L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Daniel DELBOURGO & Tom WARD

Non-abelian congruences betweenL-values of elliptic curves Tome 58, n^o3 (2008), p. 1023-1055.

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Article mis en ligne dans le cadre du

NON-ABELIAN CONGRUENCES BETWEEN L-VALUES OF ELLIPTIC CURVES

by Daniel DELBOURGO & Tom WARD (*)

Abstract. — LetEbe a semistable elliptic curve overQ. We prove weak forms of Kato’sK1-congruences for the special valuesL 1, E/Q(µpⁿ,^pn√

∆)

.More pre- cisely, we show that they are true modulopⁿ⁺¹, rather than modulop²ⁿ. Whilst not quite enough to establish that there is a non-abelian L-function living in K1 Zp[[Gal(Q(µp^∞,^p∞√

∆)/Q)]]

, they do provide strong evidence towards the ex- istence of such an analytic object. For example, ifn= 1these verify the numerical congruences found by Tim and Vladimir Dokchitser.

Résumé. — SoitE une courbe elliptique définie surQ. Nous démontrons des versions faibles des congruences K1 de Kato, pour les valeurs spéciales L 1, E/Q(µpⁿ,^pn√

∆)

.Plus précisément, nous vérifions que les congruences sont vraies modulopⁿ⁺¹, plutôt que modulop²ⁿ. Bien que ça ne suffise pas pour établir l’existence d’une fonctionL p-adique qui vit dansK1 Z^p[[Gal(Q(µp^∞,^p∞√

∆)/Q)]]

, elles fournissent beaucoup d’indices de l’existence de cet objet analytique. Par exemple, sin= 1les congruences trouvées numériquement par Tim et Vladimir Dokchitser sont vraies.

1. Introduction

In this paper, we study the behaviour of the Hasse-Weil L-functions of elliptic curves, over the so-called “False Tate Curve” extensions of Q. These are non-abelianp-adic Lie extensions of dimension two, which may be constructed as follows.

Fix a prime number p 6= 2. Let ∆ > 1 denote a p-power free integer.

We suppose that∆ is coprime top, which ensures all the primes above∆

Keywords:Iwasawa theory, modular forms,p-adicL-functions.

Math. classification:11R23, 11G40, 19B28.

(*) To form part of the second author’s PhD thesis.

tamely ramify in the false Tate curve tower. For each integern>0, we set Kn =Q(µpⁿ)and writeFn=Q(µpⁿ)⁺for the maximal real subfield. Then

QF T = [

n>1

Q

µpⁿ, ^pn

√

∆ .

Basic Galois theory informs us that Gal(QF T/Q)∼=

Z^×p Zp

0 1

CGL₂(Zp).

In other words, the Galois group is a semi-direct product of twop-adic Lie groups of dimension one. In terms of a field diagram,

ZpoZ^×p

QF T

Q(µ_pn, ^pn√

∆)

pⁿ Z/pⁿZ

Kn

rrrrrr2rrrrr

(Z/pⁿZ)^×

Fn

(pⁿ−pⁿ⁻¹)/2

p

Q p

In this situation the representation theory is very well understood. It is proved in [6] that Gal(QF T/Q) has a unique self-dual representation of dimensionp^k−p^k−1, which we denote byρk,Qfor eachk>1. This may be written

ρk,Q= Ind^Q_K

kχρ_k

for a characterχ_ρk of Gal(QF T/K_k). Puttingρ_0,Q =1, every irreducible representation ofGal(QF T/Q)has the formρk,Q⊗ψfor somek>0, and some character

ψ: Gal Q(µp^∞)/Q

→C^×. For the rest of this section, we setG= Gal(QF T/Q).

Remark. — LetEbe an elliptic curve overQ. As part of a more general

“GL2-Main Conjecture”, Coates et al [3] predict the existence of a non- abelianp-adicL-function

L^anal_p (E/QF T)∈K1(Zp[[G]]S^∗)

whose evaluation at Artin representations ρ : G → GL(V) essentially yield the ρ-twisted L-values L(1, E, ρ). Here Zp[[G]]_S^∗ is the localisation ofZp[[G]] at a certain Ore setS^∗=S

n>0p⁻ⁿS.

In his beautiful paper [9], Kato reduced the question of existence for L^anal_p into a sequence of congruence relations, amongst the abelian p-adic L-functions interpolatingEover the false Tate curve extension. More pre- cisely, ifU⁽ⁿ⁾= ker(Z^×p →(Z/pⁿZ)^×)then there exists an injection

Θ_G,S^∗ :K1(Z^p[[G]]_S^∗) Q_(ρ

n)∗

−→ Y

n>0

Quot(Z^p[[U⁽ⁿ⁾]])^×.

Kato calculated that a sequence(an)n>0 lies in the image ofΘG,S^∗ if and only if

Y

16i6n

Ni,n

ai

N_0,i(a₀).φ◦N_0,i−1(a0) φ(a_i−1)

^pi

≡1 modp²ⁿ for alln∈N. Here we should point out thatNi,j:Zp[[U⁽ⁱ⁾]]^×→Zp[[U^(j)]]^× denotes the norm map, andφ:Zp[[Z^×p]]→Zp[[Z^×p]]is the ring homomorphism induced by thep-power map onZ^×p.

Let E be an elliptic curve defined over the rationals, in particular, it is modular by the work of Wiles et al. Moreover, we shall assume thatE is semistable (otherwise our distributions turn out to be identically zero). We writef_E for the newform of weight two and conductorN_Eassociated toE.

In order to state our full results, we are forced to make three assumptions about the primep, and the fieldFn=Q(µpⁿ)⁺.

Hypothesis (Ord). — The elliptic curveEhas good ordinary reduction atp, with∆ andNE coprime integers.

Hypothesis (p-Irr). — The base changef_/Fnoff_E to the fieldF_n is not congruent modulop to any other Hilbert modular form of the same level.

Hypothesis (Per). — Conjecture 1.3 from [4] holds for the cusp formf_E and the fieldFn. This implies the automorphic and motivic periods ofE are the same, up to ap-adic unit.

The third hypothesis has been verified numerically by Doi-Hida-Ishii in many cases. It is intimately connected to the orders of congruence modules for the space of Hilbert automorphic forms overF_n (see [4] for the precise statement).

Our first result is an integral version of a theorem of Shai Haran from [8], concerning the existence of abelianL-functions which are attached to the

motives h¹(E)⊗_ZM(ρ_n). Recall that for an Artin representation ρ over Fn, one defines the ρ-twistedL-functionL(s, E, ρ)by the infinite product

L(s, E, ρ) =Y

v

det

1−N_Fn/Q(v)^−sΦ_v

H_l¹(E)⊗

Qlρ^Iv−1

whereΦvis a geometric Frobenius element forv, andIvis the inertia group.

This Euler product converges to an analytic function on Re(s) >3/2. In general, the L-functions of E and its twists are conjectured to have an analytic continuation to all ofC.

Theorem 1.1. — Let p denote the unique prime of F_n above p. As- sume that E satisfies Hypothesis (Ord), and that in addition both of (p-Irr) and (Per) hold forf_E over F_n. Then there exists a unique element Lp(E, ρn)∈Zp[[U⁽ⁿ⁾]]such that

ψ(Lp(E, ρn)) = Fn(ρn⊗ψ)p

αp^{f(ρn⊗ψ,p)}

× Pp(ρn⊗ψ, α^−[Fp ^n:Q]) Pp(ρn⊗ψ⁻¹, α⁰p^−[Fn:Q])

×L_S(1, E, ρ_n⊗ψ⁻¹) (Ω⁺_EΩ⁻_E)^φ(pn)/2 for all finite charactersψofU⁽ⁿ⁾.

Note. — Here Pp(ρn⊗ψ, X)denotes the characteristic polynomial of Φ_pon the inertia invariant subspace, andα_pdenotes thep-adic unit root of

X²−a_p(E)X+p

withα⁰_p being the non-unit root. Further, _Fn(ρ_n⊗ψ)_p denotes the local -factor atp. This factor depends on the choice of a local Haar measure and an additive character atp(see [14] for details). We choose the Haar measure dxwhich givesZp measure 1, and the additive characterτ: (Qp,+)→C^× given byτ(ap^−m) = exp(2πia/p^m)witha∈Zp (these are the choices used in [3]).

It is vital to know thep-integrality of the aboveL-function when tackling Kato’s higher congruences. We now puta_n=L_p(E, ρ_n), and consider both

bn=an/N0,n(a0) and cn=bn/φ(bn−1).

Kato’s calculations in [9] of the image ofK1 predict that Y

16i6n

N_i,n(c_i)^pi ≡1 modp²ⁿ.

Theorem 1.2. — Under the same hypotheses as the previous result, the non-abelian congruences

Y

16i6n

N_i,n(c_i)^pi ≡1 modpⁿ⁺¹ hold true.

Whenn >1these congruences are (conjecturally) not the best possible.

For example, ifn= 2we expect a congruence modulo p⁴ notp³, if n= 3 we expect one modulop⁶ notp⁴, and so on. However whenn= 1, we have proved the congruences found numerically by Tim and Vladimir Dokchitser.

Theorem 1.3. — Under the same hypotheses, the congruences com- puted in[5]always hold, i.e.

a1≡N0,1(a0) modp.

Remarks.

(i) Theorem 1.3 was proved at p= 3by T. Bouganis in [1], using prop- erties of the 3-adic Eisenstein measure. Likewise, he had a non-integral version of Theorem 1.1.

(ii) In fact Hypothesis (Per) is not actually necessary. However, it must then be replaced with the assumption that the ρn-twisted homology of the space of modular symbols on Γ₀(N_E) is p-integral (which is an old conjecture of Glenn Stevens from [13]).

(iii) If we replace Hypothesis (Ord) with the condition that E has bad multiplicative reduction at p, then the congruences mentioned above still hold.

Acknowledgements. — The authors thank Vladimir Dokchitser for his very helpful comments, and also for the argument which proves Claim(?).

They are also grateful to Thanasis Bouganis, and strongly urge the reader to consult his forthcoming work on the period relations.

2. Algebraic-valued distributions Let ρk := Ind^F_K^k

kχρ_k denote the two-dimensional Artin representation over F_k, and writeρ_k/F_n := Res_Fnρ_k for its restriction to F_n. Consider the finite set of primes

S={v:v is a prime ofFn,v|p∆}.

