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 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 -adic -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 -adic -functions. In the final two sections we present numerical data in several critical slope examples and examine the Newton polygons of the associated -adic -functions.
Cet article est une exploration constructive des rapports entre les symboles modulaires classiques et les symboles modulaires -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 -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 -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 -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}, pages = {1--42}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 44}, number = {1}, year = {2011}, doi = {10.24033/asens.2139}, mrnumber = {2760194}, zbl = {1268.11075}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2139/} }
TY - JOUR AU - Pollack, Robert AU - Stevens, Glenn TI - Overconvergent modular symbols and $p$-adic $L$-functions JO - Annales scientifiques de l'École Normale Supérieure PY - 2011 SP - 1 EP - 42 VL - 44 IS - 1 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2139/ DO - 10.24033/asens.2139 LA - en ID - ASENS_2011_4_44_1_1_0 ER -
%0 Journal Article %A Pollack, Robert %A Stevens, Glenn %T Overconvergent modular symbols and $p$-adic $L$-functions %J Annales scientifiques de l'École Normale Supérieure %D 2011 %P 1-42 %V 44 %N 1 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2139/ %R 10.24033/asens.2139 %G en %F 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/articles/10.24033/asens.2139/
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