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Towards a classification of isolated $j$-invariants

Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla

TL;DR

Running this algorithm on all elliptic curves presently in the L-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that E gives rise to an isolated point on X_1(N) if and only if j(E)=-140625/8, -9317, $351/4, or $-162677523113838677.

Abstract

We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$.

Towards a classification of isolated $j$-invariants

TL;DR

Running this algorithm on all elliptic curves presently in the L-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that E gives rise to an isolated point on X_1(N) if and only if j(E)=-140625/8, -9317, -162677523113838677.

Abstract

We develop an algorithm to test whether a non-CM elliptic curve gives rise to an isolated point of any degree on any modular curve of the form . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to . Running this algorithm on all elliptic curves presently in the -functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that gives rise to an isolated point on if and only if , or .
Paper Structure (27 sections, 28 theorems, 30 equations, 1 figure, 1 table, 4 algorithms)

This paper contains 27 sections, 28 theorems, 30 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Key Components of Algorithm.
  3. Connection to Open Uniformity Problems
  4. Outline
  5. Code
  6. Background
  7. Isolated Points on Curves
  8. Modular Curves
  9. Closed Points on Modular Curves
  10. Maps Between Modular Curves
  11. Galois Representations
  12. Overview of the Main Algorithm
  13. Computing the level of the m-adic representation
  14. Primitive Points on Modular Curves
  15. Construction of P(E) and minimality
  16. ...and 12 more sections

Key Result

Theorem 2

Let $x=[E,P]\in X_1(N)$ be a non-CM isolated point with $j(E) \in \mathbf{Q}$. Fix an equation for $E/\mathbf{Q}$ and let $N_E$ denote its conductor. Suppose that one of the following holds: Then $j(E) \in \{-140625/8,-9317,351/4, -162677523113838677\}$. Moreover, each one of these $j$-invariants corresponds to a $\mathbf P^1$-isolated point on $X_1(21)$, $X_1(37)$, $X_1(28)$, or $X_1(37)$, respe

Figures (1)

  • Figure 1: Degrees of Residue Fields

Theorems & Definitions (66)

  • Theorem 2
  • Remark 3
  • Conjecture 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Theorem 10: Bourdon, Ejder, Liu, Odumodu, Viray, BELOV
  • Remark 11
  • ...and 56 more