## Duke Mathematical Journal

### Exceptional splitting of reductions of abelian surfaces

#### Abstract

Heuristics based on the Sato–Tate conjecture and the Lang–Trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces with real multiplication. As in previous work by Charles and Elkies, this shows that a density $0$ set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

#### Article information

Source
Duke Math. J., Volume 169, Number 3 (2020), 397-434.

Dates
Revised: 5 June 2019
First available in Project Euclid: 11 December 2019

https://projecteuclid.org/euclid.dmj/1576033298

Digital Object Identifier
doi:10.1215/00127094-2019-0046

Mathematical Reviews number (MathSciNet)
MR4065146

#### Citation

Shankar, Ananth N.; Tang, Yunqing. Exceptional splitting of reductions of abelian surfaces. Duke Math. J. 169 (2020), no. 3, 397--434. doi:10.1215/00127094-2019-0046. https://projecteuclid.org/euclid.dmj/1576033298

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