A geometric approach to counting norms in cyclic extensions of function fields
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RESEARCH
A geometric approach to counting norms in cyclic extensions of function fields Vlad Matei * Correspondence:
[email protected] Department of Mathematics, University of California Irvine, 340 Rowland Hall, Irvine, CA 92697, USA
Abstract 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 qn → ∞ regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.
1 Introduction Odoni obtained in [17] an asymptotic result for integers that can be expressed as norms of elements of a number field K . As a reminder for the reader for any Galois extension K /k of fields, the norm of an element x ∈ K is the product of its Galois conjugates. In [3], a function field analogue of this problem was proved; namely, the authors obtain the main term of the asymptotic for any Galois extension of function fields. The goal of this √ paper is reprove this result for norms in the case of the cyclic extension Fq [ d T ]/Fq [T ] and obtain an explicit polynomial in q that gives this count. Throughout the paper, we assume that our finite field Fq contains all 2dth roots of unity or equivalently q ≡ 1 (mod 2d) and ξ will denote a primitive root of unity of order d. We denote with Mn,q the set of monic polynomials of degree n in Fq [T ]. The Galois √ extension Fq [ d T ]/Fq [T ] is cyclic with Galois group μd , and norms can be very easily described. They are polynomials f ∈ Fq [T ] such that there exists a polynomial g ∈ Fq [T ] with f (T d ) = g(T )g(T ξ ) . . . g(T ξ d−1 ). Definition 1.1 Let bq : Mn,q → {0, 1} denote the indicator function for norms, i.e., √ bq (f ) = 1 if and only if f is a norm in the extension Fq [ d T ]/Fq [T ].
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Our first result is about counting squarefree monic polynomials of degree n that are norms. Theorem 1.2 Consider the counting function Sq (n) = f ∈Mn,q bq (f ). For every n ≥ n
2 and q ≡ 1 (mod 2d), we can write Sq (n) = bk,n q k=0 2k 1/d + (n − j − k − 1) (a) bk,n = δk,j,n ; j=k n−j−k (b) We have that δk,j,n = δk,j,n+1 for n ≥ 2k and k (1.1)k |δk,j,n | ≤ C j−k
f squarefree n−k
such that
for some absolute constant C. Remark 1.3 For an explicit computa
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