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Improved bounds for integral points on modular curves using Runge's method

David Zywina

Abstract

Consider a modular curve $X_G$ defined over a number field $K$, where $G$ is a subgroup of $GL_2(\mathbb{Z}/N\mathbb{Z})$ with $N>2$. The curve $X_G$ comes with a morphism $j: X_G\to \mathbb{P}^1_K=\mathbb{A}^1_K \cup\{\infty\}$ to the $j$-line. For a finite set of places $S$ of $K$ that satisfies a certain condition, Runge's method shows that there are only finitely many points $P \in X_G(K)$ for which $j(P)$ lies in the ring $\mathfrak{O}_{K,S}$ of $S$-units of $K$. We prove an explicit version which shows that if $j(P)\in \mathfrak{O}_{K,S}$ for some $P\in X_G(K)$, then the absolute logarithmic height of $j(P)$ is bounded above by $N^{12} \log N$. Explicits upper bounds have already been obtained by Bilu and Parent though they are not polynomial in $N$. The modular functions needed to apply Runge's method are constructing using Eisenstein series of weight $1$.

Improved bounds for integral points on modular curves using Runge's method

Abstract

Consider a modular curve defined over a number field , where is a subgroup of with . The curve comes with a morphism to the -line. For a finite set of places of that satisfies a certain condition, Runge's method shows that there are only finitely many points for which lies in the ring of -units of . We prove an explicit version which shows that if for some , then the absolute logarithmic height of is bounded above by . Explicits upper bounds have already been obtained by Bilu and Parent though they are not polynomial in . The modular functions needed to apply Runge's method are constructing using Eisenstein series of weight .
Paper Structure (27 sections, 36 theorems, 115 equations)

This paper contains 27 sections, 36 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Notation
  4. Background: modular curves
  5. Modular functions
  6. Modular curves
  7. Cusps
  8. The modular curve $X(N)$
  9. Analytic expansions at the cusps
  10. The modular curve $X(N)$
  11. The sets $\Omega_{c,v}$ for a general modular curve
  12. Properties of $\varphi$
  13. Proof of Theorem \ref{['T:main']}
  14. Background: modular forms
  15. Notation
  16. ...and 12 more sections

Key Result

Theorem 1.1

Fix an integer $N>2$ and a subgroup $G$ of $\operatorname{GL}_2(\mathbb Z/N\mathbb Z)$ containing $-I$. Let $\mu$ be the degree of the morphism $j\colon X_G\to \mathbb P^1_{K_G}$. Assume that $(K,S)$ satisfies Runge's condition for $X_G$, where $K\supseteq K_G$ is a number field and $S$ is a finite

Theorems & Definitions (68)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 58 more