Composite images of Galois for elliptic curves over $\mathbf{Q}$ & Entanglement fields

TL;DR

This paper constructs models for modular curves that classify elliptic curves over Q with specific non-surjective Galois representations at multiple primes and determines their rational points, with applications to entanglement fields.

Contribution

It develops models for composite level modular curves and determines rational points, advancing understanding of Galois image entanglements for elliptic curves over Q.

Findings

Models for composite level modular curves constructed

Rational points on these curves determined

Applications to entanglement fields provided

Abstract

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \text{GL}_2(\mathbf{F}_{\ell})$ that describes the Galois action on the $\ell$-torsion points of $E$. Building on recent work of Rouse--Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over $\mathbf{Q}$ with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields.