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Compatibility of arithmetic and algebraic local constants (the case $\ell \neq p$)

Part of:Arithmetic algebraic geometryAlgebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  08 April 2015

Jan Nekovář
Jan Nekovář*
Affiliation:
Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu, Théorie des Nombres, Case 247, 4 place Jussieu, F-75252, Paris cedex 05, France email jan.nekovar@imj-prg.fr
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Abstract

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We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

Keywords

local constantsSelmer groupsparity conjectureelliptic curves

MSC classification

Primary:11G05: Elliptic curves over global fields11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture11S40: Zeta functions and $L$-functions

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 9 , September 2015, pp. 1626 - 1646
Copyright
© The Author 2015 

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