A geometric approach to counting norms in cyclic extensions of function fields


In this paper, we prove an explicit version of a function field analogue of a classical result of Odoni (Mathematika 22(1):71–80, 1975) about norms in number fields, in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result Bary-Soroker et al. (Finite Fields Appl 39:195–215, 2016) which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky (Mathematika 63(2):622–665, 2017), and reproved by the author in his Ph.D. thesis using the method of this paper. The main tool is a twisted Grothendieck–Lefschetz trace formula, inspired by the paper (Church et al. in Contemp Math 620:1–54, 2014). Using a combinatorial description of the cohomology, we obtain a precise quantitative result which works in the \(q^n\rightarrow \infty \) regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.

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


  1. 1.

    Bank, E., Bary-Soroker, L., Fehm, A.: Sums of two squares in short intervals in polynomial rings over finite fields. Am. J. Math. 140(4), 1113–1131 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bary-Soroker, L., Smilansky, Y., Wolf, A.: On the function field analogue of Landau’s theorem on sums of squares. Finite Fields Appl. 39, 195–215 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bary-Soroker, L., Gorodetsky, O., Karidi, T.-L., Sawin, W.: Chebotarev density theorem in short intervals for extensions of \({{\mathbb{F}}_{q}(T)}\). Trans. Am. Math. Soc. 373, 597–628 (2020)

    Article  Google Scholar 

  4. 4.

    Erdös, P., Mahler, K.: On the number of integers which can be represented by a binary form. J. Lond. Math. Soc. 1, 134–139 (1938)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Casto, K.: FI\(_G\)-modules and arithmetic statistics. Online preprint arXiv:1703.07295

  6. 6.

    Church, T., Ellenberg, J.S., Farb, B.: Representation stability in cohomology and asymptotics for families of varieties over finite fields. Contemp. Math. 620, 1–54 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245, 250–314 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    de Brujin, N.G.: On Mahler’s partition problem. Indag. Math. X, 210–220 (1948)

    Google Scholar 

  9. 9.

    Douglass, J.M.: On the cohomology of an arrangement of type \(B_l\). J. Algebra 146, 265–282 (1992)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gadish, N.: A trace formula for the distribution of rational G-orbits in ramified covers adapted to representation stability. N. Y. J. Math. 23, 987–1011 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Geck, G.P.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras. London Mathematical Society Monographs New Series, vol. 21. The Clarendon Press, New York (2000)

    Google Scholar 

  12. 12.

    Gorodetsky, O.: A polynomial analogue of Landau’s theorem and related problems. Mathematika 63(2), 622–665 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Henderson, A.: Bases for certain cohomology representations of the symmetric group. J. Algebraic Comb. 24(9), 361–390 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kim, M.: Weights in cohomology groups arising from hyperplane arrangements. Proc. Am. Math. Soc. 120(3), 697–703 (1994)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Krenn, D., Ralaivaosaona, D., Wagner, S.: On the number of multi-base representations of an integer. In: 25th International Conference on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA14), DMTCS-HAL Proceedings, vol. BA (2014)

  16. 16.

    Lehrer, I.: The l-adic cohomology of hyperplane complements. Bull. Lond. Math. Soc. 24(1), 76–82 (1992)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Odoni, R.W.K.: On the norms of algebraic integers. Mathematika 22(1), 71–80 (1975)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Jimenez Rolland, R., Wilson, J.C.H.: Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields. Q. J. Math. 70(2), 565–602 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Stanley, R.: Enumerative Combinatorics. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  20. 20.

    Wilson, J.: \(FI_W\)-modules and stability criteria for representations of classical Weyl groups. J. Algebra 420, 269–332 (2014)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zhou: Restricted partition problem into parts with a given set of prime factors. MathOverflow. https://mathoverflow.net/q/368676

Download references

Author information



Corresponding author

Correspondence to Vlad Matei.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Matei, V. A geometric approach to counting norms in cyclic extensions of function fields. Res Math Sci 7, 31 (2020). https://doi.org/10.1007/s40687-020-00229-0

Download citation