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

## TL;DR

This paper proves Serre's uniformity conjecture for elliptic curves over rationals with rational cyclic isogenies, showing that for primes greater than 37, the associated Galois representations are surjective.

## Contribution

It establishes the surjectivity of mod p Galois representations for a broad class of elliptic curves with rational cyclic isogenies, confirming a case of Serre's conjecture.

## Key findings

- Surjectivity of residual mod p Galois representations for p > 37.
- Validation of Serre's uniformity conjecture in this context.
- Elliptic curves with rational cyclic isogenies exhibit maximal Galois image for large primes.

## 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.