Hyperelliptic and trigonal modular curves in characteristic $p$

TL;DR

This paper classifies primes for which certain modular curves are hyperelliptic or trigonal in characteristic p, using bounds and canonical ideal analysis, and shows these curves are not smooth plane quintics.

Contribution

It provides an explicit prime classification for hyperelliptic and trigonal properties of intermediate modular curves in characteristic p, employing Castelnuovo-Severi bounds and canonical ideal methods.

Findings

Identifies all primes p where X_Δ(N) is hyperelliptic or trigonal.

Establishes bounds on N for hyperelliptic or trigonal curves in characteristic p.

Shows X_Δ(N) is not a smooth plane quintic for any N and p.

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 $p$ 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 $N_0$ 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 the $X_\Delta(N)_{ {\overline{\mathbb F}_p}}$ is trigonal or hyperelliptic. Furthermore, using similar methods, we show that $X_\Delta(N)_{{\overline{\mathbb F}_p}}$ is not a smooth plane quintic, for any $N$ and any $p$.