  Let E and E' be two elliptic curves over a number field. We prove that the
reductions of E and E' at a finite place p are geometrically isogenous for
infinitely many p, and draw consequences for the existence of supersingular
primes. This result is an analogue for distributions of Frobenius traces of
known results on the density of Noether-Lefschetz loci in Hodge theory. The
proof relies on dynamical properties of the Hecke correspondences on the
modular curve.
