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Quaternions, polarizations and class numbers

Victor Rotger
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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 $π_0(A)$ 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 $π_0(A)$ is uniformly bounded for simple abelian varieties of odd square-free dimension.

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Victor Rotger. 2002-11-06. Quaternions, polarizations and class numbers. https://arxiv.org/abs/math/0211120

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