Overconvergent modular symbols and $p$-adic $L$-functions

Pollack, Robert; Stevens, Glenn

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 1-42.

Cet article est une exploration constructive des rapports entre les symboles modulaires classiques et les symboles modulaires $p$-adiques surconvergents. Plus précisément, nous donnons une preuve constructive d’un théorème de contrôle (Théorème 1.1) du deuxième auteur [19] ; ce théorème démontre l’existence et l’unicité des « liftings propres » des symboles propres modulaires classiques de pente non-critique. Comme application, nous décrivons un algorithme en temps polynomial pour le calcul explicite des fonctions $L$ $p$-adiques associées dans ce cas-là. Dans le cas de pente critique, le théorème de contrôle échoue toujours à produire des « liftings propres » (voir Théorème 5.14 et [16] pour un succédané), mais l’algorithme « réussit » néanmoins à 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.

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 [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 $p$-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 [16] 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.

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