Serre's Uniformity Conjecture for Elliptic Curves with Rational Cyclic Isogenies

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation $\bar{\rho}_{E,p}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ is surjective.