Duke Mathematical Journal

Exceptional splitting of reductions of abelian surfaces

Ananth N. Shankar and Yunqing Tang

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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

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

Received: 5 March 2018
Revised: 5 June 2019
First available in Project Euclid: 11 December 2019

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Mathematical Reviews number (MathSciNet)

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

Arakelov intersection theory Hilbert modular surfaces Borcherds theory


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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