AUTOMORPHIC SYMBOLS, p-ADIC L-FUNCTIONS AND ORDINARY COHOMOLOGY OF HILBERT MODULAR VARIETIES

AUTOMORPHIC SYMBOLS, p-ADIC L-FUNCTIONS AND ORDINARY COHOMOLOGY OF HILBERT MODULAR VARIETIES

MLADEN DIMITROV

Abstract. We introduce the notion of automorphic symbol generalizing the clas- sical modular symbol and use it to attach very generalp-adicL-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for thep-adically completed cohomology of a Hilbert modular variety localized at a suit- able nearly ordinary maximal ideal of the Hecke algebra. We also show its freeness over the corresponding Hecke algebra which turns out to be a universal deformation ring. In the last part of the paper we combine the above results to constructp-adic L-functions for Hida families of Hilbert automorphic forms in universal deformation rings of Galois representations.

Contents

Introduction 2

0.1. Automorphic symbols and p-adicL-functions 2

0.2. Ordinary cohomology 3

0.3. p-adicL-functions in families 4

Acknowledgments 5

1. Automorphic symbols on Hilbert Modular Varieties 5

1.1. Hilbert modular varieties 6

1.2. Automorphic symbols in level K1 6

1.3. Automorphic symbols in arbitrary level 10

1.4. Cohomology of Hilbert modular varieties 11

1.5. Evaluation of cohomology classes on automorphic symbols 13 2. p-adicL-functions for nearly ordinary automorphic forms on GL₂ 15

2.1. Automorphic representations 15

2.2. p-adicL-functions attached to newforms 16

2.3. Σ-stabilizedp-adicL-functions 19

3. Exact control theorem for the nearly ordinary cohomology of Hilbert

modular varieties 21

3.1. Towers of Hilbert modular varieties 21

3.2. Hida’s stabilization lemma 22

3.3. Exact control theorem 23

4. Freeness and R=T theorems 25

4.1. Universal nearly ordinary deformation rings 25

4.2. Universal nearly ordinary Hecke rings 25

Research of the author supported by Agence Nationale de la Recherche grant ANR-10-BLAN-0114.

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4.3. Nearly ordinary cohomology modules 26

4.4. Freeness theorem 27

4.5. Variants 28

5. Analytic p-adicL-function for Hida families 29

5.1. Universal nearly ordinaryp-adic automorphic symbol 29 5.2. Analytic p-adicL-functions for ordinary families 30 5.3. Analytic p-adicL-function for nearly ordinary families 31

References 32

Introduction

Throughout this paper F will denote a totally real number field of degree d > 1 and ring of integers o. LetI be the set of its infinite places. Letp be a prime number and let O be the ring of integers of a p-adic field E which is sufficiently large (to be specified later). Let Q denote the field of algebraic numbers in C and let us fix an embedding of QinQp.

0.1. Automorphic symbols and p-adic L-functions. Modular symbols arise clas- sically as first homology classes of modular curves, represented by geodesics connecting rational points on the boundary of the upper half plane. In general, the closure of the image of a standard Shintani cone in the compactified Hilbert modular variety yields a d-cycle, hence an element in the d-th homology group that we call a Manin-Oda modular symbol (see Oda’s book [Od] for the case of Hilbert modular surfaces). The relations between those symbols remain mysterious even in the simplest case of a Hilbert modular surface.

In the spirit of Tate’s thesis, we define in §1 an “adelic” counterpart of the Manin- Oda modular symbols, called automorphic symbols. Their importance (in particular their non-vanishing) will become visible in§2 where they are related toL-values. When F 6= Q the presence of units and non-trivial class groups complicates substantially the evaluation of cohomology classes on automorphic symbols. Those difficulties are addressed and overcome in §1.5 which makes it a key part of our construction. Our definition of automorphic symbol is group-theoretic and can be easily transposed to other groups than GL2.

Analyticp-adicL-functions for classical modular forms have been extensively studied by Mazur, Manin, Vishik, Amice-Veluet al. by means of modular symbols (we refer to [MTT] for a complete exposition of their results). For Hilbert modular forms, besides the precursory work of Manin [Ma], the question has been studied by Dabrowski [Da], Panchishkin [Pa] and Mok [M1] using Rankin-Selberg convolution, and more recently by Januszewski [Ja]. Nevertheless, prior to the present article several aspects of the theory remained unavailable in the Hilbert modular setting, namely:

• allowing horizontal twists, that is, twists by characters ramified at some finite set Σ of primes of F not dividingp;

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• allowing nearly ordinary forms; note that a nearly ordinary Hilbert automor- phic form is not necessarily a twist of an ordinary one by a Hecke character unramified outsidep∞;

• allowing forms that are not necessarily new, namely forms whose Euler factors are 1 at primes in Σ;

• in [Da, Pa] the level is prime to p, whereas in [M1] the weight is parallel and in [Ja] the class number is not divisible by p.

We use the flexibility of our construction of automorphic symbols to prove the fol- lowing theorem (see theorem 2.11 for more precise statement).

Theorem 0.1. Letπbe a cohomological cuspidal automorphic representation ofGL2/F whose weight(w, w0) is critical (see definitions 1.16 and 2.1). Assume thatπ is nearly ordinary at all places dividing p (see definition 2.2). Fix a finite set Σ of primes not dividing p and let F^(pΣ) be the maximal abelian pro-p extension of F unrami- fied outside pΣ∞. Then there exists a p-adic L-function L^Σ_p(π) ∈ O[[Gal(F^(pΣ)/F)]], uniquely determined by the following interpolation property: for every finite order char- acter φ : Gal(F^(pΣ)/F) → O^× the image of L^Σ_p(π) by the resulting homomorphism O[[Gal(F^(pΣ)/F)]]→ O equals:

L^(pΣ)(π⊗φ,1)Γ(π,1) Ω^Σ_π

Y

v|pΣ

Zv,

where L^(pΣ)(π⊗φ, s) denotes the L-function whose Euler factors are 1 at primes di- viding pΣ, Ω^Σ_π is an Archimedean period (see definition 2.9) and Zv are local terms.