Our main goal is to show for every integern>k, there exists aQ-valued distribution interpolating

simple factors×L_S(1, E, ρ_k/F_n⊗ψ⁻¹) a period

for a suitable family of Hecke charactersψ(see Theorem 3.4 for the precise statement).

2.1. Hilbert modular forms

Let F be a totally real field such that F/Q is abelian. Following the notation from [10], leth=|Cl^†(F)|be the narrow class number ofF, and choose idelest1, . . . , thsuch that ˜tλCOF (the ideals generated by the tλ) are all prime top, and form a complete set of representatives for Cl^†(F).

We also denote the different ofF/Qbyd_F.

Hilbert automorphic forms overFare holomorphic functionsf : GL2(AF)

→Csatisfying certain automorphy properties (see [10] or [12] for details).

They also correspond toh-tuples(f1, . . . , fh)of Hilbert modular forms on H^d (where d= [F :Q]). Iff ∈ M_k(c, ψ)(the set of Hilbert automorphic forms of parallel weightk, levelcand characterψ) then

fλ|kγ=ψ(γ)fλ

for allγ∈Γλ(c), where Γ_λ(c) =

a b c d

:b∈t˜⁻¹_λ d⁻¹_F , c∈˜t_λcd_F, a, d∈ O_F, ad−bc∈ O^×_F

. We define

eF(ξz) = exp

2πi X

16a6d

ξ^τaza

where z = (z1, . . . , zd) ∈ H^d, ξ ∈ F and τ1, . . . , τd are the embeddings F ,→R. Then, each componentf_λhas a Fourier expansion of the form

fλ(z) =X

ξ

aλ(ξ)eF(ξz),

where the sum is taken over all totally positiveξ∈˜tλ and ξ= 0. If f is a cusp form, thenaλ(0) = 0for allλ. The set of cusp forms of parallel weight k, levelcand characterψis written Sk(c, ψ).

The form f itself also has Fourier coefficients C(m,f)which satisfy C(m,f) =

(a_λ(ξ)N_{F /Q}(˜t_λ)^−k/2 if the idealm=ξ˜t⁻¹_λ is integral;

0 ifmis not integral.

We will use certain linear operators on the space of Hilbert automorphic forms. Letqbe an integral ideal ofOF, andqan idele such thatq˜=q. We define the operatorsqandU(q)onf ∈ M_k(c, ψ):

(f|q)(x) =N_{F /Q}(q)^−k/2f

x q 0

0 1

(f|U(q))(x) =N_{F /Q}(q)^k/2−1 X

v∈OF/q

f

x 1 v

0 q

.

These operators may also be described by their effect on the Fourier coef- ficients off, namely

C(m,f

q) =C(mq⁻¹,f) and C(m,f|U(q)) =C(mq,f).

We also use the involutionJc, which is defined by (f|Jc)(x) =ψ(det(x)⁻¹)f

x

0 1 c₀ 0

where c₀ is an idele such that ˜c₀ = cd²_F. Then, f|Jc ∈ Mk(c, ψ⁻¹). This map has the property

f|Jmc=NF /Q(m)^k/2(f Jc)

m.

Further, whenf is a primitive form inM_k(c, ψ), we have f|Jc= Λ(f)f^ι,

where Λ(f) is a root of unity, and f^ι is the form defined by C(m,f^ι) =C(m,f).

Remark. — If fE ∈ S₂^new(Γ0(NE))is the newform associated to E_/Q, then we writef for the base change off_E to the totally real field F. As- sumingF/Qis abelian, this is the Hilbert automorphic form whoseL-series satisfies

L(s,f) = Y

ψ∈Gˆ

L(s, f_E, ψ)

whereG= Gal(F/Q).

We introduce the following notation: for a characterχ: Gal(QF T/Fn)→C^×, we writeχ^†:IFn→C^×for the character of ideals associated toχvia com- position with the reciprocity map. Specificallyχ^† is normalised by

χ^†(q) =χ(Frob_q)

for all primesqofF_n, whereFrob_qdenotes an arithmetic Frobenius element atq.

Now, let K/F be a totally imaginary quadratic extension. We have the following theorem due to Serre [11]:

Theorem 2.1. — If ρ is an Artin representation over F which is in- duced from the Hecke character χ_ρ over K, then there exists a Hilbert automorphic formgρ overF such thatgρ∈ S1(c(gρ),(detρ)^†)and

L(s,gρ) =L(s, ρ).

Further,g_ρis primitive if and only if χ_ρ is a primitive character.

It is easily checked that the Fourier coefficients ofgρ are C(m,gρ) = X

aCOK , a¯a=m

χ^†_ρ(a).

Also, in the caseF =Fk, K=Kkandρ=ρk, we assumegcd(p∆, NE) = 1 which implies thatc(f)andc(gρ_k)are coprime ideals ofOF_k.

The character(detρ)^† can be written as

(detρ)^†(a) =θ_K/F(a)χ^†_ρ(aOK)

whereθ_K/F is the quadratic character ofK/F, given on primes ofOF by

θ_K/F(q) =











1 ifq splits inK/F

−1 ifq is inert inK/F 0 ifq ramifies inK/F .

