Regulator constants and the parity conjecture

Details Regulator constants and the parity conjecture Regulator constants and the parity conjecture

2007-09-18

arxiv.org/abs/0709.2852v3

Mathematics

Number Theory

50 pages; minor corrections; final version, to appear in Invent. Math

Scientific paper

10.1007/s00222-009-0193-7

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/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 Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not 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.

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