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Critical slope p ‐adic L ‐functions
Author(s) -
Pollack Robert,
Stevens Glenn
Publication year - 2013
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jds057
Subject(s) - mathematics , modular form , interpolation (computer graphics) , function (biology) , property (philosophy) , upper and lower bounds , combinatorics , pure mathematics , image (mathematics) , mathematical analysis , computer science , philosophy , epistemology , artificial intelligence , evolutionary biology , biology
Let g be an eigenform of weight k +2 on Γ 0 ( p )∩Γ 1 ( N ) with p ∤ N . If g is non‐critical (that is, of slope less than k +1), using the methods of Amice–Vélu and Višik, one can attach [‘Distributions p ‐adiques associées aux séries de Hecke’, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) , Astérisque 24–25 (Soc. Math. France, Paris, 1975) 119–131 (French)] and Višik [ Mat. Sb. (N.S.) 99 (1976) 248–260], then one can attach a p ‐adic L ‐function to g which is uniquely determined by its interpolation property together with a bound on its growth. However, in the critical slope case, the corresponding growth bound is too large to uniquely determine the p ‐adic L ‐function with its standard interpolation property. In this paper, using the theory of overconvergent modular symbols, we give a natural definition of p ‐adic L ‐functions in this critical slope case. If, moreover, the modular form is not in the image of theta, then the p ‐adic L ‐function satisfies the standard interpolation property.