Worked solution: Potential automorphy of symmetric powers proves the Sato–Tate conjecture (2011)

An automorphic representation $\pi$ of $\mathrm{GL}(n+1)$ over $\mathbb{Q}$ is, roughly, a higher-rank generalization of a classical modular form, with its own $L$-function $L(\pi, s)$ built from an Euler product. Deep general theorems — the Godement–Jacquet meromorphic continuation, and non-vanishing on $\mathrm{Re}(s) = 1$ via Rankin–Selberg integrals and results of Jacquet, Shalika, and Shahidi — already establish exactly the holomorphic continuation and non-vanishing on $\mathrm{Re}(s) = 1$ that Serre's criterion (Step 2) demands, for any automorphic $\pi$.

So the entire remaining difficulty of the Sato–Tate conjecture is concentrated into a single, purely representation-theoretic question: is the symmetric power $L$-function $L(\mathrm{Sym}^n E, s)$ actually the $L$-function of an automorphic representation of $\mathrm{GL}(n+1)$, for every $n \ge 1$?

This question — automorphy of $\mathrm{Sym}^n E$ — is exactly what Clozel, Harris, Shepherd-Barron, and Taylor, and later Barnet-Lamb, Geraghty, Harris, and Taylor, set out to answer using the machinery built for Wiles's proof of Fermat's Last Theorem.