Overview of “On Λ-adic forms of half integral weight for SL(2)/Q”
Kenji MAKIYAMA
Abstract. We will see an overview of [H95]
Contents
1. Introduction 1
2. Classical and adelic cusp forms of half integral weight 2
3. Shimura’s action on classical cusp forms 4
4. Λ-adic forms and Λ-adic representations 5
4.1. On half integral weight 5
4.2. On integral weight 6
5. Main theorem 7
6. Construction of Λ-adic forms of half integral weight 8
7. A result slightly stronger main theorem 10
References 11
1. Introduction In this talk, we will get a Λ-adic version of the following.
Waldspuger’s result ([W81, Corollaire 2, p.379] or [P, Corollary 5.2] for English ver.). Let φ ∈ Sk−1new(Mφ, χ2) be a primitive form with an odd integer k ≥3 and π the automorphic representation attached to φ. Assume that for any prime`,
If π` ∼=π(α, β) (principal series), thenα(−1) =β(−1) = 1.
(H`)
Suppose that f ∈Sk/2(N, χ, φ) for some positive integerN such that Mφ dividesN/2. Then for any pair (m, n) of two square-free positive integers withm/n∈Q
`|N(Q×` )2, we have
am(f)2L(1/2, π⊗ψ−1χn)χ(n/m)nk/2−1 =an(f)2L(1/2, π⊗ψ−1χm)mk/2−1. (1-0-1)
In particular, if L(1/2, π⊗ψ−1χm)6= 0, then an(f)2
am(f)2 = L(1/2, π⊗ψ−1χn)
L(1/2, π⊗ψ−1χm)χ(n/m)(n/m)k/2−1. (1-0-2)
Remark 1.1. (1) Letf ∈Sk/2(N, χ). Assume one of the following:
(a) k≥5.
(b) N is square-free.
Date: August 18, 2016.
1
(c) N is cubic-free andχ is the trivial character.
Thenf ∈Sk/2(N, χ, φ) if and only iff|Tp2 =ap(φ)f for almost all primep-N.
(2) Hypothesis (H`) for any prime ` is equivalent to that there exists a positive integer N such that Sk/2(N, χ, φ) 6= 0 by a theorem of Flicker ([W81, Proposition 2] or [P, Theorem 5] for English ver. and proof is in [F]).
(3) Hypothesis (H`) holds for any prime`ifχ2 =1by a theorem of Vigneras ([P, Theorem 6] and proof is in [V]).
Notation and convention. Let p ≥5 be a prime, K an extension of Qp inside ¯Qp, O the closure of OK in Cp. Let W := 1 +pZp and µ the group of (p−1)-st roots of unity so that Z×p = W ×µ.
Let h i : Z×p W be the first projection and ω : Z×p µ the second projection (Teichm¨uller character). We often regard ω as a Dirichlet character (Z/pZ)× → µ by ω(z mod p) := ω(z). Let Λ :=O[[W]] be the Iwasawa algebra,L:= Frac(Λ),Ka finite extension ofL, andIthe integral closure of Λ in K. Let u := 1 +p ∈ W be a topological generator. We call P ∈ X(I) := HomΩ-alg(I,Cp) an arithmetic point if P|Λ(u) = ukε(u) for some positive integer k ≥ 2 and ε ∈ Hom(W,C×p) with [W : Kerε]<∞ and writek(P) andεP fork and ε, respectively. Let A(I) be the set of arithmetic points in X(I) and A(I;O) :={P ∈ A(I) |P(I) ⊂ O}. For a Dirichlet character ψ and P ∈ A(I), we put ψP :=ψεPω−k(P). We putG:= GL(2)/Zand S := SL(2)/Z. Let Sebe the two-fold metaplectic cover ofS (see Section 2.2). Given a (2×2)-matrix γ, we put
aγ bγ cγ dγ
:=γ.
(1-0-3)
Let Abe the adele ring of Q,A(∞):={x∈A| x∞= 0}, and A(p∞):= {x∈A| xp =x∞ = 0}. Put Zb :=Q
`Z` and Z(p):=Q
`6=pZ`. For a non-zero integer a, we let χa denote the Kronecker symbol χa(b) :=
a b (1-0-4)
defined by [MFM, (3.1.9)].
