(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 \(3\).