$2^\infty$-Selmer Rank Parities via the Prym Construction

TL;DR

This paper develops a local formula for predicting the parity of the infinite 2-Selmer rank of Jacobians of genus 2 and 3 curves with rational 2-torsion points, aiding in testing the 2-parity conjecture.

Contribution

It introduces an explicit local formula for the 2-infinity Selmer rank parity of certain Jacobians, providing new tools for parity conjecture investigations.

Findings

Derived a local parity formula for Jacobians of genus 2 and 3 curves.

Provided an explicit example illustrating the formula's application.

Applied results to support the parity conjecture for semistable genus 3 curves.

Abstract

We derive a local formula for the parity of the $2^{\infty}$-Selmer rank of Jacobians of curves of genus $2$ or $3$ with a $K$-rational $2$-torsion point. We give an explicit example to show how this local formula gives rank parity predictions against which the $2$-parity conjecture may be tested. Our results yield applications to the parity conjecture for semistable curves of genus $3$.