Quaternions, polarizations and class numbers

TL;DR

This paper investigates abelian varieties with quaternion multiplication, providing criteria for principal polarizations and expressing their counts through class numbers, revealing differences based on dimension parity.

Contribution

It offers a pure arithmetic criterion for principal polarizations and relates their number to class numbers of CM fields, with results on bounds depending on dimension.

Findings

Criteria for principal polarizations on quaternionic abelian varieties

Expression of polarization counts via class numbers of CM fields

Existence of varieties with arbitrarily many polarizations in even dimensions

Abstract

We study abelian varieties $A$ with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number $\pi_0(A)$ of isomorphism classes of principal polarizations on $A$ in terms of relative class numbers of CM fields by means of Eichler's theory of optimal embeddings. As a consequence, we exhibit simple abelian varieties of any even dimension admitting arbitrarily many non-isomorphic principal polarizations. On the other hand, we prove that $\pi_0(A)$ is uniformly bounded for simple abelian varieties of odd square-free dimension.