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

Combining Steps 1–7: for every non-CM elliptic curve $E/\mathbb{Q}$, the symmetric power $L$-functions $L(\mathrm{Sym}^n E, s)$ are holomorphic and non-vanishing on $\mathrm{Re}(s) \ge 1$ for every $n \ge 1$ (Steps 4–7), so by Serre's criterion (Step 2) the angles $\theta_p$ equidistribute on $[0,\pi]$ with respect to the measure $\frac{2}{\pi}\sin^2\theta\,d\theta$, establishing the Sato–Tate conjecture (Clozel–Harris–Shepherd-Barron–Taylor 2008; Barnet-Lamb–Geraghty–Harris–Taylor 2011).

The method — potential automorphy via auxiliary Calabi–Yau families combined with automorphy lifting theorems — proved far more broadly applicable than Sato–Tate alone: Barnet-Lamb, Gee, and Geraghty extended it to Hilbert modular forms over totally real fields, and the general potential automorphy theorems of Barnet-Lamb, Gee, Geraghty, and Taylor (2014) now underlie many results throughout the Langlands program.

Sato–Tate for higher genus curves and other generalizations, predicted by the random matrix model of Katz and Sarnak using larger compact groups such as $\mathrm{USp}(2n)$, remain open, marking the current frontier of this line of research.