On the non-existence of exceptional automorphisms on Shimura curves

TL;DR

This paper investigates the automorphism groups of Shimura curves, establishing that for genus at least 2, these groups are mostly generated by Atkin-Lehner involutions, with few exceptions.

Contribution

It proves that automorphism groups of Shimura curves are 2-elementary abelian groups containing Atkin-Lehner involutions, and provides criteria to identify when automorphisms are solely these involutions.

Findings

Automorphism groups are 2-elementary abelian groups for genus ≥ 2.

Atkin-Lehner involutions form a subgroup of the automorphism group.

Conjecture: Automorphism group equals Atkin-Lehner group except finitely many cases.

Abstract

We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is greater than or equal to 2, $\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\Aut (X_0(D, N)) = W_0(D, N)$ except for finitely many values of $(D, N)$ and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes $p\mid D$.