Unique factorization of principally polarized abelian varieties

TL;DR

This paper discusses Shimura's theorem on the unique factorization of principally polarized abelian varieties over complex numbers and extends the result to arbitrary ground fields, linking factors to theta divisor components.

Contribution

The paper provides an exposition of Shimura's factorization theorem and generalizes it to arbitrary ground fields, connecting irreducible factors to theta divisor components.

Findings

Shimura's unique factorization theorem for abelian varieties

Extension of the theorem to arbitrary ground fields

Connection between factors and theta divisor components

Abstract

Shimura proved that each principally polarized abelian variety over $\mathbf{C}$ admits a unique factorization into irreducible principally polarized abelian varieties. We give an exposition of his result, and generalize to an arbitrary ground field $k$. If $k$ is separably closed, the irreducible factors are in bijection with the irreducible components of a theta divisor over $k$ giving rise to the polarization.