## 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

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.

## References

- 1
Barnet-Lamb, T, Geraghty, D, Harris, M, Taylor, R: A family of calabi-yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011).

- 2
Cojocaru, AC: Questions about the reductions modulo primes of an elliptic curve. In:

*Number theory, CRM Proc. Lecture Notes, vol. 36*, pp. 61–79. Amer. Math. Soc., Providence, RI (2004). - 3
Fogels, E: On the zeros of

*L*-functions. Acta Arith. 11, 67–96 (1965). - 4
Graham, S: On Linnik’s constant. Acta Arith. 39(2), 163–179 (1981).

- 5
Iwaniec, H: Topics in classical automorphic forms (graduate studies in mathematics, v. 17). American Mathematical Society, Providence, RI (1997). http://amazon.com/o/ASIN/0821807773/.

- 6
Iwaniec, H, Kowalski, E: Analytic number theory. In:

*American Mathematical Society Colloquium Publications. vol. 53*. American Mathematical Society, Providence, RI (2004). - 7
Jutila, M: A new estimate for Linnik’s constant. Ann. Acad. Sci. Fenn. Ser. A I No. 471, 8 (1970).

- 8
Kaufman, RM: The geometric aspect of Ju. V. Linnik’s theorem on the least prime. Litovsk. Mat. Sb. 17(1), 111–114 (1977).

- 9
Koval’čik, FB: Density theorems for sectors and progressions. Litovsk. Mat. Sb. 15(4), 133–151 (1975).

- 10
Lemke Oliver, R, Thorner, J: Effective log-free zero density estimates for automorphic L-functions and the Sato-Tate conjecture (2015). arXiv e-prints available at 1505.03122.

- 11
Linnik, YV: On the least prime in an arithmetic progression. I. The basic theorem. Rec. Math. [Mat. Sbornik] N.S. 15(57), 139–178 (1944).

- 12
Linnik, YV: On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S. 15(57), 347–368 (1944).

- 13
Montgomery, HL: Ten lectures on the interface between analytic number theory and harmonic analysis. In:

*CBMS Regional Conference Series in Mathematics. vol. 84*. Amer. Math. Soc, Providence, RI (1994). - 14
Ono, K: The web of modularity: arithmetic of the coefficients of modular forms and

*q*-series. In:*CBMS Regional Conference Series in Mathematics. vol. 102*. Amer. Math. Soc., Providence, RI (2004). - 15
Silverman, JH: Advanced topics in the arithmetic of elliptic curves. In:

*Graduate Texts in Mathematics. vol. 151*. Springer-Verlag, New York (1994). doi:10.1007/978-1-4612-0851-8. - 16
Weil, A: Jacobi sums as “Grössencharaktere”. Trans. Amer. Math. Soc. 73, 487–495 (1952).

## 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.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

**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.

## About this article

### Cite this article

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

Received:

Accepted:

Published:

### Keywords

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