Evil Primes and Superspecial Moduli

For a quartic primitive CM field K, we say that a rational prime p is evil if at least one of the abelian varieties with CM by K reduces modulo a prime ideal p|p to a product of supersingular elliptic curves with the product polarization. We call such primes evil primes for K. In [GL], we showed that for fixed K, such primes are bounded by a quantity related to the discriminant of the field K. In this paper, we show that evil primes are ubiquitous in the sense that, for any rational prime p, there are an infinite number of fields K for which p is evil for K. The proof consists of two parts: (1) showing the surjectivity of the abelian varieties with CM by K, for K satisfying some conditions, onto the the superspecial points modulo p of the Hilbert modular variety associated to the intermediate real quadratic field of K, and (2) showing the surjectivity of the superspecial points modulo p of the Hilbert modular variety associated to a large enough real quadratic field onto the superspecial points modulo p with principal polarization on the Siegel moduli space.