The arithmetic theory of local constants for abelian varieties
type: Item
instance of: scholarly article
title: The arithmetic theory of local constants for abelian varieties
zbMATH Open document ID: 1257.14032
DOI: 10.4171/RSMUP/127-2
author: Marco Adamo Seveso
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2012-08-03T00:00:00Z
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol127.php
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 6063115
zbMATH Keywords: Selmer group of an abelian variety
zbMATH Keywords: \(p\)-perity conjecture for Selmer groups
reviewed by: A. A. Actor
MaRDI profile type: Publication
cites work: Q4455336
cites work: Q3972724
cites work: Q4290021
cites work: A generalisation of the Cassel-Tate pairing.
cites work: On the local divisibility of Heegner points
cites work: Finding large Selmer rank via an arithmetic theory of local constants
cites work: Twisting commutative algebraic groups
cites work: Q4166840
cites work: Congruences and rationality of Stark-Heegner points
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review text: In [Ann. Math. (2) 166, No. 2, 579--612 (2007; Zbl 1219.11084)] \textit{B. Mazur} and \textit{K. Rubin} proved a relative version of the \(p\)-parity conjecture for Selmer groups of elliptic curves over some special number field using their theory of local constants. Here \(p\) denotes an odd rational prime. They claim that their proofs work in greater generality in the case of an abelian variety which admits a polarization of degree prime to \(p\).NEWLINENEWLINE In the paper under review, proofs for this are worked out with emphasis to the case of an abelian variety with real multiplication. In an appendix it is shown moreover that an abelian scheme over a connected and normal scheme \(S\) with real multiplication in a totally real number field, whose degree over the rationals equals the relative dimension of the abelian scheme, admits, for every rational integer \(t\) which is invertible in \(S\), a polarization of degree prime to \(t\).
description: scientific article; zbMATH DE number 6063115