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Sato–Tate conjecture for Frobenius angles of elliptic curves

Establish that, for an elliptic curve defined by y^2 = P(x) where P is a degree-3 polynomial with integer coefficients and simple roots, and for primes p not dividing the discriminant of P, the angles θ_p ∈ [0, π] defined by a_p = 2√p cos θ_p are distributed according to the Sato–Tate measure with density (2/π) sin^2 θ on [0, π], in the case without complex multiplication.

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Background

The article highlights a famous conjecture shared by Mikio Sato and John Tate concerning the distribution of Frobenius angles associated with elliptic curves. For a cubic polynomial P with integer coefficients and simple roots, Hasse’s theorem gives the bounds |a_p| ≤ 2√p for the deviation a_p in the count of solutions to y2 = P(x) mod p, allowing the normalization a_p = 2√p cos θ_p.

The Sato–Tate conjecture asserts that, in the absence of complex multiplication, these normalized angles θ_p are equidistributed on [0, π] with respect to the measure having density (2/π) sin2 θ. The article situates this conjecture historically, noting Tate’s motivic perspective and Sato’s numerical investigations.

References

They also share a famous conjecture in number theory concerning the repartition of Frobenius angles. Let $P$ be a degree 3 polynomial with integer coefficients and simple roots. Hasse has shown that for any prime $p$ which does not divide the discriminant of $P$, the number of solutions of the congruence $y2=P(x)\pmod p$ is like $p-a_p$, with $\mathopen| a_p \mathclose |\leq 2\sqrt p$. When writing $a_p= 2\sqrt p \cos\theta_p$ with $0\leq\theta_p\leq\pi$, the Sato--Tate conjecture predicts that these angles $\theta_p$ follow the law $(2/\pi)\sin2\theta$ (in absence of complex multiplication).

— Mikio Sato, a visionary of mathematics (2402.15553 - Schapira, 23 Feb 2024) in Main text, paragraph beginning “Mikio Sato shared the Wolf prize with John Tate in 2002/03.”