Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms

Abstract

For a fixed characteristic $p$, let $\delta(p)$ denote the smallest degree needed to guarantee the existence of an isogeny from any supersingular curve $E$ to its Frobenius conjugate $E^{(p)}$. We translate existence questions concerning $\delta(p)$ into questions about positive definite integral ternary quadratic forms and use Voronoi's theory of perfect forms in dimension three to characterize when $\delta(p)$ is nearly maximal. For sufficiently large primes, we show that $\delta(p)$ attains its upper bound precisely when $p$ is represented by one of nine explicit cubic polynomials, reducing the infinitude of such primes to whether one of these cubics takes prime values infinitely often. We also prove that, for almost all primes, $\delta(p)$ lies at least $p^{1/6-o(1)}$ below its upper bound.