Rational points in a family of conics over $\mathbb{F}_2(t)$

TL;DR

This paper studies rational points on a family of conics over the function field (t), revealing new behaviors and deriving an asymptotic formula through harmonic analysis and a specialized Tauberian theorem.

Contribution

It introduces a novel analysis of conics over (t), demonstrating new phenomena and applying harmonic analysis techniques with a Tauberian theorem in this context.

Findings

Asymptotic formula for rational points on the family of conics

Identification of new behaviors over (t)

Application of harmonic analysis and Tauberian theorem in function fields

Abstract

Serre famously showed that almost all plane conics over $\mathbb{Q}$ have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over $\mathbb{F}_2(t)$ which illustrates new behaviour. We obtain an asymptotic formula using harmonic analysis, which requires a Tauberian theorem over function fields for Dirichlet series with branch point singularities.