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

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 \(q^n\rightarrow \infty \) regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.

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Correspondence to Vlad Matei.

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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

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