A note on local formulae for the parity of Selmer ranks

TL;DR

This paper offers evidence supporting a twisted parity conjecture for Jacobians by analyzing Selmer groups using arithmetic duality theorems, contributing to understanding the parity of Selmer ranks.

Contribution

It introduces a twisted version of the parity conjecture for Jacobians and employs arithmetic duality to analyze Selmer groups, providing new insights into their structure.

Findings

Supports the twisted parity conjecture for Jacobians

Uses duality theorems to study Selmer group determinants

Provides evidence for parity relations in Selmer ranks

Abstract

In this note, we provide evidence for a certain twisted version of the parity conjecture for Jacobians, introduced in prior work of V. Dokchitser, Green, Konstantinou and the author. To do this, we use arithmetic duality theorems for abelian varieties to study the determinant of certain endomorphisms acting on p-infinity Selmer groups.