Arithmetic Prym Constructions and (1,2)-Polarized Abelian Surfaces

Abstract: 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.

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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