Modular curves $X_0(N)$ of density degree $5$

Maarten Derickx; Wontae Hwang; Daeyeol Jeon; Petar Orlić

Modular curves $X_0(N)$ of density degree $5$

Number Theory 2026-02-20 v2

Authors: Maarten Derickx , Wontae Hwang , Daeyeol Jeon , Petar Orlić

Abstract

We determine all modular curves $X_0(N)$ with density degree $5$ , i.e. all curves $X_0(N)$ with infinitely many points of degree $5$ and only finitely many points of degree $d\leq4$ . As a consequence, the problem of determining all curves $X_0(N)$ with infinitely many points of degree $5$ remains open for only $30$ levels $N$ .

Keywords

algebraic curves

Cite

@article{arxiv.2503.08975,
  title  = {Modular curves $X_0(N)$ of density degree $5$},
  author = {Maarten Derickx and Wontae Hwang and Daeyeol Jeon and Petar Orlić},
  journal= {arXiv preprint arXiv:2503.08975},
  year   = {2026}
}

Comments

25 pages. Revised version

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