HEXAGONbeta

Gonality and quintic points on intermediate modular curves

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Marin Varivoda ↗

Version 1 / Oct 13, 2026 / CC BY 4.0

Abstract

For {±1}⊆Δ⊆(Z/NZ)×\{\pm 1\}\subseteq\Delta\subseteq(\mathbb{Z}/N\mathbb{Z})^\times let XΔ(N)X_\Delta(N) be the intermediate modular curve between X1(N)X_1(N) and X0(N)X_0(N). We determine the Q\mathbb{Q}-gonality of every strict intermediate curve XΔ(N)X_\Delta(N) with N≤50N\le 50, and of X1(N)X_1(N) for 41≤N≤5041\le N\le 50, where it equals the upper bound of Derickx and van Hoeij at every level. We also settle the thirty levels left open by Derickx, Hwang, Jeon and Orli\'c, completing the list of curves X0(N)X_0(N) with infinitely many points of degree 55, and we show that exactly 4343 strict intermediate curves have infinitely many points of degree 55. As a consequence, the strict intermediate curves of density degree 55 are exactly the four Q\mathbb{Q}-pentagonal curves found by Orli\'c. The proofs use published classifications, the Castelnuovo--Severi inequality, and three computational methods; all computations can be rerun from the accompanying code.

Provenance statement

The paper has two parts. The first was motivated by the 30 levels N for which Derickx, Hwang, Jeon and Orlić left open whether or not X0(N) has infinitely many points of degree 5. We settle all of them (Theorem A). Furthermore, we classify all intermediate modular curves with infinitely many points of degree 5 (Theorem B, Corollary E). In the cases that aren't resolved by theoretical criteria, we combine searches over finite fields with Mordell-Weil and lifting arguments. The second part concerns gonality. It was inspired by the author's earlier work with Filip Najman (arXiv:2602.03513) and relies on the CurveArith package by Derickx and Terao. For intermediate modular curves (Theorem C), CurveArith is used with no modification. For most X1(N) with 41 <= N <= 50 (Theorem D), the search phase of the CurveArith algorithm was replaced with an automatic cover search that generalizes the covering argument of Derickx and van Hoeij (appearing in "Gonality of X1(N)" and also in the author's earlier work), and this makes the computations feasible. LLMs (GPT-6 Astra via Codex, Claude via Claude Code) were used throughout: to explore, to develop the automated cover choice, to write most of the code, and to draft the text. Every computational claim relies on rerunnable code published in github.com/marin-varivoda/intermediate-modular-curves.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
CodexVersion GPT-6 Astra
Magma
MagmaVersion V2.29-4
SageMath
SageMathVersion 10.7

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Version history

  1. v1Submitted by Marin VarivodaInitial depositCurrentOct 13, 2026