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

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 $3$.