Mazur's isogeny theorem

Mazur’s isogeny theorem states that if $p$ is a prime for which there exists an elliptic curve $E\mathrm{/}\mathbb{Q}$ 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.