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

Abstract

Let $E/\mathbb {Q}$ be an elliptic curve with complex multiplication (CM), and for each prime p of good reduction, let $a_{E}(p) = p + 1 - \#E(\mathbb {F}_{p})$ denote the trace of Frobenius. By the Hasse bound, $a_{E}(p) = 2\sqrt {p} \cos \theta _{p}$ for a unique θ _p ∈ [0,π]. In this paper, we prove that the least prime p such that θ _p ∈ [α,β]⊂ [0,π] satisfies

$$ p \ll \left(\frac{N_{E}}{\beta - \alpha}\right)^{A}, $$

where N _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 ^L for an absolute constant L>0.