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

Step 1 of 8: State the conjecture: how do the angles

\theta_p

distribute?

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In plain words

For an elliptic curve $E$ over the rational numbers, reducing modulo a prime $p$ and counting points gives a number $\#E(\mathbb{F}_p)$ close to $p+1$ ; the error term $a_p = p + 1 - \#E(\mathbb{F}_p)$ is, by a 1930s theorem of Hasse, always small enough that $a_p / (2\sqrt{p})$ can be written as $\cos\theta_p$ for some angle $\theta_p$ .

Sato and Tate independently guessed around 1960 that, as $p$ ranges over primes, these angles do not spread out evenly like a uniform dartboard, but cluster according to a specific $\sin^2\theta$ -shaped density — the same distribution that shows up for eigenvalues of random rotation matrices.

a_p = p + 1 - \#E(\mathbb{F}_p) = 2\sqrt{p}\,\cos\theta_p, \qquad \theta_p \sim \frac{2}{\pi}\sin^2\theta\, d\theta \text{ on } [0,\pi]

Detailed analysis

For a non-CM elliptic curve $E/\mathbb{Q}$ (one without extra symmetries called complex multiplication), Hasse's theorem (1933) bounds the error term $a_p = p+1-\#E(\mathbb{F}_p)$ in counting points modulo a prime $p$ by $|a_p| \le 2\sqrt{p}$ , so one may write $a_p = 2\sqrt{p}\,\cos\theta_p$ for a unique $\theta_p \in [0,\pi]$ .

Sato and Tate conjectured around 1960 that, as $p$ varies, the angles $\theta_p$ equidistribute in $[0,\pi]$ with respect to the measure $\frac{2}{\pi}\sin^2\theta\,d\theta$ — the Sato–Tate measure — rather than the uniform measure one might naively expect; this measure is exactly the distribution of eigenangles of a random matrix in the compact group $\mathrm{SU}(2)$ , hinting at a hidden symmetry group behind the error terms (Mazur 2008, Bulletin AMS).

Proving this required turning a statement about the statistical spread of infinitely many individual numbers $a_p$ into a statement about a family of analytic functions, which is the subject of the next step.

Terms in this step

non-CM elliptic curve: An elliptic curve $E$ whose only endomorphisms are multiplication by integers, i.e. without complex multiplication (CM); most elliptic curves are of this type, and the Sato–Tate conjecture is specifically about them.
equidistribution: A sequence of numbers equidistributes with respect to a measure $\mu$ if, for every interval, the fraction of the sequence landing in it converges to $\mu$ of that interval, so the sequence 'fills up' the space according to $\mu$ rather than randomly.

Knowledge used in this step

Arithmetic geometry