Integral Tate-module comparison for higher-genus hyperelliptic Jacobians

Determine whether the integral Tate-module comparison established for elliptic curves extends to Jacobians of higher-genus hyperelliptic curves under the corresponding hypotheses on the defining equations and root differences.

Background

The paper proves that, for elliptic curves over a complete discretely valued field of odd residue characteristic, a Galois-equivariant correspondence between the roots of two monic cubic models—together with agreement of the square class of the coefficient multiplying the cubic—induces an isomorphism of their integral 2-adic Tate modules. The result strengthens an earlier rational comparison for hyperelliptic Jacobians, which gives isomorphic rational ℓ-adic representations for every ℓ different from the residue characteristic.

The authors identify the unresolved extension from genus 1 to higher genus: whether analogous hypotheses imply an isomorphism of integral Tate modules for Jacobians of higher-genus hyperelliptic curves. They mention combining the explicit halving method developed in the paper with Stoll’s algorithm for halving points on odd-degree hyperelliptic Jacobians in Mumford representation as a possible approach.