Congruence covers of triangular modular curves and their Galois groups (Q2833590)

Let \(\Gamma_{a,\infty,\infty}\) be the \((a,\infty,\infty)\)-triangle group, which is a subgroup of \(\text{SL}_2({O})\) where \(O\) is the ring of integers of the totally real field \(\mathbb{Q}(\zeta_{2a}+\zeta^{-1}_{2a})\). If \(\mathfrak{p}\) is a prime ideal of \(O\), let \(\Gamma_{a,\infty,\infty}(\mathfrak{p})\) and \(\Gamma^{(0)}_{a,\infty,\infty}(\mathfrak{p})\) be the congruence subgroups, and \(X_{a,\infty,\infty}(\mathfrak{p})\) and \(X^{(0)}_{a,\infty,\infty}(\mathfrak{p})\) be their corresponding modular curves. The subject paper computes three invariants of the Galois cover \(\varphi: X_{a,\infty,\infty}(\mathfrak{p}) \to X_{a,\infty,\infty}(1)\): the Galois group of \(\varphi\), the genera of \(X_{a,\infty,\infty}(\mathfrak{p})\) and \(X^{(0)}_{a,\infty,\infty}(\mathfrak{p})\).