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Regulator constants and the parity conjecture. (English) Zbl 1219.11083

Summary: The \(p\)-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in \(p^{\infty }\)-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if \(E/\mathbb Q\) is semistable at 2 and 3, \(K/\mathbb Q\) is abelian and \(K^{\infty }\) is its maximal pro-\(p\) extension, then the \(p\)-parity conjecture holds for twists of \(E\) by all orthogonal Artin representations of \(\text{Gal}(K^{\infty}/{\mathbb{Q}})\). We also give analogous results when \(K/\mathbb Q\) is non-abelian, the base field is not \(\mathbb Q\) and \(E\) is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

MSC:

11G05 Elliptic curves over global fields
11G07 Elliptic curves over local fields
11G10 Abelian varieties of dimension \(> 1\)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
19A22 Frobenius induction, Burnside and representation rings
20B99 Permutation groups
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