Self-duality of Selmer groups

TL;DR

This paper provides a new proof of the self-duality property of Selmer groups for abelian varieties over number fields and introduces a method to analyze Selmer rank parities using local Tamagawa numbers.

Contribution

It offers a novel proof of Selmer group self-duality and a new approach to study Selmer rank parities via local Tamagawa numbers in intermediate extensions.

Findings

Self-duality of Selmer groups is established with a new proof.

A method is developed to relate Selmer rank parities to local Tamagawa numbers.

The approach enhances understanding of Selmer groups in Galois extensions.

Abstract

The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.