Smoothness of Shortest Isogenies to Frobenius Conjugates

Abstract

Let $E$ be chosen uniformly from the geometric isomorphism classes of supersingular elliptic curves over ${\overline{\mathbb{F}}}_p$ and ${\delta }_E$ be the minimum degree of isogenies from $E$ to its Frobenius conjugate $E^{(p)}$. We establish bounds for the smoothness probability of ${\delta }_E$ and hence affirm Wesolowski's $p^{1/3+o(1)}$ attack on the supersingular endomorphism ring problem is unconditional. We also establish a uniform asymptotic formula, with a main term expressed in terms of Dirichlet $L$ values, for the probability that a connection isogeny of degree $n$ exists between $E$ and $E^{(p)}$.