Research

Open Access

Research

Open Access

- Evan Chen
^{1}Email author, - Peter S. Park
^{2}and - Ashvin A. Swaminathan
^{3}

Research in Number Theory2015**1**:28

**DOI: **10.1007/s40993-015-0028-0

© The Author(s) 2015

**Received: **30 June 2015

**Accepted: **16 October 2015

**Published: **29 December 2015

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 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.

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