Skip to main content
SpringerLink
Log in

A note on the least prime that splits completely in a nonabelian Galois number field

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We prove a nontrivial estimate for the size of the least rational prime that splits completely in the ring of integers of certain families of nonabelian Galois number fields. Our result complements results of Linnik and Vinogradov and of Pollack who studied this problem in the quadratic and abelian number field settings, respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We say an integer n is power-full or square-full if \(p\mid n\) implies that \(p^2\mid n\) for any prime p.

  2. To deduce this bound, we apply [17, Theorem 1] to the L-function \(L(s) = \zeta _K(s)/\zeta (s)\) which satisfies conditions (1.5a)–(1.5e) and (1.6a)–(1.6c) in that paper if \(K/{\mathbb {Q}}\) is a Galois extension.

References

  1. Blomer, V., Harcos, G., Michel, P.: Bounds for modular \(L\)-functions in the level aspect. Ann. Sci. Écol. Norm. Sup. (4) 40(5), 697–740 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brauer, R.: On the zeta-function of algebraic number fields. Am. J. Math. 69, 243–250 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burgess, D.A.: On character sums and \(L\)-series. II. Proc. Lond. Math. Soc. (3) 13, 524–536 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  4. Conrey, J.B., Iwaniec, H.: The cubic moment of central values of automorphic \(L\)-functions. Ann. Math. (2) 151, 1175–1216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duke, W., Friedlander, J.B., Iwaniec, H.: The subconvexity problem for Artin \(L\)-functions. Invent. Math. 149(3), 489–577 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Einsiedler, M., Lindenstrauss, E., Michel, P., Venkatesh, A.: Distribution of the periodic torus orbits and Duke’s theorem for cubic fields. Ann. Math. (2) 173(2), 815–885 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Heath-Brown, D.R.: Hybrid bounds for Dirichlet \(L\)-functions. Invent. Math. 47(2), 149–170 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Heath-Brown, D.R.: Hybrid bounds for Dirichlet \(L\)-functions. II. Q. J. Math. Oxford Ser. (2) 31(122), 157–167 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, X.: The smallest prime that does not split completely in a number field. Algebra Numb. Theory 6(6), 1061–1096 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Louboutin, S.: Determination of all quaternion octic CM-fields with class number 2. J. Lond. Math. Soc. (2) 54, 227–238 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Michel, P., Venkatesh, A.: The subconvexity problem for \(\rm GL_2\). Publ. Math. Inst. Hautes Études Sci. 111, 171–271 (2010)

    Article  MATH  Google Scholar 

  12. Murty, V.K.: The least prime which does not split completely. Forum Math. 6(5), 555–565 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Murty, V.K., Patankar, V.M.: Tate cycles on Abelian varieties with complex multiplication. Can. J. Math. 67(1), 198–213 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pintz, J.: Elementary methods in the theory of \(L\)-functions. VI. On the least prime quadratic residue (mod \(p\)). Acta Arith. 32(2), 173–178 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pollack, P.: The smallest prime that splits completely in an abelian number field. Proc. Am. Math. Soc. 142(6), 1925–1934 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pollack, P.: Prime splitting in abelian number fields and linear combinations of Dirichlet characters. Int. J. Numb. Theory 10(4), 885–903 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Soundararajan, K.: Weak subconvexity for central values of \(L\)-functions. Ann. Math. (2) 172(2), 1469–1498 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Stark, H.M.: Some effective cases of the Brauer-Siegel theorem. Invent. Math. 23, 135–152 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Thorner, J., Zaman, A.: An explicit bound for the least prime ideal in the Chebotarev density theorem. Algebra Numb. Theory 11(5), 1135–1197 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vaaler, J.D., Voloch, J.F.: The least nonsplit prime in Galois extensions of \({\mathbb{Q}}\). J. Numb. Theory 85(2), 320–335 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Venkatesh, A.: Sparse equidistribution problems, period bounds and subconvexity. Ann. Math. (2) 172, 989–1094 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vinogradov, A.I., Linnik, JuV: Hypoelliptic curves and the least prime quadratic residue. Dokl. Akad. Nauk SSSR. 168, 258–261 (1966)

    MathSciNet  Google Scholar 

  23. Zaman, A.: The least unramified prime which does not split completely. Forum. Math. 30(3), 651–661 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This project began as a result of an SEC Faculty Travel Grant that allowed the second author to visit the University of Georgia. We thank the Southeastern Conference for its support. We also thank Caroline Turnage-Butterbaugh, Jesse Thorner, and the anonymous referee for a number of useful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Mississippi, University, MS, 38677, USA

    Zhenchao Ge & Micah B. Milinovich

  2. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Paul Pollack

Authors
  1. Zhenchao Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Micah B. Milinovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul Pollack
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Micah B. Milinovich.

Additional information

Research of the second author was partially supported by the NSA Young Investigator Grants H98230-15-1-0231 and H98230-16-1-0311. Research of the third author was partly supported by NSF award DMS-1402268.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ge, Z., Milinovich, M.B. & Pollack, P. A note on the least prime that splits completely in a nonabelian Galois number field. Math. Z. 292, 183–192 (2019). https://doi.org/10.1007/s00209-018-2162-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-018-2162-6

Keywords

Mathematics Subject Classification

Search

Navigation