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Exceptional splitting of reductions of abelian surfaces. (English) Zbl 07198458

It was conjectured that an absolutely simple abelian variety over a number field has absolutely simple reduction for a density 1 set of primes if and only if its endomorphism ring is commutative. This conjecture is proved for abelian varieties of dimension 2 or 6 whose geometric endomorphism ring is \(\mathbb Z\); for abelian surfaces with real multiplication; and in full generality conditional on the Mumford-Tate conjecture.
Following the latter conjecture, one may ask whether the conjecturally zero density set of primes at which a given abelian variety does not have absolutely simple reduction is finite or infinite. The authors prove that an abelian surface with real multiplication defined over a number field has infinitely many places of split reduction. The strategy of the proof relies on using the theory of Arakelov intersection on the Hilbert modular surface.

MSC:

11G05 Elliptic curves over global fields
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11G18 Arithmetic aspects of modular and Shimura varieties
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