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 set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.
"Exceptional splitting of reductions of abelian surfaces." Duke Math. J. 169 (3) 397 - 434, 15 February 2020. https://doi.org/10.1215/00127094-2019-0046