In this thesis, we set up the basic theory of p-adic modular forms over Shimura curves over Q, parallel to the classical case over modular curves. We define and study the structure of the spaces of p-adic modular forms with respect to certain quaternion algebras over Q. We study the relation of these modular forms with classical quaternionic modular forms. We prove a canonical subgroup theorem ...

Cet article est exploration constructive des rapports entre les symboles modulaires classique et les symboles modulaires p-adiques surconvergents. Plus précisément, on donne une preuve constructive d’un theorème de controle (Theoreme 1.1) du deuxiéme auteur [20]; ce theoréme preuve l’existence et l’unicité des “liftings propres” des symboles propres modulaires classiques de pente non-critique. ...

Introduction 1 1. Notation and conventions 8 2. Shimura varieties for unitary groups 13 3. Modular forms on unitary groups 23 4. Hida theory for unitary groups 36 5. Ordinary p-adic Eisenstein series on U(2, 1) 52 6. Constant terms of the p-adic Eisenstein series 68 7. Eisenstein ideal and p-adic L-functions 76 8. Application to the main conjecture for CM fields 93 Acknowledgments 107 Reference...

‎In The endoscopic classification of representations‎‎, ‎J‎. ‎Arthur has proved the Langlands' classification for discrete series of p-adic classical groups‎. ‎This uses endoscopy and twisted endoscopy‎. ‎In this very short note‎, ‎we remark that the normalization ‎$‎rm‎grave{a}‎$‎ la Langlands-Shahidi of the intertwining operators‎, ‎allows to avoid endoscopy‎. ‎This is based on the intertwini...

A simplified model of tachyon matter in classical and quantum mechanics is constructed. p-Adic path integral quantization of the model is considered. Recent results in using p-adic analysis, as well as perspectives of an adelic generalization, in the investigation of tachyons are briefly discussed. In particular, the perturbative approach in path integral quantization is proposed.

T -adic exponential sums associated to a Laurent polynomial f are introduced. They interpolate all classical p-power order exponential sums associated to f . The Hodge bound for the Newton polygon of L-functions of T -adic exponential sums is established. This bound enables us to determine, for all m, the Newton polygons of Lfunctions of p-power order exponential sums associated to an f which i...

This paper is an expanded version of my lecture delivered at the 97 Seattle summer research conference on finite fields. It gives a quick exposition of Dwork’s conjecture about p-adic meromorphic continuation of his unit root L-function arising from a family of algebraic varieties defined over a finite field of characteristic p. As a simple illustration, we discuss the classical example of the ...