2. Classical and adelic cusp forms of half integral weight We let γ ∈Γ0(4) act on functions f :H:={z∈C| Im(z)>0} →Cby
f|k+1/2γ(z) :=f(γ(z))j(γ, z)−1J(γ, z)−k with γ(z) := (aγz+bγ)/J(γ, z), (2-0-1)
where J(γ, z) :=cγz+dγ and j(γ, z) :=θ(γ(z))/θ(z) for θ(z) :=P
n∈Zexp(2πin2z). For any congru- ence subgroup ∆ of Γ0(4), we write Pk+1/2cl (∆;C) for the space of holomorphic functions f : H→ C satisfying thatf|k+1/2γ(z) =f(z) for all γ∈∆ and thatf is homolomorphic at all cusps of ∆ in the sense of [Sh73, Section 1]. For a positive integerN divisible by 4, we put
Pk+1/2cl (N, χ;C) :={f ∈Pk+1/2cl (Γ1(N);C) |f|k+1/2γ(z) =χ(dγ)g for all γ ∈Γ0(N)}
(2-0-2)
Remark 2.1. The action (2-0-1) is a little bit different from the normalization of [Sh73, p.447]. When comparing among [H95] with [Sh73] and [W81], it is useful to keep in mind that
Pk+1/2cl (N, χ;C) = (
S2k+1(N, χ(−1)kχ) in [Sh73], S(2k+1)/2(N, χ(−1)kχ) in [W81].
(2-0-3)
We write Se for the two-fold metaplectic cover of S defined in [W81, II.4.]. Namely, we have a non-split central extension S(e A) = S(A)× {±1} corresponding to the 2-cocycle β : S(A) → {±1}
defined as follows: Let v be a place ofQ. For anyσ ∈S(Qv), we put x(σ) :=
(dσ ifcσ = 0, cσ ifcσ 6= 0.
(2-0-4) We put
sv(σ) :=
((cσ, dσ)v ifcσdσ 6= 0, v is finite, ordv(cσ) is odd,
1 otherwise,
(2-0-5)
where (cσ, dσ)v is the Hilbert symbol atv. Then we define
βv(σ, σ0) := (x(σ), x(σ0))v(−x(σ)x(σ0), x(σσ0))vsv(σ)sv(σ0)sv(σσ0), (2-0-6)
β(σ, σ0) :=Y
v
βv(σv, σv0).
(2-0-7)
By the product formula of the Hilbert symbol, β(σ, σ0) = s(σ)s(σ0)s(σσ0) for σ, σ0 ∈ S(Q). Thus σ →(σ, s(σ)) gives a section: S(Q)→S(e A). We identify S(Q) with its image inS(e A). We put
C∞:=
r(θ) :=
cos 2πθ sin 2πθ
−sin 2πθ cos 2πθ
|θ∈R/2Z (2-0-8)
Lete:A/Q→Cbe the standard additive character such that e∞(x∞) = exp(2πix∞) ([LFE, p.249]).
We defineγv(t) to be the Weil’s constant with respect toev and the quadratic formtx2 onQv ([Weil, p.161]). We put, following [W81, p.380],
˜
γv(t) := (t, t)vγv(t)γv(1)−1. (2-0-9)
Let
U0(N) :=n
u∈S(bZ) |cu ∈NZb o
, (2-0-10)
U0(N)` :=
n
u∈S(Z`) |cu ∈`ord`(N)Z`
o . (2-0-11)
Defining forσ ∈U0(4)2
˜ ε2(σ) :=
(γ˜2(dσ)−1(cσ, dσ)2s2(σ) ifcσ 6= 0,
˜
γ2(dσ) ifcσ = 0, (2-0-12)
we can check that ˜ε2 extends to a character U0(4)2× {±1} in S(e Q2) non-trivial on {±1}. For any open subgroup U of U0(4), we write Pk+1/2(U;C) for the space of functions f :S(e A)→ Csatisfying the following (m’1) and (m2):
f(αx(u, )r(θ)) = ˜ε2(u2, )f(x) exp((k+ 1/2)θ) (m’1)
forα∈S(Q),(u, )∈U × {±1}and r(θ)∈C∞. Df =
k0(k0−2) 2
f (m2)
fork0 :=k+ 1/2 and the Casimir operator Dat∞. Then
Pk+1/2(U;C)−∼→Pk+1/2cl (U ∩S(Z);C);f 7→fcl, (2-0-13)
where
fcl(z) :=f(g∞,1)J(g, i)k+1/2 (2-0-14)
for g ∈ S(R) such that z = g(i). For a sub algebra A of C, we define Pk+1/2(U;A) as the image of Pk+1/2cl (U∩S(Z);A) via (2-0-13).