Moreover, if (w, w₀−2) is critical too, then the automorphism [a]7→ (χω⁻¹)(a)[a]

of O[[Gal(F^(pΣ)/F)]] sends L^Σ_p(π) to L^Σ_p(π ⊗ | · |_Aω⁻¹), where χ denotes the p-adic cyclotomic character and ω is the Teichm¨uller character.

One reason to look for p-adic L-functions of Hilbert automorphic forms in that generality is for the construction of p-adic L-functions of Hida families of Hilbert automorphic forms (see theorems 0.2 and 0.3).

0.2. Ordinary cohomology. To any cohomological cuspidal automorphic represen- tationπof GL2/F, one can attach a two-dimensionalp-adic representationρπ,pdefined over a p-adic field. In [H1, H3] Hida developed the theory of nearly ordinary families of Hilbert automorphic forms and proved a control theorem for the nearly ordinary p-adic Hecke algebras, which allowed him to construct two-dimensional Galois repre- sentations over these algebras interpolating the ρ_π,p’s when π varies in a Hida family.

Hida’s proof relies, via the Jacquet-Langlands correspondence, on control theorems for the nearly ordinary cohomology of Shimura varieties of dimension zero or one, coming from quaternion algebras over F which are totally definite, or indefinite but yielding Shimura curves. The introduction of [H1] concludes with the hope that those results could be extended to other quaternionic Shimura varieties. In §3 we extend Hida’s results to the case of the indefinite quaternion algebra M₂(F) by proving an exact control theorem for the nearly ordinary cohomology of Hilbert modular varieties (see theorem 3.8). The proof proceeds by specialization to a well chosen finite level and weight where the main results of [D1, D2] can be applied.

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The last decade has seen the emergence of p-adic and mod p Langlands programs.

Although they are widely conjectural for a general reductive algebraic group over a number field, precise statements for GL₂ overQhave been proved by Breuil, Colmez, Emerton and Kisin culminating in the proof of many cases of the Fontaine-Mazur conjecture in dimension two (see [E2] and [K]). One of the main tools in these programs is the completed cohomology introduced by Emerton [E1] and the associated spectral sequence. Little is known about it in dimension > 1, in particular for GL₂ over a totally real number field. Our theorem 3.8 implies the degeneracy of the simplest yet non-trivial piece of Emerton’s spectral sequence for GL2 over F, which is its nearly ordinary part (on the Galois side, nearly ordinary means locally reducible at p).

0.3. p-adicL-functions in families. In this work, we call Hida family a local compo- nent of the universal Hecke algebra. It turns out that under certain assumptions on the residual representation ρ_π,p mod p such components can be identified with universal deformation rings. As explained in [EPW], in the case F =Q, the original definition of a Hida family would then correspond to a branch of our family.

These families are parametrized by irreducible representations

¯

ρ: Gal_F,pΣ→GL₂(Fp),

which are nearly ordinary at all places dividing p and totally odd at infinity, where Gal_F,pΣdenotes the Galois group of the maximal extension ofF inQunramified outside pΣ∞.

Consider the following assumptions on ¯ρ:

(?) no twist of ¯ρextends to a representation of Gal(Q/F⁰) for any strict subfield F⁰ of F and its image contains SL₂(Fp).

(??) p is unramified in F and ¯ρ ' ρ_π,p¯ mod p for some cuspidal automorphic repre- sentation ¯πof GL₂/F which is nearly ordinary and unramified at all places dividingp and has cohomological weight ( ¯w,w¯0) such that ¯wτ >0 for allτ ∈I, ¯w0 = maxτ∈I( ¯wτ) and p−1>P

τ∈I(^{w¯0+ ¯}₂^wτ + 1).

Note that, if a cuspidal automorphic representation π of GL₂/F is neither a theta series, nor a twist of a base change, then ρπ,p mod p satisfies (?) for all but finitely many primes p.

The following theorems prove the existence of analyticp-adicL-functions for (nearly) ordinary families of Hilbert automorphic forms, extending results of Kitagawa [Ki], Greenberg-Stevens [GS] and Emerton-Pollack-Weston [EPW] for Q. A novelty of our approach (even for F = Q) is that the p-adic L-functions are naturally elements of universal deformation rings of Galois representations, rather than abstract Iwasawa algebras, which confirms an expectation of Greenberg [Gr, §4].

LetR^n.o_ρ,Σ¯ be the universalO[[Gal(F^(pΣ)/F)×(o⊗Zp)^×p−part]]-algebra parametrizing nearly ordinary deformations of ¯ρ (see §4.1) and let R^det_ρ,Σ¯ be the O[[(o⊗Zp)^×p−part]]- algebra parametrizing those deformations having determinant det(ρ_¯π,p) (see (20)).

Theorem 0.2. Under the assumptions (?) and (??), there exists a p-adic L-function L^n.o_p ( ¯ρ,Σ)∈ R^n.o_ρ,Σ¯ =R^det_ρ,Σ¯ [[Gal(F^(pΣ)/F)]],

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uniqueup to an element inR^det,×_ρ,Σ¯ , whose specialization by any homomorphismR^det_ρ,Σ¯ → O whose restriction to(o⊗Zp)^×p−part is a finite order character, yields the p-adic L- functionL^Σ_p(π) of a nearly ordinary cuspidal automorphic representationπ onGL₂/F of parallel weight w¯0.

