Serre curves relative to obstructions modulo 2

Abstract

We consider elliptic curves $E/\mathbb{Q}$ for which the image of the adelic Galois representation $\ddot{I}{}_E$ 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 $\ddot{I}{}_E$, and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.