3. Shimura’s action on classical cusp forms By the strong approximation theorem, we have a bijection:
congruence subgroups of S(Z) of level prime top
←→n
open subgroups of S(Z(p)) o
=:Z (3-0-1)
∆ =∆b ∩S(Z)←→∆ : the closure of ∆ inb S(Z(p)), where Z(p):=Q
`6=pZ`. For∆b ∈ Z, we put
∆1(pr) := ∆∩Γ1(pr), (3-0-2)
Sκcl(∆;b A) := [
r≥1
Sκcl(∆1(pr);A) and Pk+1/2cl (∆;b A) := [
r≥1
Pk+1/2cl (∆1(pr)∩Γ0(4);A).
(3-0-3)
We fix two embeddings ¯Q,→Cand ¯Q,→Cp. Put
Sκcl(∆;b O) :=Sκcl(∆;b O ∩Q¯)⊗O∩Q¯ O and Pk+1/2cl (∆;b O) :=Pk+1/2cl (∆;b O ∩Q¯)⊗O∩Q¯ O.
(3-0-4)
For eachf ∈Sκcl(∆;b O)∪Pk+1/2cl (∆;b O), we define the uniform norm
|f|p := sup
n≥1
|an/N(f)|p. (3-0-5)
We write S(b∆;b O) (resp. Pb(∆;b O)) for the completion of Sκcl(∆;b O) (resp. Pk+1/2cl (∆;b O)) under the norm (3-0-5).
Fact ([H88b, Corollary 5.4]). Ifκ≥2, thenS(b ∆;b O) is independent of κ.
Theorem 3.1 ([H95, Theorem 1]). If k≥2, then Pb(∆;b O) is independent of k.
LetQ(p)ab :=Q[ζn |p-n] be the maximal abelian extension ofQunramified at p. Put Sκcl(Q(p)ab) := [
∆∈Zb
Sκcl(∆;b Q(p)ab) andPk+1/2cl (Q(p)ab) := [
∆∈Zb
Pk+1/2cl (∆;b Q(p)ab), (3-0-6)
S(O) :=b [
∆∈Zb
S(b∆;b O) andPb(O) := [
∆∈Zb
Pb(∆;b O).
(3-0-7)
Fact. In [Sh78a], Shimura defined smooth the action ofG(A(p∞)) (resp. S(e A(p∞))) onSκcl(Q(p)ab) (resp.
Pk+1/2cl (Q(p)ab)). Using Katz’s theory ofp-adic modular forms (see [H92, Chapter 2]), we can check that the action of G(A(p∞)) (resp. S(e A(p∞))) preserves on Sκcl(O) (resp. Pk+1/2cl (O)) and extends toSb(O) (resp. P(O)) byb p-adic continuity.
Remark 3.2. In fact, Shimura defined the action of GA+:= G(A(∞))G+(R) on meromorphic Siegel modular forms which is stable on holomorphic cusp forms ([Sh78a, Theorem 1.2]). In our case, the action is described as follows: For f ∈ Sκcl(∆;b Q(p)ab), let ∆b ∈ Z and r ∈ Z>0 such that f ∈ Sκcl(∆1(pr);Q(p)ab). For x ∈ G(A(∞)), we can take α ∈ G(Q)+ and u·diag[1, t]−1 ∈ ∆ for someb t ∈ (Z(p))× such that x = αu by the strong approximation theorem. Then the action of x on f is given by
fx(z) :=J(α, z)−kfσ(α(z)), (3-0-8)
where σ:= ArtQ(det(u)−1)∈Gal(Qab/Q) and fσ is defined by an(fσ) :=an(f)σ for all n.
4. Λ-adic forms and Λ-adic representations 4.1. On half integral weight
Let Abe aZp-algebra, ∆b ∈ Z and N its level. We letz= (zp, zN)∈ZN :=Z×p ×(Z/NZ)× act on Pk+1/2cl (∆;b A) by
f|z:=zpkf|k+1/2σz (4-1-1)
for σz ∈S(Z) with σz ≡diag[z−1, z] (mod N pr). This action of Z×p extends by continuity to Pb(O).