For ordinary ¯ρ, let R^ord_ρ,Σ¯ be the universal ordinary deformation O[[Gal(F^(pΣ)/F)]]- algebra (see (19)). A homomorphism R^ord_ρ,Σ¯ → O is algebraic if its restriction to Gal(F^(pΣ)/F) is the product of a finite order character with a non-positive integer power of χω⁻¹.

Theorem 0.3. Assume that ρ¯ satisfies (?) and (??) with an ordinary π¯ of paral- lel weight. Then there exists a p-adic L-function L^ord_p ( ¯ρ,Σ) ∈ R^ord_ρ,Σ¯ [[Gal(F^(pΣ)/F)]]

uniquely determined, up to an element in R^ord,×_ρ,Σ¯ , by the following universal property:

the specialization of L^ord_p ( ¯ρ,Σ) by any algebraic homomorphism R^ord_ρ,Σ¯ → O yields the p-adicL-functionL^Σ_p(π)of a parallel weight, ordinary, cuspidal automorphic represen- tation π of GL2/F.

The proof of theorem 0.3 (resp. theorem 0.2) relies on the fact that R^ord_ρ,Σ¯ (resp.

R^n.o_ρ,Σ¯ ) is canonically isomorphic to Hida’s universal (nearly) ordinary Hecke algebra and that certain modules of (nearly) ordinary cohomology of Hilbert modular varieties are free over these rings (see theorem 4.6 and corollaries 4.8, 4.9).

We believe that using some ideas of Ash-Stevens [AS] and Urban [Ur] would allow to extend our results to the finite slope case, or at least to relax the assumptions (?) and (??) in the nearly ordinary case as in [GS] (see also [BL]).

Finally we hope that our p-adic L-functions will be useful for the study of higher order partial derivatives as in [M2], and for a formulation and proof of Iwasawa Main Conjectures for GL₂ over totally real fields.

Acknowledgments. The author is indebted to T. Ochiai for pointing out the relevance of the results in [D1] for the construction of analytic p-adicLfunctions for Hida families of Hilbert modular forms.

The author has greatly benefited from a weekly p-adicL-function seminar that took place in Paris in 2009, and would like to thank the other participants and especially J. Nekov´aˇr for their constructive remarks. It is a pleasure to thank P. Charollois, M. Emerton, O. Fouquet, K. Fujiwara, R. Greenberg, H. Hida, L. Merel, C.-P. Mok, G. Stevens, J. Tilouine and E. Urban for their interest in this work and for helpful discussions.

The automorphic symbols were born during a visit at the Tata Institute for Fundamental Research in 2008 and the author would like to thank the institution for its hospitality. The author would like to thank the CNRS for granting him a delegation in 2009-2010, as well as the Centre de Recerca Matem`atica in Barcelona for providing him with excellent conditions for work during the same academic year.

Finally, the author would like to thank the anonymous referee for his helpful comments and suggestions to improve the presentation of this paper.

1. Automorphic symbols on Hilbert Modular Varieties

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1.1. Hilbert modular varieties. The ring A of adeles of F is the product of finite adelesAf =F⊗bZand infinite adelesF∞=F⊗_QR. We denote byF_∞⁺ the connected component of 1 inF_∞^×.

Definition 1.1. For an open compact subgroup K of GL₂(Af) we define the analytic Hilbert modular variety of level K as

YK= GL2(F)\GL₂(A)/KK_∞⁺,

whereK_∞⁺ is the subgroup of GL2(F∞) generated by its centerF_∞^×and by the connected component of identity in the standard maximal compact subgroup.

The minimal (or Baily-Borel-Satake) compactification YK of YK is obtained by adding finitely many points (the cusps). A basis of neighborhoods for the cusp at infinity is given by the sets

(1)

GL2(F)

y∞ 0

0 1

KK_∞⁺

y∈F_∞⁺,N_{F /Q}(y∞)> H

, forH∈R^×₊,

and neighborhoods of other cusp are obtained by translating those by the group action.

The adjoint Hilbert modular variety of level K is defined as Y_K^ad= GL2(F)\GL₂(A)/A^×KK_∞⁺. We will only consider K factoring as a product Q

vK_v over the finite places v ofF and such that Y_K^ad is smooth. ThenYK is a finite abelian cover of Y_K^ad with group the class group A^×_f /F^×(A^×_f ∩K).

Since our interest is in p-adic aspects, usuallyKv will be fixed for v not dividing p and forα≥1,Y0(p^α),Y1(p^α),Y11(p^α) andY(p^α) will denote, respectively, the Hilbert modular varieties whose level K_p =Q

v|pK_v atp equals:

K₀(p^α) =

u∈GL₂(o⊗Zp)|u≡ ∗ ∗

0 ∗

(modp^α)

, K1(p^α) =

u∈GL2(o⊗Zp)|u≡ ∗ ∗

0 1

(modp^α)

, K₁₁(p^α) =

u∈GL₂(o⊗Zp)|u≡ 1 ∗

0 1

(modp^α)

, K(p^α) =

u∈GL₂(o⊗Zp)|u≡ 1 0

0 1

(modp^α)

. (2)

Remark 1.2. Classically Hilbert modular varieties are defined as quotients of the d- fold product of upper half planes by congruence subgroups of SL2(o). These occur as connected components of Y_K, but are not preserved by Hecke correspondences, hence the importance of the above adelic definition.

1.2. Automorphic symbols in level K₁. The automorphic symbols considered in this section will be sufficient for the construction of thep-adicL-function of a nearly or- dinary Hilbert automorphic newform (see theorem 2.5). The definition of automorphic symbols in arbitrary level is postponed to §1.3 and can be skipped at first reading.

