Congruences of elliptic curves arising from nonsurjective mod 𝑁 Galois representations

Author:

Frengley Sam

Abstract

We study N N -congruences between quadratic twists of elliptic curves. If N 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 GL 2 ( Z / N Z ) \operatorname {GL}_2(\mathbb {Z}/N\mathbb {Z}) . By computing explicit models for these double covers we find all pairs ( N , r ) (N, r) such that there exist infinitely many j j -invariants of elliptic curves E / Q E/\mathbb {Q} which are N N -congruent with power r r to a quadratic twist of E E . We also find an example of a 48 48 -congruence over Q \mathbb {Q} . We make a conjecture classifying nontrivial ( N , r ) (N,r) -congruences between quadratic twists of elliptic curves over Q \mathbb {Q} .

Finally, we give a more detailed analysis of the level 15 15 case. We use elliptic Chabauty to determine the rational points on a modular curve of genus 2 2 whose Jacobian has rank 2 2 and which arises as a double cover of the modular curve X ( n s 3 + , n s 5 + ) X(\mathrm {ns\,} 3^+, \mathrm {ns\,} 5^+) . As a consequence we obtain a new proof of the class number 1 1 problem.

Funder

Cambridge Trust

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference44 articles.

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Modular curves with many points over finite fields;Journal of Algebra;2023-12

2. On 12-congruences of elliptic curves;International Journal of Number Theory;2023-11-23

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