Overconvergent modular symbols and p-adic L-functions
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 1, p. 1-42

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.

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.

@article{ASENS_2011_4_44_1_1_0,
     author = {Pollack, Robert and Stevens, Glenn},
     title = {Overconvergent modular symbols and $p$-adic $L$-functions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 44},
     number = {1},
     year = {2011},
     pages = {1-42},
     doi = {10.24033/asens.2139},
     zbl = {1268.11075},
     mrnumber = {2760194},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2011_4_44_1_1_0}
}
Pollack, Robert; Stevens, Glenn. Overconvergent modular symbols and $p$-adic $L$-functions. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 1, pp. 1-42. doi : 10.24033/asens.2139. http://www.numdam.org/item/ASENS_2011_4_44_1_1_0/

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