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1.2.1. Automorphic cycles and the mirabolic group. The mirabolic groupM is defined as the semi-direct productGmn Ga, whereGmacts onGaby multiplication. A natural embedding of M in GL₂ is given by (y, x)7→(^{y x}_{0 1}).

Fix an integral ideal a of F and denote by av the valuation of a at a finite place v. Put M(a) =U(a)n(o⊗bZ), where U(a) is the open compact subgroup of (o⊗bZ)^× whose elements are congruent to 1 modulo a. For all v, choose an uniformizer $v of F_v.

The map:

(3) C(a) :A^×/F^×U(a)−→M(F)\M(A)/M(a) , y7→ y,(yv$^−a_v ^v)v|a

is well defined, since for all z∈F^× and u∈U(a) we have

zyu,((zyu)_v$^−a_v ^v)_v|a

= (z,0) y,(y_v$_v^−av)_v|a

u,((u_v−1)$_v^−av)_v|a and (z,0)∈M(F) whereas u,((u_v−1)$^−a_v ^v)_v|a

∈M(a).

Denote by E(a) the group of totally positive units in o, congruent to 1 modulo a, and by Cl⁺_F(a) the narrow ray class groupA^×/F^×U(a)F_∞⁺.

Definition 1.3. Forη∈Cl⁺_F(a) we defineC(η) as the restriction ofC(a) to the inverse image eη ofη using the short exact sequence

(4) 1→F_∞⁺/E(a)→A^×/F^×U(a)→Cl⁺_F(a)→1.

Definition 1.4. Letη ∈Cl⁺_F(a) and letK ⊂GL₂(Af) be an open compact subgroup containing the image ofM(a) by the natural inclusion ofM in GL₂. Theautomorphic cycle C_K(η) on the Hilbert modular variety Y_K is defined as the composed map of C(η) with the map coming from the natural inclusion M ⊂GL₂:

ι_K:M(F)\M(A)/M(a)−→GL₂(F)\GL₂(A)/KK_∞⁺ =Y_K.

1.2.2. Automorphic symbols for Hilbert modular varieties. As mentioned in the intro- duction, the natural generalization in the Hilbert modular case of the geodesic between the cusps 0 and ∞ in the upper half plane is the Shintani coneF_∞⁺/E(a). In order to define the corresponding automorphic symbol, it needs to be compactified.

By (4), for any choice of an idele ξ ∈ ηe∩A^×_f, the map y∞ 7→ y∞ξ yields an isomorphism F_∞⁺/E(a)'η, hence there is a continuous map:e

(5) ηe−→R^×₊ ,y7→N_{F /Q}(yξ⁻¹).

By Dirichlet’s unit theorem F_∞⁺/E(a)'(R/Z)^d−1×R^×₊.

Definition 1.5. Denote by η the compactification ofηe−→^∼ (R/Z)^d−1×R^×₊ obtained by adding two points (zero and infinity).

If d= 2 thenη is homeomorphic to a sphere. In general, it is homeomorphic to the suspension of the torus (R/Z)^d−1, hence

H_i(η)'







Z fori= 0,

0 fori= 1,

Hi−1((R/Z)^d−1) for 2≤i≤d.

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Any ξ induces the same orientation on η, in particular, for any d ≥ 2 we can canonically identify H_d(η) = H_d(η,{0,∞}) andZ.

Lemma 1.6. CK(η) extends uniquely to a continuous proper map CK(η) :η→YK. Proof. The uniqueness is clear. For the existence of a continuous mapCK(η) one has to show that the image byC_K(η) of any sequence (y_n)n≥1of points inηetending to zero or infinity, converges to a cusp in YK. Suppose first that limnN_{F /Q}(ynξ⁻¹) = +∞.

It follows then from (1) that the sequenceCK(η)(yn) converges to the cusp at infinity (on the connected component of Y_K corresponding toξ).

Suppose next that limnN_{F /Q}(ynξ⁻¹) = 0 and consider the following diagram:

(F ∩ξ(o⊗bZ))\F_∞ ^{∼ //}

A/(F +ξ(o⊗bZ))

E(a)n(F∩ξ(o⊗bZ))\M_∞^{+ ·(ξ,0) //}

M(F)\M(A)/M(a) ^////

M(F)\M(A)/M(a)M_∞⁺ E(a)\F_∞^{+  ·ξ //}A^×/F^×U(a) ^////Cl⁺_F(a)

where M_∞⁺ =F_∞⁺nF∞.

Note that for any y=y∞ξ∈ηethe elementyaa⁻¹ = (ξv$_v^−av)v|a has finite order in A/(F+ξ(o⊗bZ))'(F∩ξ(o⊗bZ))\F_∞,

and denote byx∞∈F∞any element in its coset. Clearlyx∞∈F and we will show that CK(η)(yn) converges to the corresponding cusp. In fact, since limnN_{F /Q}(ynξ⁻¹) = 0, the sequence

0 1

−1 x∞

yn (ξv$_v^−av)v|a

0 1

=

0 1

−y_n 0

converges to cusp at infinity, hence C_K(η)(y_n) converges to the cusp determined by x∞.

Since the mapC_K(η) is finite, to show its properness it is enough to prove that it is closed. This follows easily from the fact that C_K(η) is continuous andη compact.

Definition 1.7. Theautomorphic symbolS_K(η)∈H_d(Y_K) is defined as the image of 1∈Z'H_d(η) by C_K(η).

1.2.3. p-adic automorphic symbols. Letα≥1 be an integer such thatKp⊃K11(p^α).

