Inventiones mathematicae

, 178:23

Regulator constants and the parity conjecture

Article

DOI: 10.1007/s00222-009-0193-7

Cite this article as:
Dokchitser, T. & Dokchitser, V. Invent. math. (2009) 178: 23. doi:10.1007/s00222-009-0193-7

Abstract

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)

11G0511G0711G1011G4019A2220B99

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

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