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

For an elliptic curve $E/\mathbb{Q}$ and a prime $p$ of good reduction, Hasse's theorem (1933) gives $|a_p|\le 2\sqrt p$ for the trace of Frobenius $a_p = p+1-\#E(\mathbb{F}_p)$, so there is a unique angle $\theta_p\in[0,\pi]$ with $a_p = 2\sqrt p\,\cos\theta_p$ — the eigenvalues of Frobenius on $H^1$ are $\sqrt p\,e^{\pm i\theta_p}$. The hypothesis 'non-CM' matters: for curves with complex multiplication the distribution is provably different (a mixture of a uniform part and a discrete part), so the conjecture is stated precisely for non-CM $E$.

Around 1960 Mikio Sato (numerically) and John Tate (theoretically, via algebraic cycles) conjectured that the $\theta_p$ are equidistributed for the measure $\frac{2}{\pi}\sin^2\theta\,d\theta$ on $[0,\pi]$. This is the pushforward of Haar measure on $\mathrm{SU}(2)$ to conjugacy classes — the 'random matrix' signature indicating that the Frobenius conjugacy classes in the $\ell$-adic representation $\rho_{E,\ell}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2$ behave like a maximally random family in a compact unitary group.

The statement is statistical: it concerns the distribution of infinitely many individual numbers $a_p$, not any single one. The entire difficulty is to convert this counting statement into an analytic one about functions — a conversion executed by Serre's criterion in the next step.