We use a non-standard normalisation of the Petersson inner product (from [10]), namely

hF,Gi_c:=

h

X

λ=1

Z

Γ_λ(c)\H^d

Fλ(z)Gλ(z)N(y)^kdν(z) whered= [F:Q], and

dν(z) = Y

16j6d

y_j⁻²dx_jdy_j.

FinallyΩ^Aut_E/F

n= (2π)^φ(pn)

f_/Fn,f_/Fn

c(f)denotes the automorphic period.

2.2. Integrality

We will study the value ats= 1of the normalised Rankin-Selberg prod- uct

Ψ(s,f,g_ρ) = Γ(s)

(2π)^s 2[F:Q]

L_c(2s−1,(detρ)^†)L(s,f,g_ρ)

wherec=c(f)c(g_ρ), and L(s,f,gρ) =X

a

C(a,f)C(a,gρ)NF /Q(a)^−s. Our first goal is to prove that

_F(0, ρ)·Ψ(1,f,g^ι_ρ) hf,fi_c(f) ∈ O_Cp,

i.e.that this quantity isp-integral (it is already known to be algebraic by results of Shimura et al).

We will need a few preparatory lemmas, starting with a result about the epsilon factor_F(s, ρ). The Artin L-functionL(s, ρ)obeys the functional equation

Γ_∞(s)L(s, ρ) =F(s, ρ)Γ_∞(1−s)L(1−s, ρ^∨) whereρ^∨ is the contragredient representation, and

Γ∞(s) := ((2π)^−sΓ(s))^[F:Q].

The global-factor at zero may be decomposed into an infinite product _F(0, ρ) = Y

all placesv

_Fv(ρ_v, ψ_ν, dx_ν)

where each local factor depends on the normalisation of additive characters ψν, and Haar measuresdxν (however the product does not).

Lemma 2.2. — Setting F(ρ) =F(0, ρ), we have Λ(gρ) =i^−[F:Q]N_{F /Q}(cd²_F)^−1/2F(ρ).

Proof. — Following Shimura in [12], we define R(s,g) :=N_{F /Q}(cd²_F)^s/2Γ_∞(s)L(s,g)

wheregis a Hilbert automorphic form of parallel weight 1 and conductorc.

Then from [12], (2.48) there is a functional equation R(s,g) =i^[F:Q]R(1−s,g|Jc).

Supposinggis primitive, we haveg|Jc= Λ(g)g^ι, so the functional equation becomesR(s,g) =i^[F:Q]Λ(g)R(1−s,g^ι). However, takingg=gρwe obtain L(s, ρ) =L(s,g)andL(s, ρ^∨) =L(s,g^ι). Therefore

Γ∞(s)L(s, ρ) =F(s, ρ)Γ∞(1−s)L(1−s, ρ^∨) can be rewritten as

R(s,g) =F(s, ρ)NF /Q(cd²_F)^s−1/2R(1−s,g^ι).

Comparing this with the functional equation from [12], it follows that there is an equalityi^[F:Q]Λ(g) =F(s, ρ)N_{F /Q}((cd²_F)^s−1/2, which gives the result.

We use the following integral representation, a special case of [12], (4.32):

Proposition 2.3.

Ψ(1,f,g^ι) =D_F^1/2π^−[F:Q]hf^ι, V(0)i_c whereD_F is the discriminant ofF/Qand

V(0) =g^ι·K₁⁰(0;c,OF; (detρ)^†−1),

with K₁⁰ the Eisenstein series given in [10] equation (4.5) whose λ-components are:

K₁⁰(0;c,OF;ω)λ(z)

=N_{F /Q}(˜t_λ)^1/2X

c,d

sign(N_{F /Q}(d))ω^∗(dOF)N_{F /Q}(cz+d)⁻¹. Note that the sum is taken over the set of equivalence classes

(c, d)∈ ˜tλdFc× OF

∼

where the relation∼is defined by(c, d)∼(uc, ud)for allu∈ O^×_F.

It is useful to convertK₁⁰to an Eisenstein series which has a user-friendly Fourier expansion; we can do this via the involution J_c. Using [10] (4.6), one can show:

K₁⁰(0;c,OF; (detρ)^†−1)

Jc= (4πi)^[F:Q] D_F^1/2NF /Q(c(g)d²_F)^1/2

E1(0,c,(detρ)^†−1).

HereE1 is the Eisenstein series (4.13) in [10], withλ-components E1(0,c, ω)λ(z)

=N_{F /Q}(˜t_λ)^−1/2D_F^1/2 (−4πi)^[F:Q]

X

c,d

sign(N_{F /Q}(c))ω^∗(cOF)N_{F /Q}(cz+d)⁻¹ such thatωis an ideal character modulo c, and the sum ranges over

(c, d)∈O_F טt⁻¹_λ d⁻¹_F

∼ .

The Fourier expansion of eachλ-component is computed in [10], Prop 4.2:

E₁(0,c,(detρ)^†−1)_λ(z) =N_{F /Q}(˜t_λ)^−1/2 X

0ξ∈˜t_λ

a_λ(ξ)e_F(ξz)

with

aλ(ξ) = X

ξ=˜˜ b˜c, c∈OF ,

b∈˜tλ

(detρ)^†−1(˜c).

We are now in a good position to prove our integrality result.

