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

## Abstract

*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

*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.

## 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.

## 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).MathSciNetCrossRefzbMATHGoogle Scholar
- 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).Google Scholar - 3.Fogels, E: On the zeros of
*L*-functions. Acta Arith. 11, 67–96 (1965).MathSciNetzbMATHGoogle Scholar - 4.Graham, S: On Linnik’s constant. Acta Arith. 39(2), 163–179 (1981).MathSciNetzbMATHGoogle Scholar
- 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/.Google Scholar
- 6.Iwaniec, H, Kowalski, E: Analytic number theory. In:
*American Mathematical Society Colloquium Publications. vol. 53*. American Mathematical Society, Providence, RI (2004).Google Scholar - 7.Jutila, M: A new estimate for Linnik’s constant. Ann. Acad. Sci. Fenn. Ser. A I No. 471, 8 (1970).MathSciNetGoogle Scholar
- 8.Kaufman, RM: The geometric aspect of Ju. V. Linnik’s theorem on the least prime. Litovsk. Mat. Sb. 17(1), 111–114 (1977).MathSciNetzbMATHGoogle Scholar
- 9.Koval’čik, FB: Density theorems for sectors and progressions. Litovsk. Mat. Sb. 15(4), 133–151 (1975).MathSciNetGoogle Scholar
- 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.Google Scholar
- 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).MathSciNetGoogle Scholar
- 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).MathSciNetGoogle Scholar
- 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).Google Scholar - 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).Google Scholar - 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.Google Scholar - 16.Weil, A: Jacobi sums as “Grössencharaktere”. Trans. Amer. Math. Soc. 73, 487–495 (1952).MathSciNetzbMATHGoogle Scholar

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