Definition 1.8. Forβ ≥α we define SK,β= X

η∈Cl⁺_F(p^β)

SK(η)[η]∈Hd(YK,O)[Cl⁺_F(p^β)].

Remark 1.9. (i) Here one can see one advantage of the adelic approach. The automorphic symbol is defined on the whole class group Cl⁺_F(p^β) in contrast with a collection of modular symbols, indexed by Cl⁺_F(o), each one defined on (o/p^β)^×/E(o).

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(ii) Although some of the constructions are similar to those in [Ma] in order to make the relations precise one has to choose representatives in class groups, which would bring some cumbersome notations.

The compatibility of S_K,β with respect to β, known as distribution property, is governed by the Hecke operatorU_p=Q

v|pU_v^ev, wheree_v denotes the inertia degree at v, so that (p) =Q

v|pv^ev. Assume for the rest of this section that for allv dividing p we have $^e_v^v =p.

Lemma 1.10. For β ≥α, the image of S_K,β+1 by the natural projection induced by pr : Cl⁺_F(p^β+1)→Cl⁺_F(p^β) equals U_p·S_K,β.

Proof. It is enough to show that for allη ∈Cl⁺_F(p^β), U_p·C_K(η) and `

pr(η⁰)=ηC_K(η⁰) define the same cycles on Y_K. We will check this by a computation in the mirabolic group M(F)\M(A)/M(p^α) on which U_p acts according to the following double coset decomposition:

M(p^α)(p,0)M(p^α) = a

b (modp)

(p, b)M(p^α).

For y∈A we denote byyp the adele (yv)_v|p. We have U_p·

y∈A^×/F^×U(p^β)7→M(F)(y, y_pp^−β)M(p^α)

=

= a

b (modp)

y∈A^×/F^×U(p^β)7→M(F)(yp, y_p(1 +bp^β)p^−β)M(p^α)

=

= a

b (modp)

y∈A^×/F^×U(p^β)7→M(F)(y(1 +bp^β), yp(1 +bp^β)p^−(β+1))M(p^α)

=

=

y∈A^×/F^×U(p^β+1)7→M(F)(y, y_pp^−(β+1))M(p^α) .

Remark 1.11. When F =Qthe image by Up of the geodesic from x ∈Qto ∞ is a union ofpgeodesics each joining ^x+b_p to∞(b= 0, ..., p−1). WhenF 6=Qthe situation is quite different. For example ifd= 2, forβlarge enough the image byUpof the 2-cycle F_∞⁺/E(p^β) is a union of p² 2-chains piecing together into [Cl⁺_F(p^β+1) : Cl⁺_F(p^β)] = p copies of the 2-cycle F_∞⁺/E(p^β+1). Note that, in some sense, applying Up reduces the monodromy of the units.

By lemma 1.10 we can define the p-adic automorphic symbolof level K:

S_K = lim←−

β

U_p^−β·S_K,β∈e^∗_pH_d(Y_K,O)[[Cl⁺_F(p^∞)]], where e^∗_p denotes Hida’s ordinary idempotent on the homology.

It is clear that the natural projection induced byY1(p^α+1)→Y1(p^α) sendsS_K1(p^α+1₎ toS_K1(p^α₎. The universal ordinary p-adic automorphic symbol is defined as:

lim←−

α

S_K1(p^α₎ ∈lim←−

α

e^∗_pH_d(Y₁(p^α),O)⊗b_OO[[Cl⁺_F(p^∞)]].

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One can construct similarly automorphic symbols on adjoint Hilbert modular varieties and define the universal nearly ordinary p-adic automorphic symbol as:

lim←−

α

S_K^ad_11(pα)∈lim←−

α

e^∗_pHd(Y₁₁^ad(p^α),O)⊗bOO[[Cl⁺_F(p^∞)]].

1.3. Automorphic symbols in arbitrary level.

1.3.1. Automorphic cycles. We keep the notations from the previous section. In particular we denote by av the valuation of a at v. For any ideal a⁰ ⊂ a we let M(a,a⁰) =U(a⁰)n(a⁰a⁻¹⊗bZ). As in (3), the following map is well defined:

(6) C(a,a⁰) :A^×/F^×U(a⁰)−→M(F)\M(A)/M(a,a⁰) ,y 7→ y,(yv$_v^−av)v|a

. For any ideal a⁰⁰ ⊂ a⁰, the following diagram is commutative, where the vertical maps are the natural projections:

A^×/F^×U(a⁰⁰) ^C(a,a

00) //

M(F)\M(A)/M(a,a⁰⁰)

A^×/F^×U(a⁰) ^C(a,a

0) //M(F)\M(A)/M(a,a⁰).

Forη⁰ ∈Cl⁺_F(a⁰) we denote byC(a, η⁰) the restriction ofC(a,a⁰) to the inverse image ηe⁰ ofη⁰ using the short exact sequence (4).

For a prime w dividing a, the following lemma describes the action of U_w on an automorphic cycle of level a.

Lemma 1.12. For allη∈Cl⁺_F(a⁰)the cyclesUw·C(a, η)and`

pr($w⁻¹η⁰)=ηC(aw, η⁰)on M(F)\M(A)/M(a,a⁰) are cohomologically equivalent, where pr : Cl⁺_F(a⁰w) →Cl⁺_F(a⁰) denotes the natural projection.

Proof. Leta₀ = val_w(a⁰a⁻¹). For v dividingaputa⁰_v =a_v+ 1, ifv=w, and a⁰_v =a_v, otherwise. We have the following double coset decomposition:

M(a,a⁰)($_w,0)M(a,a⁰) = a

b (modw)

($_w, b$_w^a0)M(a,a⁰).

