Sato–Tate conjecture

Solved, 2011Arithmetic and number theory

Statement

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.

The Sato–Tate measure is the pushforward of Haar measure on $\mathrm{SU}(2)$ to its conjugacy classes. The generalized Sato–Tate conjecture predicts that for any smooth projective variety (such as an abelian surface of genus $g = 2$ ), the normalized Frobenius eigenvalues are equidistributed according to the Haar measure of a compact Lie subgroup of $\mathrm{USp}(2g)$ —the Sato–Tate group—determined by the algebraic cycles on powers of the variety.

References

Thomas Barnet-Lamb, David Geraghty, Michael Harris, Richard Taylor (2011). A family of Calabi–Yau varieties and potential automorphy II · DOI:10.2977/PRIMS/40
Laurent Clozel, Michael Harris, Richard Taylor (2008). Automorphy for some l-adic lifts of automorphic mod l Galois representations · DOI:10.1007/s10240-008-0016-1

History of the problem

1960Mikio Sato (via numerical computations) and John Tate (via algebraic cycles and representation theory) independently conjecture the $\sin^2 \theta$ law for Frobenius angles of non-CM elliptic curves.
1968Jean-Pierre Serre shows that the Sato–Tate equidistribution follows if all symmetric power $L$ -functions $L(\mathrm{Sym}^n E, s)$ extend holomorphically and do not vanish on $\mathrm{Re}(s) = 1$ .
2008Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor prove the Sato–Tate conjecture for elliptic curves over totally real fields with non-integral $j$ -invariant.
2011Thomas Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor complete the proof for all non-CM elliptic curves over totally real fields and generalize it to holomorphic modular forms.