Automorphisms of general modular curves

Abstract

We prove that for a geometrically connected modular curve $X_H$ of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of $X_H$. The proof relies on a general criterion involving the gonality of $X_H$ and the dimension of the CM part of the Jacobian of $X_H$. For arbitrary level, we give algorithms to compute the modular automorphisms and, in many cases, to prove that there are no other automorphisms. Finally, we apply these algorithms to the curves in the database LMFDB.