HenceUw·C(a,a⁰) =Uw· y∈A^×/F^×U(a⁰)7→M(F) y,(yv$^−a_v ^v)v|a

M(a,a⁰)

=

= a

b (modw)

y∈A^×/F^×U(a⁰)7→M(F) y$w, ξ_w,b(yv$_v^−av)_v|a

M(a,a⁰) ,

where ξ_w,b is the idele equal to 1 +b$^a_w^0+aw at w and 1 at all other places. Since a0+aw = valw(a⁰), we haveξw,b∈U(a⁰), hence Uw·C(a,a⁰) equals:

a

b (modw)

y ∈A^×/F^×U(a⁰)7→M(F) ξ_w,by$w, ξ_w,b(yv$_v^−av)_v|a

M(a,a⁰) .

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By the change of variabley 7→y$⁻¹_w this is homologically equivalent to a

b (modw)

y∈A^×/F^×U(a⁰)7→M(F)

ξw,by,(ξw,byv$_v^−a0v)v|a

M(a,a⁰)

=

=

y∈A^×/F^×U(a⁰w)7→M(F)

y,(y_v$_v^−a0v)_v|a

M(a,a⁰) , which is the composition of C(wa, wa⁰) with the natural projection:

M(F)\M(A)/M(wa, wa⁰)−→M(F)\M(A)/M(a,a⁰).

1.3.2. Automorphic symbols for Hilbert modular varieties. LetK be any open compact subgroup of GL2(Af). Given integral ideals a⁰ ⊂ a such that K contains the image of M(a,a⁰) =U(a⁰)n(a⁰a⁻¹⊗bZ) in GL₂, we denote by C_K(a,a⁰) the composition of C(a,a⁰) with the following map coming from the natural inclusion M ⊂GL2:

ιK :M(F)\M(A)/M(a,a⁰)−→GL2(F)\GL₂(A)/KK_∞⁺ =YK.

Definition 1.13. Let a_K be an integral ideal of F, such that for any a, K contains the image of M(a,a a_K) in GL2, so that the map C_K(a,a a_K) is well defined.

For η⁰ ∈Cl⁺_F(a⁰) let CK(a, η⁰) =ιK◦C(a, η⁰) be the restriction ofCK(a,a⁰) to ηe⁰. As in lemma 1.6 one can prove the existence and uniqueness of a proper continuous map CK(a, η⁰) : η⁰ → YK extendingCK(a, η⁰) and define SK(a, η⁰) ∈ Hd(YK) as the image of 1∈Hd(η⁰) by CK(a, η⁰).

1.3.3. p-adic automorphic symbols. SinceR^n.o_ρ,Σ¯ is naturally aO[[Gal(F^(pΣ)/F)]]-algebra, it will be useful to define automorphic symbols indexed by Cl⁺_F(p^∞Σ) and not only by Cl⁺_F(p^∞) as in §1.2.3. For this, we fix a finite set Σ of auxiliary primes not dividing p and let Σ also denote their product.

LetK ⊂GL₂(Af) be an open compact subgroup and fix an ideala_K as in definition 1.13.

Definition 1.14. Given an integer β≥1, we define:

S_K,β^Σ = X

η∈Cl⁺_F(p^βΣaK)

S_K(p^βΣ, η)[ηΣ⁻¹Y

v|p

$_v^−evβ]∈H_d(Y_K,O)[Cl⁺_F(p^βΣa_K)]

Lemma 1.12 has the following consequence:

Corollary 1.15. The image of S_K,β+1^Σ by the natural projection induced by pr : Cl⁺_F(p^β+1Σa_K)→Cl⁺_F(p^βΣa_K) equals U_p·S_K,β^Σ .

By corollary 1.15 we can define the p-adic automorphic symbol of levelK: (7) S_K^Σ = lim←−

β

U_p^−β·S_K,β^Σ ∈e^∗_pH_d(Y_K,O)[[Cl⁺_F(p^∞Σa_K)]], where e^∗_p denotes Hida’s ordinary idempotent on the homology.

1.4. Cohomology of Hilbert modular varieties.

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1.4.1. Cohomological weights. The characters of the torus Res^F_QGm can be identified with Z[I] as follows: for any w=P

τ∈Iwττ ∈Z[I] and for any Q-algebra A splitting F^×, we consider the characterx∈(F ⊗_QA)^×7→x^w =Q

τ∈Iτ(x)^wτ ∈A^×. The norm character N_{F /Q}: Res^F_QGm→Gmthen corresponds to the element t=P

τ∈Iτ ∈Z[I].

Definition 1.16. (i) A weight (w, w0)∈Z[I]×Ziscohomologicalif for all τ ∈I we have w_τ ≥0 andw_τ ≡w₀ (mod 2).

(ii) A cohomological weight (w, w0) is critical if|w₀| ≤ minτ∈I(wτ). It is parallel ifw=w₀t.

Remark 1.17. The correspondence with the classical notion of weight of a Hilbert modular form is as follows. Letfbe a Hilbert modular newform of weight (k_τ)τ∈I,k_τ ≥ 2 of the same parity, generating a cuspidal automorphic representation π. According to [Da] and [Pa], the value L(f, m) is critical, if for all τ we have ^k0−k₂ ^τ + 1 ≤ m ≤

k0+kτ

2 −1, where k₀ = max_τ(k_τ). Putw₀ =k₀−2m andw_τ =k_τ −2 for τ ∈I. Then L(f, m) =L(π⊗ | · |

k0−w0 2 −1

A ,1) where the automorphic representation π⊗ | · |

k0−w0 2 −1

has cohomological weight (w, w0) which is critical (see definition 2.1). A

An irreducible algebraic representation of Res^F_QGL₂ with central action factoring through the norm is necessarily of the form

O

τ∈I

Sym^w_τ^τ ⊗det^(w_τ ^0−wτ)/2,

for some cohomological weight (w, w₀)∈Z[I]×Z. For an O-moduleA we denote by L(w, w₀;A) the correspondingA[GL2(o⊗ O)]-module, where we use the fixed embed- ding of QinQp to identify GL2(o⊗ O) with GL₂(O)^I. It can be realized as the space of polynomials in (X= (X_τ)τ∈I, Y = (Y_τ)τ∈I) which are homogeneous of degreew_τ in the variables (Xτ, Yτ).

