In 1987, Elkies proved that every elliptic curve defined over Q has infinitely many supersingular primes. In this talk, I will present an extension of this result to certain abelian fourfolds in Mumford’s families and more generally, to certain families of Kuga-Satake abelian varieties. I will review Elkies’ proof and explain how his strategy of intersecting with CM cycles can be adapted to our setting. I will also discuss some of the techniques in our proof to study the local properties of the CM cycles.