For ε∈Hom(W, A×) with Kerε=Wpr−1 and a Dirichlet character ψmodulo N p, we put
∆(pr) := ∆1(p)∩Γ0(pr), (4-1-2)
Pk+1/2cl (∆(pr), ψε;A) :={f ∈Pk+1/2cl (∆1(pr);A) |f|z=ψ(z)ε(hzpi)zpkf forz∈ZN}.
(4-1-3)
As shown in [Sh73, Theorem 1.7], the action of the Hecke operatorT`2 for each prime`onPk+1/2cl (∆;b C) is described by
an(f|T`2) =a`2n(f) +χn(`)`−1(`)an(f|`) +`−1an/`2(f|`2) (4-1-4)
in our notation, where we regard a prime ` as an element in ZN and note that RHS equals a`2n(f) if `|N p. The formula (4-1-4) combined with [H90, Theorem 2.2] shows that Pk+1/2cl (∆;b O) is stable under T`2. In particular, we can define the idempotent e0 ∈EndO(Pk+1/2cl (∆;b O)) by taking the limit:
e0 := lim
n→∞(Tp2)n!. (4-1-5)
Hereafter we allow as a base ring O a finite extension of the ring of Witt vectors with coefficients in F¯p. LetP(∆;I) be the space of I-adic cusp forms, i.e.,f ∈P(∆;I) is a formal q-expansion:
f =X
n≥1
an/N(f)qn/N ∈I[[q1/N]]
(4-1-6)
whose specialization fP ∈P(I)[[q1/N]] defined by an/N(fP) := P an/N(f)
at P ∈ A(I) is a classical cusp form inPk(Pcl )+1/2(∆(pr(P)), εP;Cp) for all P ∈ A(I) withk(P)0. We put
P(N;I) :=P(Γ1(N);I).
(4-1-7)
Remark 4.1. If f ∈P(N;I), then f has a formalq-expansion of the form:
f =X
n≥1
an(f)qn∈I[[q]].
(4-1-8)
Since Λ is a regular local ring of dimension 2, I is Λ-free. Fixing a base {ij} of I over Λ, we can write formally that f = P
jfjij and see that fj is a Λ-adic form. Thus P(∆;I) = P(∆; Λ) ⊗Λ I. Let C(W,O) be the space of continuous functions on W having values in O and Meas(W,O) :=
HomcO(C(W,O),O). Recall that there exists an O-algebra isomorphism Λ −∼→ Meas(W,O);a 7→ da such that for any P ∈X(Λ;O), we have P(a) = R
W P da:= da(P|W). Then to each f ∈P(∆; Λ), we attach df ∈Meas W,O[[q1/N]]
by Z
W
φdf :=X
n≥1
Z
W
φdan/N(f)
qn/N (4-1-9)
for any φ∈C(W, O). For each P ∈ A(Λ) withk(P)0, we have Z
W
P df =fP ∈Pk(P)+1/2cl (∆(pr(P)), εP;Cp).
(4-1-10)
Since {P|W | P ∈ A(Λ), k(P) 0} spans a dense subspace of C(W,O), as a measure, df has values inPb(O), and hence
P(∆; Λ)−→∼ Meas(W,P(O));b f 7→df.
(4-1-11)
In particular, the new measure φ 7→ R
Wφdf|s for s ∈ S(Ae (p∞)) again comes from a Λ-adic form f|s ∈ P(∆s; Λ) for a suitable congruence subgroup ∆s corresponding to ∆bs ∈ Z. Thus, we have a natural action ofS(e A(p∞)) on P(I) :=S
∆∈Zb P(∆;I).
Proposition 4.2 ([H95, Proposition 1]). For any prime ` prime to the level of ∆, we have the Hecke operators T`2 and the ordinary projector e0 in EndI(P(∆;I)). The group S(Ae (p∞)) acts on P(I) smoothly. Here the smoothness means that the stabilizer of each v∈P(I) is open.
