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Trivial zeros of $p$-adic $L$-functions at near-central points

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms (Co)homology theory

Published online by Cambridge University Press:  24 September 2013

Denis Benois
Denis Benois*
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, F-33400 Talence, France ([email protected])
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Abstract

Using the $\ell $-invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of $p$-adic $L$-functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

Keywords

modular formp-adic representation(φ,Γ)-modulep-adic L-function

MSC classification

Secondary: 11R23: Iwasawa theory 11F80: Galois representations 11F85: $p$-adic theory, local fields 11S25: Galois cohomology 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 3 , July 2014 , pp. 561 - 598
Copyright
©Cambridge University Press 2013 

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