Enumerating D₄ Quartics and a Galois Group Bias Over Function Fields

2020/09/19 by Daniel Keliher, Keliher, Daniel · 1 citation
Computer Science · Mathematics · #11R11 #11R16 #11R45 #11R58 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2009.09274

openalex publication_date 2020/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an asymptotic formula for the number of D₄ quartic extensions of a function field with discriminant equal to some bound, essentially reproducing the analogous result over number fields due Cohen, Diaz y Diaz, and Olivier, but with a stronger error term. We also study the relative density of D₄ and S₄ quartic extensions of a function field and show that with mild conditions, the number of D₄ quartic extensions can far exceed the number of S₄ quartic extensions

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• Non-vanishing of Artin L-functions associated with D₄-quartic function fields ordered by conductor

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