[Paper Review] Gonality of the modular curve $X_0(N)$

Filip Najman, Petar Orlić|arXiv (Cornell University)|Jul 24, 2022

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper determines the $ℚ$-gonality and $ℂ$-gonality of modular curves $X_0(N)$ for all $N < 145$, and extends results to larger $N$, identifying all $X_0(N)$ of gonality 4, 5, and 6 over $ℚ$. It provides the first known examples of pentagonal curves over $ℂ$, specifically for $N = 97$ and $169$, and fully classifies modular curves of low gonality over $ℚ$, resolving long-standing gaps in the classification of modular curves with few rational points.

ABSTRACT

In this paper we determine the $\mathbb Q$-gonalities of the modular curves $X_0(N)$ for all $N&lt;145$. We determine the $\mathbb C$-gonality of many of these curves and the $\mathbb Q$-gonalities and $\mathbb C$-gonalities for many larger values of $N$. Using these results and some further work, we determine all the modular curves $X_0(N)$ of gonality $4$, $5$ and $6$ over $\mathbb Q$. We also find the first known instances of pentagonal curves $X_0(N)$ over $\mathbb C$.

Motivation & Objective

• To determine the $ℚ$-gonality and $ℂ$-gonality of $X_0(N)$ for all $N < 145$, extending to larger $N$ where possible.
• To classify all modular curves $X_0(N)$ that are tetragonal, pentagonal, or hexagonal over $ℚ$, resolving a key arithmetic question about degree $d$ points on modular curves.
• To identify the first known examples of modular curves $X_0(N)$ that are pentagonal over $ℂ$, filling a notable gap in the literature.
• To provide precise gonality data for $X_0(N)$ to support the LMFDB database and future arithmetic studies of modular curves.
• To develop and apply computational techniques tailored to $X_0(N)$, overcoming challenges such as limited modular units and lack of natural plane models.

Proposed method

• The authors compute $ℚ$-gonality using a combination of group action analysis, modular units, and rational maps to $ℑ^1$, leveraging the abundance of involutions in $X_0(N)$ for highly composite $N$.
• They derive upper bounds for gonality by constructing explicit rational maps via modular parametrizations and Hecke operators, and lower bounds using genus and Clifford’s theorem.
• For $ℂ$-gonality, they apply results from algebraic geometry, including the Castelnuovo–Severi bound and known classifications of trigonal and tetragonal curves.
• They use computational algebra systems to analyze function fields and verify the existence of degree-$d$ maps, particularly for $d = 4,5,6$, by testing the existence of rational functions of bounded degree.
• They compare their results with prior work on $X_1(N)$ and $X_0(N)$, adapting methods from Derickx and van Hoeij but modifying them to account for the scarcity of cusps and modular units in $X_0(N)$.
• They validate results via consistency checks with known classifications of hyperelliptic, trigonal, and tetragonal $X_0(N)$, and use bounds from Abramovich to refine estimates.

Experimental results

Research questions

• RQ1 Which modular curves $X_0(N)$ are tetragonal over $ℚ$, and what is the complete list of such $N$?
• RQ2 Are there any known examples of modular curves $X_0(N)$ that are pentagonal over $ℂ$, and if so, for which $N$?
• RQ3 What is the complete classification of $X_0(N)$ of gonality 5 and 6 over $ℚ$?
• RQ4 How do the $ℚ$-gonality and $ℂ$-gonality of $X_0(N)$ relate, and in which cases do they differ?
• RQ5 Can effective computational techniques be developed to determine the gonality of $X_0(N)$ despite the lack of natural plane models and limited modular units?

Key findings

• The paper fully classifies $X_0(N)$ of gonality 4 over $ℚ$, showing that $X_0(N)$ is tetragonal over $ℚ$ if and only if $N$ is in the set $\{38,42,44,51,52,53,55,56,57,58,60,61,62,63,65,66,67,68,69,70,72,73,74,75,77,78,79,80,83,85,87,88,89,91,92,94,95,96,98,100,101,103,104,107,111,119,121,125,131,142,143,167,191\}$.
• The paper identifies $X_0(109)$ as the only $X_0(N)$ that is pentagonal over $ℚ$, resolving a long-standing open question about the existence of such curves.
• It presents the first known examples of $X_0(N)$ that are pentagonal over $ℂ$, specifically for $N = 97$ and $N = 169$, marking a significant advancement in the classification of modular curves by gonality.
• For gonality 6 over $ℚ$, the paper provides a complete list of $N$ including $76,82,84,86,90,93,97,99,108,112,113,115-118,122-124,127-129,135,137,139,141,144,146,147,149,151,155,159,162,164,169,179,181,215,227,239$.
• The authors compute and document $ℚ$-gonality and $ℂ$-gonality for all $X_0(N)$ with $N < 145$, and provide bounds for many larger $N$, significantly expanding the known data in the LMFDB database.
• The study confirms that $ℚ$-gonality is strictly less than or equal to $ℂ$-gonality for all $X_0(N)$, and identifies cases where the two differ, such as for $N = 169$, where $ℚ$-gonality is 6 and $ℂ$-gonality is 5.