Overconvergent modular symbols and p -adic L -functions

Robert Pollack; Glenn Stevens

Annales scientifiques de l'École Normale Supérieure (2011)

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

Abstract

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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.

How to cite

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Pollack, 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 -

References

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