1.4.2. Sheaf cohomology. Let (w, w₀) be a cohomological weight. Then GL₂(F)\GL₂(A)× L(w, w₀;A)/KK_∞⁺,

with KK_∞⁺ acting onL(w, w₀;A) on the left via Kp, is a local system onYK, and we denote by L_K(w, w₀;A) the corresponding sheaf of locally constant sections.

For K⁰ ⊂K, there is a natural projection pr : YK⁰ → YK and pr^∗L_K(w, w0;A) = L_K⁰(w, w0;A). We denote H^•(Y_K,L_K(w, w0;A)) the corresponding singular (or Betti) cohomology groups, and by H^•_c(Y_K,L_K(w, w₀;A)) the cohomology with compact sup- port.

1.4.3. Hecke operators. For g ∈GL₂(Af) we define the Hecke correspondence [KgK] on Y_K by the usual diagram:

Y_K∩g⁻¹_Kg

pr₁

wwooooooo

·g //Y_gKg⁻¹_∩K

pr₂

''O

OO OO OO

YK YK

12

We define the standard Hecke operators Tv = [Kv 1 0 0$v

Kv], Sv = [Kv $v 0 0 $v

Kv], for v outside a certain finite set of bad primes, and U_v = [K_v ^{1 0}_0$v

K_v], for the remaining v.

The Hecke correspondences at infinity are [K_∞⁺g∞K_∞⁺], where g∞ is an element of the group ^{±1 0}_{0 1}I

⊂GL₂(F∞).

The Betti cohomology groups H^•(Y_K,L_K(w, w₀;A)) admit a natural action of all the Hecke correspondences and the induced endomorphisms commute with each other.

Definition 1.18. Lete∞ denote the idempotent on H^•(Y_K,L_K(w, w₀;A)) which cuts out the part fixed by all the Hecke correspondences at infinity.

1.5. Evaluation of cohomology classes on automorphic symbols. As mentioned in the introduction, when F 6= Q, the evaluation of cohomology classes on modular symbols requires special care.

By definition the cycleC_K,β^Σ =C_K(p^βΣ, p^βΣa_K) yields a homomorphism:

C_K,β^Σ∗ : H^d_c(Y_K,L_K(w, w₀;O))→H^d_c(A^×/F^×U(p^βΣa_K), C_K,β^Σ∗ L_K(w, w₀;O)).

The first difficulty comes from the fact that the local systemC_K(a,a⁰)^∗L_K(w, w₀;O) on A^×/F^×U(a⁰) is not trivial, because of the monodromy action of the units. Ex- plicitely, it is given by

F^×\ A^××L(w, w₀;O)

/U(a⁰), whereu∈U(a⁰) acts on L(w, w₀;O) by

up((uv−1)$_v^−av)v|p

0 1

whereas F^×acts trivially (we recall that u_p denotes the adele (u_v)_v|p). In particular

C_K,β^Σ∗ L_K(w, w₀;O) =F^×\ A^××L(w, w₀;O)

/U(p^βΣa_K), where u∈U(p^βΣa_K) acts onL(w, w₀;O) by

up(up−1)p^−β

0 1

.

Nevertheless, one can extract one coefficient (the critical one) as follows. We first need to untwist this sheaf, an operation which for classical modular symbols is implicit in the definition. The map (y, v) 7→ (y,

1−p^−β 0 1

·v) induces a homomorphism of sheaves on A^×/F^×U(p^βΣa_K):

tw^w,w_β ⁰ :C_K,β^Σ∗ L_K(w, w0;O)−→F^×\ A^××L(w, w₀;E)

/U(p^βΣa_K), where u∈U(p^βΣa_K) acts onL(w, w₀;E) simply by

1−p^−β 0 1

up(up−1)p^−β

0 1

1p^−β 0 1

=

u_p 0 0 1

.

Suppose now that (w, w₀) is critical, so that X^(w−w0t)/2Y^(w+w0t)/2∈ L(w, w₀;O).

A direct computation shows that for any 0≤j≤w, ^u_{0 1}^p0

acts byu^(w+wp ^0t)/2−j on X^w−jY^j ∈ L(w, w₀;O). In particular,

(8) ^u_{0 1}^p0

acts trivially on X^(w−w0t)/2Y^(w+w0t)/2.

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Therefore evaluating at the coefficient in front of X^(w−w0t)/2Y^(w+w0t)/2 induces the following homomorphism of sheaves:

crit^w,w_β ⁰ :F^×\ A^××L(w, w₀;E)

/U(p^βΣa_K)−→A^×/F^×U(p^βΣa_K)×E.

Since p^β

w−w0t

2 (X−p^−βY)^(w−w0t)/2Y^(w+w0t)/2 ∈ L(w, w₀;O), we have the following:

Lemma 1.19. Assume that(w, w₀) is critical. Then the mapp^βw−w20t crit^w,w_β ⁰◦tw^w,w_β ⁰ takes values in A^×/F^×U(p^βΣa_K)× O. Further composing withC_K,β^Σ∗ induces

(9) H^d_c(Y_K,L_K(w, w0;O))−→H^d_c(A^×/F^×U(p^βΣa_K),O) =O[Cl⁺_F(p^βΣa_K)].

