Open Access

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

Research in Number Theory20151:28

DOI: 10.1007/s40993-015-0028-0

Received: 30 June 2015

Accepted: 16 October 2015

Published: 29 December 2015

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 pa (mod q) for (a,q)=1 satisfies pq L for an absolute constant L>0.