4.2. On integral weight
Let S(∆;I) be the space ofI-adic cusp forms, i.e., f ∈S(∆;I) is a formal q-expansion f ∈I[[q1/N]]
whose specialization fP ∈P(I)[[q1/N]] atP ∈ A(I) is a classical cusp form in Sk(Pcl )(∆(pr(P)), εP;Cp) for allP ∈ A(I) withk(P)0. Then similar to Proposition 4.2, we have Hecke operatorsTn and the ordinary projector e:= limn→∞(Tp)n! on S(∆;I) (cf. [LFE, Chapter 7]). The group S(A(ptinf ty)) acts on S
∆∈Zb S(∆;I). We actually need to haveG(A(p∞))-action. Note that G(A) =G(Q)G(Zb)G+(R) for the identity connected component G+(R). For any open subgroup U of G(bZ), we write Sk(U;C) for the space of functionsf :G(A)→Csatisfying the following (M1), (M2) and (M3):
f(αxu) =f(x) det(u∞)J(u∞, i)−k (M1)
forα∈G(Q) andu∈U C∞R×.
Df =
k(k−2) 2
f (M2)
for the Casimir operatorD at∞.
Z
Q\A
f
1 u 0 1
x
= 0 (M3)
for all x∈G(A). Let R(U) be a complete representative set forG(Q)\G(A)/U G+(R) in G(Z). Thenb Sk(U;C)−∼→ ⊕t∈R(U)Sclk(ΓtU t−1;C);f 7→(ftcl)t∈R(U),
(4-2-1) where
ΓtU t−1 :=S(Q)∩tU t−1S(R), (4-2-2)
ftcl(z) :=f(tg) det(g)−1J(g, i)k (4-2-3)
forg∈G+(R) such thatz=g(i). We defineSk(U;A) as the image of⊕t∈R(U)Skcl(ΓtU t−1;A). We can take R(U) inside
R:={diag[a,1]|a∈Z(p)}.
(4-2-4)
We always choose R(U) in this way. Then we haveeand Tp well defined onSk(U;Cp). ForU in U :={open subgroups of G(Z(p)},
(4-2-5)
we write U0 :=U ×G(Zp). Taking R(U0) inR so that R(U0)⊃R(V0) ifU ⊂V for allU, V ∈ U, we define
S(U;I) :=⊕t∈R(U0)S(ΓtU0t−1;I), (4-2-6)
S(I) := [
U∈U
S(U;I).
(4-2-7)
Using the stability of S
∆∈Zb S(∆;I) under S(A(p
tinf ty)), we see that S(I) is stable underS(A(p
tinf ty)).
Since diag[a,1] with a∈ (A(p∞))× basically permutates the direct summand S(ΓtU0t−1;I) of S(U;I), we have
Proposition 4.3 ([H95, Proposition 2]). For any prime`prime to the level of U, we have the Hecke operatorsT`and the ordinary projectoreinEndI(S(U;I)). The groupG(A(p∞))acts onS(I)smoothly.
5. Main theorem
Theorem 5.1 ([H95, Theorem 2]). The automorphic representation of S(e A(p∞)) on Pord(I) :=eP(I) is smooth and after extending scalar to K, is a direct sum of irreducible admissible reapersentations with multiplicity at most one.
Roughly speaking, the Λ-adic Shimura correspondence Sh is as follows:
Λ-adic ordinary automorphic representations ofS(e A)
mod P
Sh //
Λ-adic ordinary automorphic representations ofG(A)
mod P2
p-ordinary automorphic representations ofS(e A) of weight k(P) + 1/2
Shcl //
p-ordinary automorphic representations ofG(A)
of weight 2k(P)
(5-0-1)
forP ∈ A(I), whereP2 :=P ◦σ2 and σ2 is an automorphism on a normal extensionI of Λ (i.e.,K/L is Galois) induced by the automorphismu7→u2 on W. Note that
k(P2) = 2k(P) andεP2 =ε2P. (5-0-2)
Let Π be a Λ-adic ordinary automorphic representation ofe S(e A), Π := Sh(Π), ande πP := Π modP2. Then
(1) c(Π) :=p−ordp(c(πP))c(πP) is independent of P.
(2) Π|Z(A(p∞))=ιψ2 for some finite order even characterψ modulo 4pc(Π) and ι:Z(A(p∞)) = (A(p∞))×−→n Z×p
−−−h iW →Λ× (5-0-3)
with n(x) :=|x|−1
A ω(x)−1.
