Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication

Abstract

Let E/ℚ be an elliptic curve with complex multiplication (CM), and for each prime p of good reduction, let a[subscript E](p) = p + 1 − #E(𝔽[subscript p]) denote the trace of Frobenius. By the Hasse bound, a[subscript E] (p) = 2 √pcosθ[subscript p] for a unique θ[subscript p] ∈ [0,π]. In this paper, we prove that the least prime p such that θ[subscript p]∈ [α,β]⊂ [0,π] satisfies p ≪ (N[subscript E]/β − α)[superscript A], where N[subscript E] is the conductor of E and the implied constant and exponent A>2 are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik’s Theorem for arithmetic progressions, which states that the least prime p≡a (mod q) for (a,q)=1 satisfies p≪q[superscript L] for an absolute constant L>0.