Research in Number Theory

, 1:28

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

  • Evan Chen
  • Peter S. Park
  • Ashvin A. Swaminathan
Open Access


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 θp∈ [0,π]. In this paper, we prove that the least prime p such that θp∈ [α,β]⊂ [0,π] satisfies
$$ p \ll \left(\frac{N_{E}}{\beta - \alpha}\right)^{A}, $$
where NE 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 pqL for an absolute constant L>0.


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© The Author(s) 2015

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Evan Chen
    • 1
  • Peter S. Park
    • 2
  • Ashvin A. Swaminathan
    • 3
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUnited States of America
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUnited States of America
  3. 3.Department of MathematicsHarvard CollegeCambridgeUnited States of America

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