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2010•Mathematical Research LettersOpen access

On finiteness of endomorphism rings of abelian varieties

Chia‐Fu Yu

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Abstract

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 characteristic 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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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 characteristic 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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Available abstract

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 characteristic 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.

Key concepts: Mathematics, Endomorphism, Endomorphism ring, Abelian group, Pure mathematics, Arithmetic of abelian varieties, Abelian variety of CM-type, Abelian variety

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