Operations and cooperations in elliptic cohomology. I: Generalized modular forms and the cooperation algebra.
Optimal levels for modular mod 2 representations over totally real fields.
Orbites unipotentes et spectre discret non ramifié. Le cas des groupes classiques déployés
Ordinary -adic Eisenstein series and -adic -functions for unitary groups
The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for by the method of Eisenstein congruence on , where is an imaginary quadratic field. We construct a -adic family of ordinary Eisenstein series on the group of unitary similitudes with the optimal constant term which is basically the product of the Kubota-Leopodlt -adic -function and a -adic -function for . This construction also provides a different point of view of -adic...
Ordre de Bruhat et transfert : le cas réel
Oscillations of Fourier Coefficients of Modular Forms.
Oscillations of Hecke eigenvalues at shifted primes.
In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is √n at prime arguments.
Overconvergent modular forms
We give a geometric definition of overconvergent modular forms of any -adic weight. As an application, we reprove Coleman’s theory of -adic families of modular forms and reconstruct the eigencurve of Coleman and Mazur without using the Eisenstein family.
Overconvergent modular symbols and -adic -functions
This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent -adic modular symbols. Specifically, we give a constructive proof of acontrol theorem (Theorem 1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of non-critical slope. As an application we describe a polynomial-time algorithm for explicit computation of associated -adic -functions in this case. In...
Oб акцессорных коэффициентах уравнения Фукса второго порядка с вещественными особыми точками
-adic Clifford algebras
-adic Differential Operators on Automorphic Forms on Unitary Groups
The goal of this paper is to study certain -adic differential operators on automorphic forms on . These operators are a generalization to the higher-dimensional, vector-valued situation of the -adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the -adic case of the -differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain -adic...
-adic interpolation of convolutions of Hilbert modular forms
In this paper we construct -adic measures related to the values of convolutions of Hilbert modular forms of integral and half-integral weight at the negative critical points under the assumption that the underlying totally real number field has class number . This extends the result of Panchishkin [Lecture Notes in Math., 1471, Springer Verlag, 1991 ] who treated the case that both modular forms are of integral weight. In order to define the measures, we need to introduce the twist operator...
-adic -functions for modular forms
-adic -functions for unitary Shimura varieties. I: Construction of the Eisenstein measure.
-adic -functions of Hilbert modular forms
We construct -adic -functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.
-adic measures attached to Siegel modular forms
We study the critical values of the complex standard--function attached to a holomorphic Siegel modular form and of the twists of the -function by Dirichlet characters. Our main object is for a fixed rational prime number to interpolate -adically the essentially algebraic critical -values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of...
-adic ordinary Hecke algebras for
We study the -adic nearly ordinary Hecke algebra for cohomological modular forms on over an arbitrary number field . We prove the control theorem and the independence of the Hecke algebra from the weight. Thus the Hecke algebra is finite over the Iwasawa algebra of the maximal split torus and behaves well under specialization with respect to weight and -power level. This shows the existence and the uniqueness of the (nearly ordinary) -adic analytic family of cohomological Hecke eigenforms...
-adic representations of a local field. (Représentations -adiques d'un corps local.)
p-Adic Congruences and Modular Forms of Half Integer Weight.