Theorem 2.4. — Letg=g_ρ. If there exists no non-trivial congruence modulop betweenf and another automorphic form inM2(c(f)), then

F(ρ)·Ψ(1,f,g^ι) hf,fi_c(f) isp-integral, whereF(ρ) =F(0, ρ)as before.

Proof. — Letc=c(f)c(g). By Proposition 2.3, Ψ(1,f,g^ι) =D_F^1/2π^−[F:Q]hf^ι, V(0)i_c whereV(0) =g^ι·K₁⁰(0;c,OF; (detρ)^†−1). ConsiderV(0)

Jc; by our earlier formula forK₁⁰

J_c we have V(0)|Jc= (g^ι

Jc)·(K₁⁰ Jc)

= Λ(g^ι)N_{F /Q}(c(f))^1/2(g

c(f))·(K₁⁰ Jc)

= Λ(g^ι)(4πi)^[F:Q]D^−1/2_F NF /Q(c(f))^1/2NF /Q(c(g)d²_F)^−1/2(g c(f))

·E1(0,c,(detρ)^†−1).

We now employ the trace mapTr^c_c(f): M2(c, ψ)→ M2(c(f), ψ), which is defined by

H

Tr^c_c(f)

(x) =X

v∈T

H

x 1 0

cv 1

.

wherecis an idele such thatc˜=c(f), andT is a set of coset representatives forOF/c(g). This map has the property

hF,Hi_c=D F,H

Tr^c_c(f)E

c(f)

for any two Hilbert automorphic forms H ∈ M2(c, ψ),F ∈ S2(c(f), ψ).

Further, from [10] equation (4.11) we have the formula H

Tr^c_c(f)=H

J_c◦U(c(g))◦J_c(f).

This arises from the definitions of the operators, and the matrix identity 1 0

cv 1

= (cm)⁻¹

0 1 cm 0

1 v

0 m

0 1 c 0

which holds for anyc,mandv.

Remark. — SettingΘ = (g|c(f))·E₁(0,(detρ)^†−1), we calculate Ψ(1,f,g^ι) =D_F^1/2π^−[F:Q]hf^ι, V(0)i_c

=D_F^1/2π^−[F:Q]D

f^ι, V(0) Tr^c_c(f)E

c(f)

=D_F^1/2π^−[F:Q]

f^ι, V(0)

J_c◦U(c(g))◦J_c(f)

c(f)

= Λ(g^ι)(4i)^[F:Q]N_{F /Q}(c(f))^1/2N_{F /Q}(c(g)d²_F)^−1/2

· f^ι,Θ

U(c(g))◦J_c(f)

c(f). Observe thatΛ(g^ι)(4i)^[F:Q]NF /Q(c(f))^1/2is ap-adic unit, asgcd(p,4c(f))=1.

Also, from Lemma 2.2 we know thatordp(NF /Q(c(g)d²_F)^1/2) = ordp(F(ρ)).

Therefore,

ordp F(ρ)·Ψ(1,f,g^ι) hf,fi_c(f)

!

= ordp

f^ι,Θ

U(c(g))◦J_c(f)

c(f)

hf,fi_c(f)

!

so it suffices to prove thep-integrality of the quantity on the right hand side. As the operatorJ is an involution,

f^ι,Θ

U(c(g))◦J_c(f)

c(f)= f^ι

J_c(f),Θ|U(c(g))

c(f)

= Λ(f^ι) f,Θ

U(c(g))

c(f).

By choosing a basis for the spaceM2(c(f))which includesf, we may express Θ

U(c(g)) =cf+X

f_i6=f

cifi

bi

for algebraic numbersci (which are almost all zero), and primitive forms f_i of levela_i such thata_ib_i divides c(f).

We deduce that f^ι,Θ

U(c(g))◦Jc(f)

c(f)

hf,fi_c(f) = Λ(f^ι) c+X

f_i6=f

ci

f,fi

bi

c(f)

hf,fi_c(f)

! .

Quoting Dünger’s paper [7], Section 5.5, f,fi

bi

c(f)

hf,fii_c(f) =

L(s,f,f_i^ι|bi) L(s,f,f_i^ι)

s=2

. One may therefore write

f,fi

bi

c(f)

hf,fi_c(f) =

L(s,f,f_i^ι|bi) L(s,f,f_i^ι)

s=2

×hf,f_ii_c(f) hf,fi_c(f).

Buthf,f_ii_c(f)= 0 asf andf_i are distinct primitive forms, whence f,fi

bi

c(f)

hf,fi_c(f) = 0 for eachi. As a consequence,

f^ι,Θ|U(c(g))◦J_c(f)

c(f)

hf,fi_c(f) = Λ(f^ι)c.

LastlyΛ(f^ι)is a root of unity, thus it suffices to provecisp-integral. Sup- pose not; thenc⁻¹ ≡0 modp. We know that both E1(0,(detρ)^†−1) and gρ havep-integral Fourier coefficients, soΘ = (gρ|c(f))·E1(0,c,(detρ)^†−1) does too. Therefore

c⁻¹Θ

U ≡0 modp, whence

f =c⁻¹Θ

U−X

f_i6=f

c⁻¹cifi

bi

≡ −X

fi6=f

c⁻¹cifi

bi modp.

