2∞-Selmer rank parities via the Prym construction

Abstract

We derive a local formula for the parity of the $2^{\infty }$-Selmer rank of Jacobians of curves of genus 2 or 3 which admit an unramified double cover. 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.