Quaternions, polarizations and class numbers

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.