Regulator constants and the parity conjecture

Abstract

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.

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

References in corpus (2)

• Parity of ranks for elliptic curves with a cyclic isogeny
• Self-duality of Selmer groups

Cited by in corpus (29)

• On the Birch-Swinnerton-Dyer quotients modulo squares
• Root numbers and parity of ranks of elliptic curves
• Arithmetic of hyperelliptic curves over local fields
• Disparity in Selmer ranks of quadratic twists of elliptic curves
• Self-duality of Selmer groups
• Notes on the Parity Conjecture
• Models of curves over DVRs
• Index formulae for integral Galois modules
• Torsion homology and regulators of isospectral manifolds
• Large Selmer groups over number fields
• Group representations in the homology of 3-manifolds
• Rational representations and permutation representations of finite groups
• Comparing local constants of elliptic curves in dihedral extensions
• Regulator constants of integral representations of finite groups
• The Brauer-Kuroda formula for higher S-class numbers in dihedral extensions of number fields
• Parity conjectures for elliptic curves over global fields of positive characteristic
• On Brauer-Kuroda type relations of S-class numbers in dihedral extensions
• Elliptic curves with p-Selmer growth for all p
• Factor equivalence of Galois modules and regulator constants
• Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
• Euler factors determine local Weil representations
• On a BSD-type formula for L-values of Artin twists of elliptic curves
• The $\ell$-parity conjecture over the constant quadratic extension
• Generalised Burnside Rings, G-categories and Module Categories
• Relations between permutation representations in positive characteristic
• Distributions of rational points on Kummer Varieties
• An abelian subfield of the dyadic division field of a hyperelliptic Jacobian
• Growth of Selmer Groups over function fields
• Finiteness of the Tate-Shafarevich Groups for Elliptic Curves over the Field of Rational Numbers