Inventiones mathematicae

, 178:23

Regulator constants and the parity conjecture



The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-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 is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of \(\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})\) . 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.

Mathematics Subject Classification (2000)

11G05 11G07 11G10 11G40 19A22 20B99 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Robinson CollegeCambridgeUK
  2. 2.Gonville & Caius CollegeCambridgeUK

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