Serre curves relative to obstructions modulo 2

TL;DR

This paper characterizes elliptic curves over rationals with maximal adelic Galois image constrained by modulo 2 conditions, providing explicit examples, detailed images, and applications to cyclicity, thus extending foundational Serre curve results.

Contribution

It offers a new characterization of Serre curves with specific modulo 2 constraints, computes explicit examples, and applies findings to the cyclicity problem.

Findings

Characterization of elliptic curves with large Galois image under modulo 2 constraints

Explicit examples of such curves with conductor ≤ 500,000

Detailed description of the Galois image for these curves

Abstract

We consider elliptic curves $E / \mathbb{Q}$ for which the image of the adelic Galois representation $\rho_E$ is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their $\ell$-adic images, compute all examples of conductor at most 500,000, precisely describe the image of $\rho_E$, and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.