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.

arXiv comment50 pages; minor corrections; final version, to appear in Invent. Math

References in corpus (4)

• On the parity of ranks of Selmer groups II
• Parity of ranks for elliptic curves with a cyclic isogeny
• Rational points of the group of components of a Neron model
• 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