Theorem 5.2 ([H95, Theorem 3]). Let Π be a Λ-adic ordinary automorphic representation of G(A) and ψ a finite order even character ψ modulo4pc(Π) such that Π|Z(
A(p∞)) =ιψ2. Suppose that there exists a Λ-adic ordinary automorphic representation ofS(A)e such that Π = Sh(Π). Then for any paire (m, n) of two square-free positive integers withm/n∈Q
`|N p(Q×`)2, we find two elementsΦandΨ6= 0 in I such that for P ∈ A(I) with k(P)≥2 or ψ2P 6=1, if L(1/2, π⊗ψ−1P χm)6= 0, then we have
Φ(P)2
Ψ(P)2 = L(1/2, πP ⊗ψ−1P χn)
L(1/2, πP ⊗ψP−1χm)ψP(n/m)(n/m)k(P)−1/2, (5-0-4)
where πP := Π mod P2.
6. Construction of Λ-adic forms of half integral weight Fix a Dirichlet character ψ moduloN p.
Proposition 6.1 ([H95, Proposition 3]). The dimension of Pk(Pord)+1/2(∆(pr(P)), ψP;Cp) is bounded independent of P ∈ A(Λ) if k(P)≥1.
To prove the proposition, we prepare several lemmas. Let `be a prime and put Ur,`:={γ ∈S(Z`) |cγ≡0 (mod`r)}.
(6-0-1)
For a character χof Z×` modulo `r and aUr,`-moduleV, we writeV(`r, χ) for theχ-eigenspace, i.e., V(`r, χ) :={v∈V |γv =χ(dγ)v forγ ∈Ur,`}.
(6-0-2)
We denote by Irr(∗) the set of irreducible admissible representation of ∗and for π∈Irr(∗), V(π) the representation space of π.
Lemma 6.2 ([H95, Lemma 1 and 2]). Let π˜∈Irr(S(e Q`))and V :=V(π).
(1) Suppose thatπ˜ appears as a local factor of a holomorphic automorphic representation of weight k+ 1/2 for some k≥2. Then the dimension of V(`r, χ) is bounded independent of V and χ (but it depends on r).
(2) Suppose thatπ˜ is supercuspidal. Then for sufficiently largem, the Hecke operatorT`m annihi- lates V(`r, χ) if r >0.
Lemma 6.3 ([H95, Lemma 3]). Suppose ` ≥ 3. Let V = Bµ,e` be the induced representation space of a quasi character µ of the standard Borel subgroup of S(Qe `), as in [W80, II.2] and [W81, II], for the `-part e` of the standard additive character of A/Q, and χ ∈ Homc(Q×` ,C×). Suppose that r ≥sup(ord`c(χ),ord`(c(µχ)c(µχ−1))), wherec(χ)is the conductor ofχ. Then we have the following:
(1) If µχ|
Z×`
6=1, µχ−1|
Z×`
6=1, then T`2 is nilpotent on V(`r, χ) if r >0.
(2) If µχ|
Z×` 6=1, µχ−1|
Z×` =1, then we can decompose V(`r, χ) =N ⊕V(c(χ), χ) so that T`2 is nilpotent on N, dimCV(c(χ), χ) = 1, andT`2 =χ(`)`k−1/2µ(`−1) on V(c(χ), χ).
(3) If µχ|
Z×` =1, µχ−1|
Z×` 6=1, then we can decompose V(`r, χ) =N ⊕V(c(χ), χ) so that T`2 is nilpotent on N, dimCV(c(χ), χ) = 1, andT`2 =χ(`)`k−1/2µ(`) on V(c(χ), χ).
(4) If µχ|
Z×` = µχ−1|
Z×` = 1, then we can decompose V(`r, χ) = N ⊕V(`, χ) so that T`2 is nilpotent on N, dimCV(`, χ) = 2, and we have a base {v1, v2} of V(`, χ) such that the matrix presentation of T`2 is
χ(`)`k−1/2µ(`−1) χ(`)`k−1/2µ(`)
0 c
(6-0-3)
with some constantc.
