OVERCONVERGENT MODULAR SYMBOLS AND p-ADIC L-FUNCTIONS

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. En tant qu’application nous décrivons un algorithme en temps polynomial pour la calculation explicite des fonctions L p-adiques associées dans ce cas-là. Dans le cas de pente critique, le theorème de controle échoue toujours de produire des “liftings propres” (voire Theoreme 5.14 et [17] pour une récupération), mais l’algorithme “réussit” néanmoins de produire des fonctions L p-adiques. Dans les deux dernières sections nous présentons des données numériques pour plusieurs exemples de pente critique et examinons le polygone de Newton des fonctions L p-adiques associées. Abstract. This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent p-adic modular symbols. Specifically, we give a constructive proof of a control theorem (Theorem 1.1) due to the second author [20] 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 p-adic L-functions in this case. In the case of critical slope, the control theorem fails to always produce eigenliftings (see Theorem 5.14 and [17] for a salvage), but the algorithm still “succeeds” at producing p-adic L-functions. In the final two sections we present numerical data in several critical slope examples and examine the Newton polygons of the associated p-adic L-functions. This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent p-adic modular symbols. Specifically, we give a constructive proof of a control theorem (Theorem 1.1) due to the second author [20] 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 p-adic L-functions in this case. In the case of critical slope, the control theorem fails to always produce eigenliftings (see Theorem 5.14 and [17] for a salvage), but the algorithm still “succeeds” at producing p-adic L-functions. In the final two sections we present numerical data in several critical slope examples and examine the Newton polygons of the associated p-adic L-functions.