On finiteness of endomorphism rings of abelian varieties

The endomorphism ring $\End(A)$ of an abelian variety $A$ is an order in a semi-simple algebra over $\Q$. The co-index of $\End(A)$ is the index to a maximal order containing it. We show that for abelian varieties of fixed dimension over any field of \ch $p >0$, the $p$-exponents of the co-indices of their endomorphism rings are bounded. We also give a few applications to this finiteness result.

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Published 1 January 2010