Research in Number Theory

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

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

Keywords

Prime Ideal Elliptic Curve Complex Multiplication Elliptic Curf Class Number 

Notes

Acknowledgements

This research was supervised by Ken Ono at the Emory University Mathematics REU and was supported by the National Science Foundation (grant number DMS-1250467). We would like to thank Ken Ono and Jesse Thorner for offering their advice and guidance and for providing many helpful discussions and valuable suggestions on the paper. We would also like to thank Professor Jean-Pierre Serre for pointing us to the reference [9]. Finally, we would like to thank the referees for their helpful comments.

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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 (http://creativecommons.org/licenses/by/4.0/), 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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