Exceptional isogenies between reductions of pairs of elliptic curves

Abstract

Let $E$ and $E\text{'}$ be two elliptic curves over a number field. We prove that the reductions of $E$ and $E\text{'}$ at a finite place $\mathfrak{p}$ are geometrically isogenous for infinitely many $\mathfrak{p}$, and we 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.