For A an abelian variety defined over a number field, a conjecture of Serre predicts that the set S of primes of ordinary reduction for A has positive density. If A is an elliptic curve without CM, Serre proved that S has density 1, and Elkies showed that, if L admits a real embedding, then the complement of S, that is the set of supersingular primes, is infinite.
In this talk, I will discuss generalizations of both Serre’s and Elkies’s theorems to abelian varieties of type IV, that is with multiplication by a CM field, which are parametrized by unitary Shimura curves.
This talk is based on joint work in progress with Victoria Cantoral-Farfan, Wanlin Li, Rachel Pries, and Yunqing Tang.