The arithmetic theory of local constants for abelian varieties
\(p\)-perity conjecture for Selmer groups
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 \(t\).
scientific article; zbMATH DE number 6063115
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