On local Galois representations associated to ordinary Hilbert modular forms

Let F be a totally real field and p be an odd prime which splits completely in F. We show that a generic p-ordinary non-CM primitive Hilbert modular cuspidal eigenform over F of parallel weight two or more must have a locally non-split p-adic Galois representation, at at least one of the primes of F lying above p. This is proved under some technical assumptions on the global residual Galois representation. We also indicate how to extend our results to nearly ordinary families and forms of non-parallel weight.     

Critical slope p-adic L-functions of CM modular forms

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the critical-slope L-function arising from Kato’s Euler system and comparing this with results of Bellaïche on the critical-slope L-function defined using overconvergent modular symbols.     

On p-adic families of Hilbert cusp forms of finite slope

Let p be a prime number and F a totally real field. In this article, we obtain a p-adic interpolation of spaces of totally definite quaternionic automorphic forms over F of finite slope, and construct p-adic families of automorphic forms parametrized by affinoid Hecke varieties. Further, as an application to the case where [F:Q] is even, we obtain p-adic analytic families of Hilbert eigenforms having fixed finite slope parametrized by weights. This is an analogue of Coleman's analytic families in [R.F. Coleman, p-Adic Banach spaces and families of modular forms, Invent. Math. 127 (1997) 417-479].     

On special zeros of p-adic L-functions of Hilbert modular forms

Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the $\mathcal {L}$ -invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.     

Galois representations for holomorphic Siegel modular forms

We prove local–global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert–Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local–global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case.     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

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control of the image of Galois representations modulo p,

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Hida's congruence criterion outside an explicit set of primes,

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freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

Local constancy of dimensions of Hecke eigenspaces of automorphic forms

We use a method of Buzzard to study p-adic families of Hilbert modular forms and modular forms over imaginary quadratic fields. In the case of Hilbert modular forms, we get local constancy of dimensions of spaces of fixed slope and varying weight. For imaginary quadratic fields we obtain bounds independent of the weight on the dimensions of such spaces.     

On the weights of mod <Emphasis Type="Italic">p</Emphasis> Hilbert modular forms

We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.     

Congruences between Siegel modular forms II

Using a p-adic monodromy theorem on the affine ordinary locus in the minimally compactified moduli scheme modulo powers of a prime p of abelian varieties, we extend Katz?s results on congruence and p-adic properties of elliptic modular forms to Siegel modular forms of higher degree.     

Zero loci of differential modular forms

The theory of p-adic modular forms initiated by Serre, Dwork, and Katz (p-Adic Properties of Modular Schemes and Modular Forms, Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973) “lives” on the complement (in the p-adic completion of the appropriate modular curve) of the zero locus of the Eisenstein form E_p−1. On the other hand, most of the interesting phenomena in the theory of differential modular forms (J. Reine Angew. Math. (520) (2000) 95) take place on the complement of the zero locus of a fundamental differential modular form called f_jet. We establish that the zero locus of the reduction modulo p for p not congruent to one modulo 12 of the Eisenstein form E_p−1 is not contained in the zero locus of the reduction modulo p of the differential modular form f_jet implying that the theory of differential modular forms is applicable in certain situations not covered by the theory of p-adic modular forms.     

On p-adic quaternionic Eisenstein series

We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U(p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).     

On l-adic representations attached to Hilbert and Picard modular surfaces

In this paper we compute the l-adic Lie algebra of a product of l-adic representations associated to a Hilbert modular surface and a Picard modular surface.     

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

p-adic periods and modular symbols of elliptic curves of prime conductor

Under certain assumptions, we prove a conjecture of Mazur and Tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction atp, and itsp-adic period. We generalize this relation to modular forms of weight 2 with coefficients not necessarily in.Oblatum 24-XI-1993 & 8-VI-1994     

Irrationality of some p-adic L-values

We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.     

Congruences between Siegel modular forms

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.     

Stark-Heegner points on modular Jacobians

We present a construction which lifts Darmon's Stark-Heegner points from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an elliptic curve E of conductor Np with the universal modular symbol for Γ₀(Np). We then construct a certain torus T over Qp and lattice L⊂T, and prove that the quotient T/L is isogenous to the maximal toric quotient J₀(Np)^p-new of the Jacobian of X₀(Np). This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens. As a by-product of our theorem, we obtain an efficient method of calculating the p-adic periods of J₀(Np)^p-new.     

Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL₂n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.     

An analogue of the BGG resolution for locally analytic principal series

Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.