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

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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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Correspondence to Evan Chen.

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Chen, E., Park, P.S. & Swaminathan, A.A. Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication. Res. number theory 1, 28 (2015). https://doi.org/10.1007/s40993-015-0028-0

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Keywords

  • Prime Ideal
  • Elliptic Curve
  • Complex Multiplication
  • Elliptic Curf
  • Class Number