2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

Abstract

We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove it over quadratic extensions of the base field, providing essentially the first examples of the 2-parity conjecture in dimension greater than one. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. Particularly surprising is the appearance in the formula of terms that govern whether or not the Cassels-Tate pairing on the Jacobian is alternating, which first appeared in a paper of Poonen and Stoll. We prove this local formula in many instances and show that in all cases it follows from standard global conjectures.

Quadratic twists of abelian varieties and disparity in Selmer ranks

“…Lemma 10.9 (ii) enables one to evaluate the local terms δ v for archimedean places. For nonarchimedean places of odd residue characteristic, the dimension of the cokernel of the norm map may be expressed in terms of Tamagawa numbers: see [17,Lemma 2.5].…”

The 2-parity conjecture for elliptic curves with isomorphic 2-torsion

“…In particular the p-parity conjecture is still open for elliptic curves over number fields in general. In higher dimensions, the p-parity conjecture is known for principally polarized abelian surfaces subject to local conditions ( [11]), for Jacobians of hyperelliptic curves base-changed from a subfield of index 2 ( [16]) and abelian varieties admitting a suitable isogeny ( [5]). In this paper, we consider two elliptic curves whose 2-torsion groups are isomorphic as Galois modules and prove the following.…”