Infinitely many primes of basic reduction for abelian varieties

Elkies proved that an elliptic curve over the field of rational numbers has infinitely many supersingular reductions. The generalization of the 0-dimensional supersingular locus of the modular curve is the basic locus of a Shimura curve at a good prime.  In this talk, we generalize Elkies’s theorem  to some abelian varieties over totally real fields parametrized by certain unitary Shimura curves  arising as moduli spaces of cyclic covers of the projective line ramified at 4 points. 

This talk is based on joint work in progress with Wanlin Li, Rachel Pries, Yunqing Tang and in part also with Victoria Cantoral Farfan.