Rational points on X0(N)∗ when N is non-squarefree

Publikation: Arbeitspapier/Preprint › Preprint

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  • Sachi Hashimoto
  • Timo Keller
  • Samuel Le Fourn

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OriginalspracheEnglisch
PublikationsstatusElektronisch veröffentlicht (E-Pub) - 1 Mai 2025

Abstract

Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a conjecturally complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before.

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Rational points on X0(N)∗ when N is non-squarefree. / Hashimoto, Sachi; Keller, Timo; Fourn, Samuel Le.
2025.

Publikation: Arbeitspapier/Preprint › Preprint

Hashimoto, S., Keller, T., & Fourn, S. L. (2025). Rational points on X0(N)∗ when N is non-squarefree. Vorabveröffentlichung online. https://doi.org/10.48550/arXiv.2505.00680
Hashimoto S, Keller T, Fourn SL. Rational points on X0(N)∗ when N is non-squarefree. 2025 Mai 1. Epub 2025 Mai 1. doi: 10.48550/arXiv.2505.00680
Hashimoto, Sachi ; Keller, Timo ; Fourn, Samuel Le. / Rational points on X0(N)∗ when N is non-squarefree. 2025.
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N2 - Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a conjecturally complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before.

AB - Let $N$ be a non-squarefree integer such that the quotient $X_0(N)^*$ of the modular curve $X_0(N)$ by the full group of Atkin-Lehner involutions has positive genus. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, we provide a conjecturally complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$. In the process we identify exceptional rational points on $X_0(147)^*$ and $X_0(75)^*$ which were not known before.

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