It follows thatf is congruent modulopto some other automorphic form in M2(c(f)), contradicting the hypothesis of the theorem. This completes the

proof.

2.3. Constructing the distribution

Having established our integrality result, we can now go on to construct the distribution. First we work over a totally real fieldF.

For a finite placev6=pofF, we label rootsα(v),α⁰(v)of the polynomial X²−C(v,f)X+N_{F /Q}(v) = (X−α(v))(X−α⁰(v)).

We also defineα(p)andα⁰(p)as the roots of

X²−C(p,f)X+p= (X−α(p))(X−α⁰(p)).

where α(p) is thep-adic unit, andα⁰(p) is the non-unit root. From these definitions, we extendα(m),α⁰(m)multiplicatively to all idealsm ofOF.

Definition 2.5. — Setl0 :=Q

q|∆q. Then thepl0-stabilisation off is defined to be

f0:= X

a|pl0

M(a)α⁰(a)·f a

whereM is the Möbius function on ideals.

Following (3.14) of [10], we also define g_ρ,pl

0 := X

n|pl₀

M(n)·g_ρ

U(n)◦n.

In particularg_ρ,pl0∈ M₁(c(g_ρ)p²l²₀,(detρ)^†).

Setc0=pl0c(f). We shall choose idealsm⁰ andl⁰ such thatm⁰ is a power ofp,supp(l⁰) = supp(l0), and thatc(gρ)p²l²₀|m⁰l⁰. Clearly

f0∈ S2(c0)⊂ S2(c(f)m⁰l⁰) and

gρ,pl₀∈ M1(c(gρ)p²l²₀,(detρ)^†)⊂ M1(c(f)m⁰l⁰,(detρ)^†).

Then the contragredient Euler factor is given by Eulpl₀(ρ^∨, s) := Y

v|pl₀

(1−α⁰(v) ˆβ(v)N(v)^−s)(1−α⁰(v) ˆβ⁰(v)N(v)^−s)

×(1−α⁻¹(v)β(v)N(v)^s−1)(1−α⁻¹(v)β⁰(v)N(v)^s−1).

N.B. We have factorised the Hecke polynomial as

X²−C(v,gρ)X+ (detρ)^†(v) = (X−β(v))(X−β⁰(v)), and similarly the dual Hecke polynomial via

X²−C(v,g_ρ)X+ (detρ)^†−1(v) = (X−β(v))(Xˆ −βˆ⁰(v)).

Remark. — Because we assumed thatE is semistable overQ, the coef- ficient

C(c(f),f) = (−1)^#TFns

whereT_F^nsdenotes the set of finite places whereEhas non-split multiplica- tive reduction. In particular,C(c(f),f)6= 0.

Lemma 2.6.

Ψ(s,f0,gρ,pl₀

J_c(f)m⁰_l⁰) =N_{F /Q}

c(f)m⁰l⁰ c(gρ)

1/2−s

Λ(gρ)α m⁰l⁰

c(gρ)

×C(c(f),f) Eulpl₀(ρ^∨, s)Ψ(s,f,g^ι_ρ).

Proof. — Recall the formula f|Jmc = N_{F /Q}(m)^k/2(f Jc)

m from Sec- tion 2.1. Sincec(g_ρ,pl0) =c(g_ρ)p²l²₀, it follows that

gρ,pl₀

Jc(f)m⁰l⁰ =NF /Q

c(f)m⁰l⁰ c(gρ)p²l²₀

1/2

· gρ,pl₀

J_c(gρ)p2l²₀

c(f)m⁰l⁰ c(gρ)p²l²₀. For brevity, we will write

h=gρ,pl₀

J_c(gρ)p2l²₀.

Then

Ψ(s,f0,gρ,pl₀

Jc(f)m⁰l⁰)

=N_{F /Q}

c(f)m⁰l⁰ c(gρ)p²l²₀

1/2

Ψ

s,f₀,h

c(f)m⁰l⁰ c(gρ)p²l²₀

=N_{F /Q}

c(f)m⁰l⁰ c(g_ρ)p²l²₀

1/2−s

Ψ

s,f0

U

c(f)m⁰l⁰ c(g_ρ)p²l²₀

,h

=N_{F /Q}

c(f)m⁰l⁰ c(gρ)p²l²₀

^1/2−s α

m⁰l⁰ c(gρ)p²l²₀

C(c(f),f)Ψ (s,f0,h). Here we have exploited the fact that

L s,f₀,g_ρ^ι|a

=N_{F /Q}(a)^−sL s,f₀|U(a),g_ρ^ι for any ideala, and also the formula

f₀ U

c(f)m⁰l⁰ c(gρ)p²l²₀

=α

m⁰l⁰ c(gρ)p²l²₀

C(c(f),f)f₀

which follows by construction of thepl₀-stabilisationf0. Furthermore, Ψ(s,f0,h) =N_{F /Q}(p²l²₀)^1−2sα(p²l²₀)Λ(gρ) Eulpl₀(ρ^∨, s)Ψ(s,f,g^ι_ρ).

Combining the two equations together yields the required result.

