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

By the Weyl equidistribution criterion, a sequence of angles $\theta_p$ equidistributes with respect to a measure $\mu$ on $[0,\pi]$ if and only if, for every $n \ge 1$, the average of the Chebyshev-type functions $U_n(\cos\theta_p)$ (essentially $\sum_{j} e^{i(n-2j)\theta_p}$) tends to $0$; these averages are exactly the coefficients controlling the symmetric power $L$-function $L(\mathrm{Sym}^n E, s) = \prod_p \prod_{j=0}^{n} \left(1 - e^{i(n-2j)\theta_p} p^{-s}\right)^{-1}$.

Serre showed in 1968 that the angles $\theta_p$ equidistribute according to the Sato–Tate measure once every symmetric power $L$-function extends to a holomorphic, non-vanishing function on $\mathrm{Re}(s) \ge 1$ for every $n \ge 1$; this reduced an equidistribution question in analytic number theory to a question in the Langlands program about the analytic behavior of an infinite family of $L$-functions.

What remained was to prove those analytic properties, and the route taken was to show each $L(\mathrm{Sym}^n E, s)$ is automorphic — a much stronger structural statement, the subject of the next steps.