] and Taylor-Wiles [TW]), we always have R a K[[Xp]]. In general, assuming the following two conditions: (ai) p = (PF mod m^) is absolutely irreducible over Gal(Q/F[jUp]); (ds) pss has a non-scalar value over Gal(F p /F p ) for all prime factors p\p, by using a result of Fujiwara (see [Fu] and [Ful]), we can prove R = K[[Xp]]p\p. The following conjecture for the arithmetic L-function is a theorem under the condition (ai) and semi-stability of pp over O except for the nonvanishing £(Ad(pF)) * 0 (see [HMI] Theorem 5.27 combined with (5.2.6) there):
27
.£-Invariant ofp-Adic L-Functions
Conjecture 1.2 (Greenberg). Suppose (ds) and that p is absolutely irrerith x Sx s n ducible. For L'SaIpith (s,Ad(p (s,Ad(pFF)))) = Qf ]. We can reverse the above argument starting a cocycle c giving an element of Sel^(V) to construct a deformation pc with values in K[e\. Thus we have {p : Gal(Q/F) -> GL2JK[s])\p satisfies the condtions (Kl^)} = SeVF(V).
29
.£-Invariant ofp-Adic L-Functions
Since the algebra structure of R over W[[Xp]]P|p is given by 6papx, the /^-derivation 5 : R —» K corresponding to a ^T[e]-defonnation p is a W[[^p]]-derivation if and only if pi|Gal(-p /F) ~ (oo)> which is equivalent to [cp] € Self (V), because we already knew that Tr(cp) = 0. Thus we have Self (V) a Derw[[Xv]](R, K) = 0. a We also have Lemma 1.4.
(V)
Self (V) = 0=>H\(S,V)sY\
^J%P-.
Indeed, by the Poitou-Tate exact sequence, the following sequence is exact: Self (V) - • H\f&M, V) -» J~| P
tf (
. ^;;.V) -> Sel F (V*(l))\ ^ w
An old theorem of Greenberg gives dim Self (V) = dim Self (V*(l))* (see [G] Proposition 2); so, we have the assertion (V). • 1.2
Greenberg's .^-invariant
Here is Greenberg's definition of £(V): The long exact sequence of TpV/TpV