Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations

TL;DR

This paper investigates N-congruences between quadratic twists of elliptic curves, classifies infinite families of such congruences over Q, and applies elliptic Chabauty to analyze specific modular curves, providing new insights into elliptic curve relationships.

Contribution

It provides a classification of N-congruences arising from non-surjective Galois representations, including explicit models and a conjecture for all such cases over Q.

Findings

Classified pairs (N, r) with infinitely many N-congruent quadratic twists over Q.

Constructed explicit models for double covers of modular curves related to these congruences.

Applied elliptic Chabauty to determine rational points on a genus 2 modular curve, offering a new proof of the class number 1 problem.

Abstract

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs $(N, r)$ such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an example of a $48$-congruence over $\mathbb{Q}$. We make a conjecture classifying nontrivial $(N,r)$-congruences between quadratic twists of elliptic curves over $\mathbb{Q}$. Finally, we give a more detailed analysis of the level $15$ case. We use elliptic Chabauty to determine the rational points on a modular curve of genus $2$ whose Jacobian has rank $2$ and which arises as a double cover of the modular curve $X(\mathrm{ns} 3^+, \mathrm{ns} 5^+)$. As a consequence we obtain a new proof of the class number $1$ problem.