Let \(A\) be an abelian variety over a number field \(K\), and let \(w(A/K)\) be the root number appearing in the functional equation of the \(L\)-function of \(A/K\). The Birch-and-Swinnerton-Dyer Conjecture implies that \(w(A/K) = (-1)^{\mathrm{rk}(A/K)}\), where \(\mathrm{rk}\) is the Mordell-Weil rank of \(A/K\), and further \(w(A/K) = (-1)^{\mathrm{rk}_p(A/K)}\), where \(\mathrm{rk}_p\) is the \(p^\infty\)-Selmer rank of \(A/K\) and \(p\) is prime. This is called the \(p\)-parity conjecture.
The \(p\)-parity conjecture is known for elliptic curves over \(\mathbb Q\), and the literature for elliptic curves is discussed in the paper under review. Much less is known for higher dimensional abelian varieties, and the author of the paper under review proves the following. Let \(K\) be a number field, and \(L/K\) be a quadratic extension. Let \(C/K\) be a hyperelliptic curve of genus \(g\ge 2\), and let \(J/K\) be the Jacobian of \(C\). Suppose that \(J\) has semistable reduction at each odd prime of \(K\) that ramifies in \(L/K\). Further assume that for each even prime of \(K\) that is inert in \(L/K\), \(J/K\) has good reduction, and that for each even prime \(\mathfrak p\) of \(K\) that ramifies in \(L/K\), \(J/K\) has good ordinary reduction, and \(K_{\mathfrak p}(J[2])/K_{\mathfrak p}\) has odd degree. Then, the \(2\)-parity conjecture holds for \(J/L\). The author also shows that there are plenty of examples that satisfy the hypothesis of the theorem.
In [K. Kramer and J. Tunnell, Compos. Math. 46, 307–352 (1982; Zbl 0496.14030); T. Dokchitser and V. Dokchitser, J. Reine Angew. Math. 658, 39–64 (2011; Zbl 1314.11041)] are considered the \(2\)-parity conjecture for an elliptic curve over quadratic extensions of a number field, and the author of the paper under review follows on the previous works. In [V. Dokchitser and C. Maistret, Proc. Lond. Math. Soc. (3) 127, No. 2, 295–365 (2023; Zbl 07740477)] is considered the \(2\)-parity conjecture for some general semistable abelian surfaces.