Mazur's isogeny theorem

TL;DR

Mazur's isogeny theorem classifies primes for which elliptic curves over rationals admit rational isogenies, using modular curves and Galois representations, and is fundamental in elliptic curve theory and Fermat's Last Theorem.

Contribution

This paper provides an overview of Mazur's original proof, highlighting the role of modular curves and Galois representations in establishing the theorem.

Findings

Prime degrees of rational isogenies are limited to specific values.

The theorem is essential for understanding elliptic curves over rationals.

It underpins key results in number theory, including Fermat's Last Theorem.

Abstract

Mazur's isogeny theorem states that if $p$ is a prime for which there exists an elliptic curve $E / \mathbb{Q}$ that admits a rational isogeny of degree $p$, then $p \in \{2,3,5,7,11,13,17,19,37,43,67,163 \}$. This result is one of the cornerstones of the theory of elliptic curves and plays a crucial role in the proof of Fermat's Last Theorem. In this expository paper, we overview Mazur's proof of this theorem, in which modular curves and Galois representations feature prominently.