Principal polarizations on products of abelian varieties over finite fields
Sergey Rybakov
TL;DR
The paper advances understanding of when products of abelian varieties over finite fields admit irreducible principal polarizations and when such products realize Jacobians. It refines Lauter and Howe’s results by leveraging endomorphism-algebra data, the Serre gluing construction, and the role of exceptional primes to characterize when a product $A\times B$ can carry an irreducible principal polarization. A key contribution is a necessary condition for a principal polarization on a variety in the isogeny class of a product of a geometrically simple abelian surface and an elliptic curve, plus an application showing that the resulting abelian threefold (or its quadratic twist) is a Jacobian. These results illuminate the interplay between polarizations, endomorphism algebras, and isogeny classes, with implications for Jacobians of genus $3$ curves and their zeta-functions over finite fields.
Abstract
We refine and generalize the results of K. E. Lauter and E. W. Howe on principal polarizations on products of abelian varieties over finite fields. Firstly, we study the reasons for the absence of an irreducible principal polarization in the isogeny class of the product of an ordinary and a supersingular abelian variety. Secondly, we provide a necessary condition for the existence of a principal polarization on an abelian variety in the isogeny class of the product of a geometrically simple abelian surface and an elliptic curve. As an application, we prove that this abelian threefold or its quadratic twist is a Jacobian.