Quadratically enriched binomial coefficients over a finite field

TL;DR

This paper introduces a quadratic enrichment of binomial coefficients over finite fields, providing new combinatorial tools for counting algebraic structures and supporting curve counting over non-closed fields using advanced homotopy theory.

Contribution

It develops a novel quadratic enrichment of binomial coefficients over finite fields, extending classical combinatorics to algebraic and geometric contexts.

Findings

Computed an analogue of Pascal's triangle enriched in bilinear forms

Derived counts of ring homomorphisms into algebraic closures

Supported curve counting over non-algebraically closed fields

Abstract

We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ ring homomorphisms into an algebraic closure from an \'etale extension of degree $n$. We also compute a quadratic twist. These (twisted) enriched binomial coefficients are defined in joint work of Brugall\'e and the second-named author, building on work of Serre. Such binomial coefficients support curve counting results over non-algebraically closed fields, using $\mathbb{A}^1$-homotopy theory.