Lemma 6.4([H95, Lemma 4] and proof is in [W80, Proposition 18, p.68]). Letρ∗ ∈Irr(PGL2(Q`))and ρ∈Irr(S(e Q`))be the corresponding representation via Weil representation with respect to the additive character eξ` for some ξ ∈ Q×` . Then the local Shimura correspondence Irr(PGL2(Q`)) Irr(S(e Q`)) gives the following:
Equivalence class ofρ∗ 7→Equivalence class of ρ (6-0-4)
π(µ, µ−1) (µ2 6=α)7→π˜µχξ (6-0-5)
σ(µ, µ−1) (µ2 =α, µ6=α1/2)7→σ˜µχξ
(6-0-6)
σ(α1/2, α−1/2)7→supercuspidal (6-0-7)
supercuspidal7→supercuspidal (6-0-8)
where eξ`(z) :=e`(ξz) and we have used the notation of[W80, Proposition 1 and 2].
An automorphic representation of G(A) attache to a primitive formf is said to beordinary atpif f is ordinary at p.
Lemma 6.5 ([H95, Lemma 5], proof is in [H89b, Section 2]). Let π be a unitary holomorphic auto- morphic representation of G(A). Suppose that π is ordinary at p. Then the p-component πp of π is either a principal series representation π(α, β) with unramified α or a special representation σ(α, β) with unramfied α. Let f ∈π be a primitive form of weight k on G(A), λ(Tp) the eigenvalue of f for Tp, and ωπ :=π|Z(G(A)).
(1) If πp =π(α, β) andβ is unramified, then α(p) +β(p) =p(1−k)/2λ(Tp) and α(p)β(p) =ωπ(p).
(2) If πp =π(α, β) andβ is ramified, then α(p) =p(1−k)/2λ(Tp) and α(p)β(p) =ωπ(p).
(3) If πp =σ(α, β), then α(p) =λ(Tp) and π∞ is of weight2.
Lemma 6.6 ([H95, Lemma 6]). Let F be a number field of finite degree. Let ρ be a cuspidal auto- morphic representation of PGL2(FA), where FA is the adele ring of F, and R the set of all cuspidal automorphic representation of S(Fe A). Define for each integral ideal N of F,
R(ρ;N) :={π∈R | π∗v ∼=ρv for all v outside N}, (6-0-9)
where π∗v denotes the corresponding representation of PGL2(Fv) via Weil representations defined in [W80, V.4] (where it is written as:T 7→ V0(e, T); see Lemma 4). Then we have
]R(ρ;N)≤]{Y
v|N
Fv×/(Fv×)2}.
(6-0-10)
Proposition 6.7 ([H95, Proposition 4 and 5 and Corollary 1 and 2]). Let ∆b ∈ Z.
(1) Pord(∆;I) is free of finite rank overI.
(2) LetP ∈ A(I;O). Then for anyf ∈Pk(Pcl )+1/2(∆(pr(P)), εP;O), there existsf ∈Pord(∆;I)such thatfP =f.
(3) ForP ∈ A(I;O) withk(P)0 depending on ∆, we have
Pk(Pcl )+1/2(∆(pr(P)), εP;O)∼=Pord(∆;I)⊗II/P.
(6-0-11)
(4) Let f1, . . . ,fr be a base of Pord(∆;I). Then there exist positive integers n1, . . . , nr such that det(ani/N(fj))∈I×.
7. A result slightly stronger main theorem
Let hord(N;O) be the p-adic ordinary Hecke algebra defined in [LFE, Section 7.3], i.e., the Λ- subalgebra of EndΛ(Sord(N; Λ)) generated by Tn for all n. Under the pairing hh,fi := a1(f|h), we know
HomΛ(hord(N;O),Λ)∼=Sord(N; Λ) and HomΛ(Sord(N; Λ),Λ)∼=hord(N;O).
(7-0-1)
Kitagawa’s two variable p-adic L-function ([K, Theorem 6.2]). Letλ∈HomI-alg(hord(N,O)⊗Λ I,I) andχ a primitive Dirichlet character of conductor ∆pm withm≥0 andp-∆. Then there exists ap-adic analytic function Lp(·,·;λ⊗χ) :X(I)×Zp→Q¯p such that for anyP ∈ A(I;O) withk(P)≥2 and ν ∈[0, k(P)−2]∩Z,
Lp(P, ν;λ⊗χ)
EP± = (−∆)ν pν
ap(fP) m
1−χ(p)pν ap(fP)
G(χ)ν!L(ν+ 1,fP ⊗χ) (−2πi)ν+1Ω±f
P
, (7-0-2)
where fP is the Hecke eigenform corresponding to λ mod P via the duality, EP± ∈ O is the error term ([K, Section 5.5]), and Ω±f
P ∈ C×p is the complex period attached to fP ([K, Section 3.8]) with
±:= sgn((−1)ν+1χ(−1)).