The second difficulty is related with the action of the Hecke operator Up on cycles (see remark 1.11).

Consider the natural projection pr^β_β+1 : Cl⁺_F(p^β+1Σa_K) → Cl⁺_F(p^βΣa_K). Lemma 1.12 implies that the cyclesU_p·C_K,β^Σ andC_K,β+1^Σ are cohomologically equivalent. This implies the commutativity of the upper square in the following diagram:

(10)

H^d_c(YK,L_K(w, w0;O)) ^p

w−w0t 2 Up=U_p⁰

//

C_K,β+1^Σ∗

H^d_c(YK,L_K(w, w0;O))

C_K,β^Σ∗

H^d_c(A^×/F^×U(p^β+1Σa_K), C_K,β+1^Σ∗ L_K)

p

w−w0t 2

“p0 0 1

”

//

p^(β+1)

w−w0t

2 crit^w,w_β+1⁰◦tw^w,w_β+1⁰

H^d_c(A^×/F^×U(p^βΣa_K), C_K,β^Σ∗ L_K)

p^β

w−w0t

2 crit^w,w_β ⁰◦tw^w,w_β ⁰

O[Cl⁺_F(p^β+1Σa_K)]

pr^β_β+1

//O[Cl⁺_F(p^βΣa_K)]

The commutativity of the lower square follows from the matrix computation:

p 0 0 1

1 −p^−β−1

0 1

=

1 p^−β

0 1

p 0 0 1

, and the fact that ^p_{0 1}⁰

acts trivially on X^(w−w0t)/2Y^(w+w0t)/2.

The ordinary idempotent e_p acting on the cohomology of quaternionic Shimura varieties (including Hilbert modular varieties) has been introduced by Hida in [H1]. In parallel weighte_p cuts out the maximal direct factor of thep-adic cohomology on which Up is invertible (i.e. acts by ap-adic unit). In arbitrary cohomological weight (w, w0) there is a shift by the lowest Hodge-Tate weight of the cohomology which is (w₀t−w)/2 and one should consider instead the operator U_p⁰ = p^w−w20tU_p which preserves the p- adic integral structure in an optimal way. Usually one reserves the word ordinary for level K₁(p^α) and uses the term nearly ordinary otherwise (some authors use the word ordinary for both).

By (10) the maps

S_K,Σ,β^w,w0 =p^βw−w20t crit^w,w_β ⁰◦tw^w,w_β ⁰◦C_K,β^Σ∗ ◦(U_p⁰)^−β form a projective system, leading to:

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Definition 1.20. For critical (w, w0), we consider the homomorphism:

S_K,Σ^w,w0 =

S_K,Σ,β^w,w0

β≥1:e_pH^d_c(Y_K,L_K(w, w₀;O))→ O[[Cl⁺_F(p^∞Σa_K)]].

It will be used in §2.3 to attach p-adicL-functions to Hilbert automorphic forms.

In order to attach p-adicL-functions to newforms (see theorem 2.5), it is enough to take K=K11(p^α)∩K1(n) and consider the homomorphism:

S^w,w_K ⁰ =

S_K,^w,w0

∅,β

β≥α:epH^d_c(YK,L_K(w, w0;O))→ O[[Cl⁺_F(p^∞)]]

whose construction only relies on the automorphic symbols introduced in §1.2.

2. p-adic L-functions for nearly ordinary automorphic forms on GL₂ We will constructp-adicL-functions for various nearly ordinary Hilbert automorphic forms encoding both the vertical and the horizontal aspects of the theory.

By global class field theory, the maximal abelian pro-p quotient Gal(F^(pΣ)/F) of Gal_F,pΣ is isomorphic to the maximal pro-pquotient Cl^(p)_F (p^∞Σ) of Cl⁺_F(p^∞Σ) and the character χω⁻¹ can be seen as character of Cl^(p)_F (p^∞Σ).

2.1. Automorphic representations.

2.1.1. Archimedean Euler factors. Letπ be a cuspidal automorphic representation of GL₂(A) and let (w, w₀)∈Z[I]×Z be a cohomological weight (see definition 1.16).

Definition 2.1. We say that π is cohomological of weight (w, w₀) if for all τ ∈I the representationπτ is parabolically induced from the following character

a ∗ 0 d

7→sgn(a)^w0|a|^{12(wτ+2−w0)}|d|^{−12(wτ+2+w0)}.

In what follows we assume thatπis cohomological of weight (w, w₀). The definition is justified by the fact that such a π contributes to H^d(Y_K,L_K(w, w₀;C)) whenever π^K_f 6= {0}. Moreover, the Harish-Chandra parameter of π∞ is given by (w+t, w₀) and it is known that such a π corresponds to a Hilbert modular newform of weight (w+ 2t, w0).

One can attach toπ a Γ-factor, depending only onπ∞, as follows:

(11) Γ(π, s) =Y

τ∈I

Γ_C

s− w0−wτ

2

, where Γ_C(s) = 2·(2π)^−sΓ(s).

The functional equation for π relates L(π, s) and L(ˇπ,1−s) = L(π, w0+ 2−s), hence the value of L(π, s) at s= 1 is critical in the sense of Deligne if neither Γ(π, s) nor Γ(π, w₀+ 2−s) has a pole at s= 1.

It is straightforward to check that L(π,1) is critical in the sense of Deligne [De] if, and only if, (w, w₀) is critical in the sense of definition 1.16, that is|w₀| ≤minτ∈I(w_τ).

15