SFU NTAG Seminar: Negarin Mohammadi

We construct explicit examples of abelian surfaces whose 2-torsion Galois representations are not self-dual. Such surfaces are necessarily not principally polarized. We consider abelian surfaces with a polarization of type (1,2) instead. Our main tool is to describe (1,2)-polarized surfaces as a Prym variety $P$ of a double cover of an elliptic curve by a bielliptic curve $C$ of genus 3. We use a Galois-theoretic construction of Donagi and Pantazis to express the dual also as a Prym variety of the same type. The 2-torsion $P[2]$ naturally embeds in $J_C[2]$, where the classical geometry of plane quartics, bitangents, and theta characteristics gives concrete access to the Galois action.

Conversely, we prove that every (1,2)-polarized abelian surface over a base field of characteristic other than 2 can be realized as a bielliptic Prym. Barth already proved this over algebraically closed base fields, and we extend it to arbitrary fields. This is achieved by factoring the polarization $\rho : A \to A^\vee$ through a principally polarized surface $J$. We then show that an appropriate plane section of the Kummer surface of $J$ yields an elliptic curve $E$ with a genus 3 double cover $C \to E$ such that $A^\vee = \mathrm{Prym}(C \to E)$. A careful Galois descent argument then allows us to deduce the general case.

This is joint work with Nils Bruin and Katrina Honigs.