Definition 2.7. — We define the complex linear functionalLF on the vector spaceM2(c0)by the rule

LF : Θ7−→

f0ι

,Θ Jc₀

c0

hf,fi_c(f) . We shall now consider the automorphic form

Φ = Φ(ρ/F,c(f)m⁰l⁰) :=gρ,pl₀·E1(0,c(f)m⁰l⁰,(detρ)^†−1) whereE1refers to the Eisenstein series of Section 2.2.

Corollary 2.8.

N_{F /Q}(c(gρ)d²_F)^1/2Λ(gρ)

α(c(g_ρ)) Eul_pl0(ρ^∨,1)Ψ(1,f,g_ρ^ι) hf,fi_c(f)

= (−4i)^[F:Q]

α(m⁰l⁰)C(c(f),f)L_F Φ

U(m⁰l⁰p⁻¹l⁻¹₀ ) . Proof. — Applying the integral representation from Proposition 2.3, then using the trace mapTr^c(f_c0 ^)m0l0 as we did to prove integrality, one obtains

Ψ(1,f0,gρ,pl₀

J_c(f)m0l⁰)

hf,fi_c(f) = (−4i)^[F:Q]

N_{F /Q}(c(f)m⁰l⁰d²_F)^1/2 L_F Φ

U(m⁰l⁰p⁻¹l⁻¹₀ ) .

Combining this formula with Lemma 2.6 gives the desired result.

Remark. — The left-hand side of the equation in the corollary does not depend onm⁰ orl⁰, so neither does the right hand side. As a consequence

(−4i)^[F:Q]

NF /Q(dF)α(m⁰l⁰)C(c(f),f) LF

Φ

U(m⁰l⁰p⁻¹l⁻¹₀ )

will satisfy the axioms of a distribution, with respect to the finitely additive functionsψ.

We now consider the Hilbert automorphic formg_ρk/Fn, the base change ofg_ρk toF_n. Just asg_ρk is associated toρ_k = Indχ_ρk, we now show that g_ρk/Fn is also associated to an induced representation.

Lemma 2.9. — Ifρk= Ind^F_K^k

k(χρ_k), then L(s,g_ρk/Fn) =L(s,Ind^F_Kⁿ

n(ResK_nχρ_k)).

Proof. — Firstly, by the properties of the base change L(s,g_ρk/Fn) = Y

ψ∈Gˆ

L(s, ρk⊗ψ) =L(s, ρk⊗R_Fn/Fk) whereG= Gal(Fn/Fk), and RF_n/F_k = Ind^F_F^k

n1denotes its regular repre- sentation. However, the Artin formalism implies

L(s, ρ/M) =L(s,Indρ/L)

wheneverρis an Artin representation over M, and L is a subfield ofM. Therefore

L(s, ρ_k⊗R_Fn/Fk) =L(s, ρ_k⊗Ind^F_F^k

n1) =L(s,Res_Fnρ_k⊗1) and the result follows becauseResF_nρk= Ind^F_Kⁿ

n(ResK_nχρ_k).

N.B. In a slight abuse of notation, we have written ρ_k/F_n as shorthand for the Artin representationResF_nρk= Ind^F_Kⁿ

n(ResK_nχρ_k).

Let us denote byGnthe topological groupGal(F_n,S^ab/F_n)whereF_n,S^ab is the maximal abelian extension ofFnunramified outside the setS={v:v|pl0} and the infinite places. For the remainder of this section, ψ : G_n → C^× will be a character with conductorfψ. We shall apply our earlier results to gρ = g_ρk/Fn⊗ψ, as the representation ρk/Fn ⊗ψ is certainly induced by the characterResKn(χρ_k⊗ψ)overKn.

Further, it is easy to check that

ResK_n(χρ_k⊗ψ)^†= (χ^†_ρk⊗ψ^†)◦NK_n/K_k.

Also, the character ofg_ρk/Fn⊗ψ is(Res_Fndetρ_k)^†⊗ψ^†2, and we have (Res_Fn(detρ_k)⊗ψ²)^† = ((detρ_k)^†◦N_Fn/Fk)⊗ψ^†2. Definition 2.10. — The parallel weight 2formΦ^n,k_ψ is given by

Φ^n,k_ψ = Φ^n,k_ψ (ρk/Fn⊗ψ,c(f/F_n)m⁰l⁰) := (g_{ρk/Fn⊗ψ,pl}

0)·E₁(0,c(f_/Fn)m⁰l⁰,(Res_Fndetρ_k)⁻¹⊗ψ⁻²) where we now assumem⁰ andl⁰ satisfyc(gρ)(pl0fψ)²

m⁰l⁰. Applying Corollary 2.8 directly tog_ρk/Fn⊗ψ produces Corollary 2.11. — For alln>k,

(−4i)^φ(pn)/2 α(m⁰l⁰)C(c(f),f)LF_n

Φ^n,k_ψ

U(m⁰l⁰p⁻¹l⁻¹₀ )

= N_Fn/Q(c(g_ρk/Fn⊗ψ)d²_F

n)^1/2

α(c(g_ρk/Fn⊗ψ)) ×Λ(g_ρk/Fn⊗ψ)

×Eulpl₀(ρk/Fn⊗ψ⁻¹,1)×Ψ(1,f_/Fn,g^ι_ρ

k/Fn⊗ψ) f_/Fn,f_/Fn

c(f)

.

