Sato–Tate conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. For each prime $p$ of good reduction, define $a_p = p + 1 - \#E(\mathbb{F}_p)$ and let $\theta_p \in [0, \pi]$ be the unique angle satisfying $\cos \theta_p = \frac{a_p}{2\sqrt{p}}$. As $p$ varies over all primes of good reduction, the angles $\theta_p$ are equidistributed on $[0, \pi]$ with respect to the Sato–Tate semicircle measure $\frac{2}{\pi} \sin^2 \theta \, d\theta$.

Following Jean-Pierre Serre's reduction of equidistribution to the non-vanishing and analytic continuation of the symmetric power $L$-functions $L(\mathrm{Sym}^n E, s)$ on the line $\mathrm{Re}(s) = 1$, Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor (2008) proved the conjecture for elliptic curves over totally real fields with non-integral $j$-invariant. In 2011, Thomas Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor removed the remaining technical hypotheses by proving the potential automorphy of $\mathrm{Sym}^n E$ for all $n \ge 1$ using a family of Calabi–Yau varieties and higher-dimensional modularity lifting theorems.