Exceptional splitting of reductions of abelian surfaces

Ananth N. Shankar, Yunqing Tang

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)397-434
Number of pages38
JournalDuke Mathematical Journal
Issue number3
StatePublished - Feb 15 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics


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