Furthermore, the Fourier coefficients of the λ-component of Φ^n,k_ψ are given by

φ^n,k_ψ,λ(ξ) = X

ξ=ξ1+ξ2

X

aCOKn , a¯a=ξ1˜t−1 λ

(χ^†_ρk◦N_Kn/Kk)(a)ψ^†(ξ1˜t⁻¹_λ )

× X

ξ˜2 =˜b˜c, c∈OFn ,

b∈˜tλ

((detρk)^†◦N_Fn/Fk)⁻¹(˜c)ψ^†(˜c)⁻².

3. Congruences

The disadvantage of Corollary 2.11 is that it is hard to decipher exactly what it has to do with Kato’s conjectures. We shall now interpret various quantites from this formula, back in terms of the arithmetic ofE_/Q.

3.1. The connection with elliptic curves

First, we will find an expression for the α-term from our main formula.

Recall that we already made a choice of α(q) for each q dividing pl0, in

order to define thepl₀-stabilisationf₀. For an Artin representation ρover a fieldF, we denote its conductor by fρ. This is an ideal of OF, and we writef(ρ,q)for the exponent of the primeqin f_ρ.

Lemma 3.1.

(i) For each prime q|p∆, there exists a root α_q of X²−a_q(E)X +q such that

α(c(g_ρk/Fn⊗ψ)) =α^f(ρ_p ^k/Fn⊗ψ,p)·Y

q|∆

α^ord_q ^q(Aχ˜)

whereχ˜= ResK_n(χρ_k⊗ψ)andAχ˜=N_Kn/Q(fχ˜).

(ii) Furthermore, if we make the stronger assumption thatψis a char- acter ofGal(F_n,{p}^ab /Fn)i.e.ψis ramified only at the prime abovep, then

ordq(Aχ˜) =pⁿ−pⁿ⁻¹ for allq|∆.

In particular,α(c(g_ρk/Fn⊗ψ))is always ap-adic unit.

Proof. — Forq6=p,α(q)is one of the eigenvalues ofFrobqacting on the Tate moduleTp(E). However, Frobq= Frob^[f_q^n,q:Fq] wherefn,q denotes the residue field ofF_n at q. Therefore

α(q) =α^[f_q^n,q:Fq] for one of the rootsαq ofX²−aq(E)X+q.

For q=p, we instead considerTp( ˜E)whereE˜ denotes the reduction of E over fn,p. In this case, rank_ZpTp( ˜E) = 1 because we have assumed E has good ordinary reduction atp, soα(p)is the unique eigenvalue ofFrobp

acting onT_p( ˜E). Applying the same argument as above,α(p) =α_p. Set c=c(g_ρk/Fn⊗ψ)for brevity. Thenα(c)is defined multiplicatively, so

α(c) = Y

q∈Spec(OFn) q|c

α(q)^ordq(c)= Y

q|N_Fn/Q(c)

α^ordq ^q(NFn/Q(c)).

Becauseρ_k/F_n⊗ψ= Ind^F_Kⁿ

nχ, by [12], Section 5˜

c(g_ρk/Fn⊗ψ) =N_Kn/Fn(fχ˜)Disc(Kn/Fn).

HoweverDisc(K_n/F_n) =p, which means

N_Fn/Q(c) =p·N_Kn/Q(fχ˜).

The primes dividingN_Kn/Q(f_χ˜)are those dividingp∆, whence α(c(g_ρk/Fn⊗ψ)) =α^1+ord_p ^p(Aχ˜)·Y

q|∆

α^ord_q ^q(Aχ˜).

LetP denote the unique prime of K_n above p. We have N_Kn/Q(P) = p, from which we deduce

1 + ord_p(A_χ˜) = 1 + ord_P(f_χ˜)

=f(ρk/Fn⊗ψ,p) using standard results on Artin conductors. This proves (i).

It remains to prove assertion (ii). We now assume ψ is ramified only abovep, and thatq|∆. We then obtain

ordq(N_Kn/Q(fχ˜)) =X

Q|q

f( ˜χ,Q)[kn,Q:Fq]

where the sum is taken over the primes ofKn above q, and kn,Q denotes the residue field of Kn at Q. Under our additional assumption on ψ, we can say

f( ˜χ,Q) =f(Res_Knχ_ρk,Q)

and the characterResKnχρ_k factors through the extensionKn(^pn√

∆)/Kn. The prime Q is totally yet tamely ramified in this extension. Therefore Res_Knχ_ρk is non-trivial on the inertia group, but is trivial on all the higher ramification groups. By definition of Artin conductor, this implies f( ˜χ,Q) = 1. Therefore,

ord_q(N_Kn/Q(f_χ˜)) =X

Q|q

[k_n,Q:Fq]

= [kn,Q:Fq]×number of primes ofKn aboveq

= [Kn:Q]

asqis unramified inK_n/Q. Observing that[K_n:Q] =pⁿ−pⁿ⁻¹completes the demonstration of (ii).

Finally, as αp was chosen to be ap-adic unit and either choice ofαq is a p-adic unit when q 6= p, it is clear α(c(gρ_k/F_n ⊗ψ)) is always a p-adic

unit.

The following result allows us to link the automorphic form Φ^n,k_ψ with Artin-twists of the Hasse-WeilL-function ofE_/Fn.