Let λ ∈HomI-alg(hord(C,O)⊗ΛI,I) be primitive ([H88a, Theorem 4.1]) andπ a unique factor of Sord(¯L) corresponding toλ, i.e., for anyP ∈ A(I) withk(P)≥2,λmodP ↔π modP via the duality (7-0-1). We define ψ0 ∈ Hom(ZC,C×p) by ψ0(zp, zC) := λ(hhω(zp)zCii), where hhzii is the image in hord(C,O)⊗ΛI of the action of z ∈(Z(p))×⊂G(A(p∞)) on S(I) defined by (4-1-1). We consider the following condition:
ψ0 =ψ2 for some even character ψmodulo N forN divisible by C and 4.
(Hp)
Here is a result slightly stronger than (5.2).
Theorem 7.1 ([H95, Theorem 4]). Let λ∈ HomI-alg(hord(C,O)⊗ΛI,I) be primitive. Suppose (H`) for any prime ` 6= p and (Hp). Then for any pair (m, n) of two square-free positive integers with m/n∈Q
`|N p(Q×` )2, there existsΦ∈K such that for all P ∈ A(I) with k(P)≥1, if Lp(P2, k(P), λ⊗ ψ−1χm)6= 0, then we have
Φ(P)2 = Lp(P2, k(P), λ⊗ψ−1χn)
Lp(P2, k(P), λ⊗ψ−1χm)ψP(n/m)(n/m)k(P)−1/2. (7-0-3)
Here note that m/n is prime to N p under our assumption on (m, n).
References
[F] Y. Flicker. Automorphic forms on covering groups GL(2).Invent. Math., 57:119–182, 1980.
[H88a] H. Hida. Ap-adic measure attached to the zeta funtions associated with two elliptic mofular forms, II.Ann.
Inst, Fourier, 38:1–83, 1988.
[H88b] H. Hida. Onp-adic Hecke algebras for GL2 over totally real fields.Ann. of Math., 128:295–384, 1988.
[H89a] H. Hida. On nearly ordinay Hecke algebras for GL(2) over totally real fields. Adv. Studies in Pure Math., 17:139–169, 1989.
[H89b] H. Hida. Nearly ordinary Hecke algbras and Galois representations of several varieties.Amer. J. of Math., pages 115–134, 1989.
[H90] H. Hida.p-adicL-functions for base change lifts of GL2 to GL3.Perspective in Math., 11:93–142, 1990.
[H92] H. Hida.Geometric modular forms. Proc. CIMPA Summer School at Nice, 1992.
[LFE] H. HidaHida. Elementary theory of L-functions and Eisenstein series, volume 26 of London Mathematical Society Student Texts. Cambridge University Press, February 1993.
[H95] H. Hida. On Λ-adic forms of half integral weitht for SL(2)/Q.in Number Theory, Paris, Lecture notes series of LMS, 215:425–475, 1995.
[K] K. Kitagawa. On standardp-adicL-functions of families of elliptic cusp forms. Comtemp. Math., 165:81–110, 1994.
[MFM] T. Miyake.Modular Forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1989.
[P] S. Purkait. Explicit application of Waldspurger’s theorem.LMS J. Comput. Math, 16:216–245, 2013.
[Sh73] G. Shimura. On modular forms of half integral weight.Ann. of Math., 97(3):440–481, May 1973.
[Sh78a] G. Shimura. On certain reciprocity laws for zeta functions and modular forms.Acta Math., 141:35–71, 1978.
[V] M. F. Vigneras. Valeur au centre de symetrie des functions L associ´ees aux formes modulaires. Progress in Mathematics, 12:331–356, 1981.
[W80] J. L. Waldspurger. Correspondence de Shimura.J. Math. Pures Appl., 59:1–133, 1980.
[W81] J. L. Waldspurger. Sur les coefficients de fourier des formes modulaires de poids demi-entier.J. Math. Pures Appl., 60(4):375–484, 1981.
[Weil] A. Weil. Sur certains groupes d’op´erateurs unitaires.Acta Math., 111:143–211.
Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto 603- 8555, Japan
E-mail address:kenji [email protected]
