Schwarzian differential equations associated to Shimura curves of genus zero
Let \(X_0^D (N)\) be the Shimura curve associated to a division quaternion algebra over \(\mathbb Q\) of discriminant \(D \neq 1\), and level \(N\) with \((D,N) = 1\). Let \(X = X_0^D (N) / W_D\), where \(W_D\) is the group of Atkin-Lehner involutions \(W_m\) on \(X_0^D (N)\) with \(m | D\). Suppose \(X\) has genus \(0\), and \(t(\tau)\) generates the field of automorphic functions on \(X\). Then \(t'(\tau)^{\frac{1}{2}}\) is a solution to a Schwarzian differential equation of the form \(f'' + Q(t)f = 0\), where \(Q(t)\) is a rational function of \(t\). The author shows that there are exactly \(13\) pairs \((D, N)\) such that \(X = X_0^D (N) / W_D\) has genus zero and \(N>1\); these pairs are explicitly listed. In addition there are \(8\) pairs \((D, 1)\) such that \(X = X_0^D (M) / W_D\) has genus zero for some \(M > 1\). For each of these \(21\) pairs, a generator \(t(\tau)\) of the field of automorphic functions on \(X\) is found, and the rational function \(Q(t)\) is explicitly determined, thus making explicit the Schwarzian differential equations associated to automorphic forms for these Shimura curves.
scientific article; zbMATH DE number 6382655