Investigation reveals new behavior of rational points in conics, implying novel properties in global function fields.
Serre famously showed that almost all plane conics over have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over which illustrates new behavior. We obtain an asymptotic formula using harmonic analysis, which requires a Tauberian theorem over function fields for Dirichlet series with branch point singularities.
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Loughran et al. (2026) studied this question.
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