2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions

TL;DR

This paper proves the 2-parity conjecture for Jacobians of hyperelliptic curves over quadratic extensions, using a generalized formula relating local invariants and the Cassels--Tate pairing.

Contribution

It generalizes Kramer and Tunnell's formula to hyperelliptic Jacobians, linking local invariants and the Cassels--Tate pairing, and proves the 2-parity conjecture under mild reduction assumptions.

Findings

Proved the 2-parity conjecture over quadratic extensions for hyperelliptic Jacobians.

Developed a new local formula involving invariants and the Cassels--Tate pairing.

Established the formula in many cases, with remaining cases relying on standard conjectures.

Abstract

We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. 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. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels--Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen--Stoll. We establish the local formula in many instances and show that in remaining cases it follows from standard global conjectures.