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

Annales scientifiques de l'École Normale Supérieure (2011)

- Volume: 44, Issue: 1, page 1-42
- ISSN: 0012-9593

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topPollack, Robert, and Stevens, Glenn. "Overconvergent modular symbols and $p$-adic $L$-functions." Annales scientifiques de l'École Normale Supérieure 44.1 (2011): 1-42. <http://eudml.org/doc/272138>.

@article{Pollack2011,

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 acontrol 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 ofcritical 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.},

author = {Pollack, Robert, Stevens, Glenn},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {modular symbols; -adic -functions},

language = {eng},

number = {1},

pages = {1-42},

publisher = {Société mathématique de France},

title = {Overconvergent modular symbols and $p$-adic $L$-functions},

url = {http://eudml.org/doc/272138},

volume = {44},

year = {2011},

}

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

PB - Société mathématique de France

VL - 44

IS - 1

SP - 1

EP - 42

AB - 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 acontrol 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 ofcritical 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.

LA - eng

KW - modular symbols; -adic -functions

UR - http://eudml.org/doc/272138

ER -

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