Hyperelliptic and Trigonal Modular Curves in Characteristic <i>p</i>

ABSTRACT Let $X_\Delta(N)$ be an intermediate modular curve of level N, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_\Delta(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p \nmid N$ such that $X_\Delta(N)_{\overline{\mathbb{F}}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes p such that $X_\Delta(N)_{\mathbb{F}_p}$ is trigonal. This is done by first using the Castelnuovo–Severi inequality to establish a bound N0 such that if $X_0(N)_{{\overline{\mathbb{F}}_p}}$ is hyperelliptic or trigonal, then $N \leq N_0$. To deal with the remaining small values of N, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_\Delta(N)$, all the primes p such that $X_\Delta(N)_{{\overline{\mathbb{F}}_p}}$ is trigonal or hyperelliptic. Furthermore, as a byproduct of the developed methods we show that $X_\Delta(N)_{{\overline{\mathbb{F}}_p}}$ is not a smooth plane quintic, for any N and any p.