Linnik’s Theorem for Sato-tate Laws on Elliptic Curves with Complex Multiplication

Let E be an elliptic curve over Q, and for each prime p, let #E(Fp) be the number of rational points of E over the finite field Fp. Taking aE(p) = p+ 1−#E(Fp) to be the trace of Frobenius as usual, we recall the following important result of Hasse, which holds when E has good reduction at p: |aE(p)| ≤ 2 √ p. It follows that for each prime p, there is a unique angle θp ∈ [0, π] (which we call the “SatoTate” angle) such that ap = 2 √ p cos θp. For a fixed elliptic curve E, it is natural to study the distribution of the angles θp as p ranges across the primes at which E has good reduction. The now-proven Sato-Tate Conjecture provides an asymptotic for this distribution that depends on whether or not E has complex multiplication (CM). While the CM case was established by Hecke, the non-CM case was recently proven in [1] by Barnet-Lamb, Geraghty, Harris, and Taylor. Theorem (Sato-Tate Conjecture). Fix an elliptic curve E/Q, and let I = [α, β] ⊂ [0, π] be a subinterval. Then we have that