Sato–Tate distribution for Frobenius angles of elliptic curves without complex multiplication
Determine whether, for an elliptic curve over the integers given by y^2 = P(x) with P a degree‑3 polynomial having integer coefficients and simple roots, and for each prime p not dividing the discriminant of P, the angles θ_p defined by a_p = 2√p cos θ_p (with |a_p| ≤ 2√p, where the number of solutions to y^2 ≡ P(x) (mod p) is approximately p − a_p) are distributed according to the probability density (2/π) sin^2 θ on [0, π], under the assumption that the elliptic curve has no complex multiplication.
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Mikio Sato shared the Wolf prize with John Tate in 2002/03. 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).