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Regulator constants and the parity conjecture

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.

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Correspondence to Tim Dokchitser.

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T. Dokchitser is supported by a Royal Society University Research Fellowship.

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Dokchitser, T., Dokchitser, V. Regulator constants and the parity conjecture. Invent. math. 178, 23 (2009). https://doi.org/10.1007/s00222-009-0193-7

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  • DOI: https://doi.org/10.1007/s00222-009-0193-7

Mathematics Subject Classification (2000)

  • 11G05
  • 11G07
  • 11G10
  • 11G40
  • 19A22
  • 20B99