Unique factorization of principally polarized abelian varieties

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.

arXiv commentThe paper "Polarisations sur les variétés abéliennes produits" (C. R. Acad. Sci. Paris, t. 323, Série 1, p. 631-635, 1996) by Olivier Debarre, of which we were initially unaware, contains stronger results than ours