Shimura curve Atkin--Lehner quotients of genus at most two

Abstract

We provide a complete enumeration of all quotients of genus $0, 1$ and $2$ of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by non-trivial subgroups of Atkin--Lehner involutions. For all $1270$ genus $1$ quotients $X$ with $N$ squarefree, we determine the isomorphism class of the Jacobian $X$. For $146$ non-elliptic genus $1$ curves $X$ and for $405$ curves genus $2$ quotients $X$, we provide a defining equation for $X$. A main tool for us is the theory of \v{C}erednik--Drinfeld uniformizations of the curves $X_0^D(N)$, which we implement in wider generality than has previously been done in the literature.