A trace formula for the distribution of rational G-orbits in ramified covers, adapted to representation stability

2017/03/06 by Nir Gadish, Gadish, Nir
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1703.01710

openalex publication_date 2017/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety \widetildeX→ X over a finite field F_q furnishes the rational points x∈ X(F_q) with additional arithmetic structure: the Frobenius action on the fiber over x. For example, in the case of the Vieta cover of polynomials over F_q this structure describes a polynomial's irreducible decomposition type. Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of \widetildeX via the Grothendieck-Lefschetz trace formula. This note presents a version of the trace formula that is suited for studying the distribution in the context of representation stability: for certain sequences of varieties (\widetildeX_n) the cohomology, and therefore the distribution of the Frobenius actions, stabilizes in a precise sense. We conclude by fully working out the example of the Vieta cover of the variety of polynomials. The calculation includes the distribution of cycle decompositions on cosets of Young subgroups of the symmetric group, which might be of independent interest.

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