  For a quartic primitive CM field $K$, we say that a rational prime $p$ is
{\it evil} if at least one of the abelian varieties with CM by $K$ reduces
modulo a prime ideal $\gerp| p$ to a product of supersingular elliptic curves
with the product polarization. We call such primes {\it evil primes for $K$}.
In \cite{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 $\gerp$ of the Hilbert modular variety associated to
the intermediate real quadratic field of $K$, and (2) showing the surjectivity
of the superspecial points modulo $\gerp$ of the Hilbert modular variety
associated to a large enough real quadratic field onto the superspecial points
modulo $\gerp$ with principal polarization on the Siegel moduli space.

Number Theory;Algebraic Geometry