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arXiv · 2412.14277

Quadratically enriched binomial coefficients over a finite field

Chongyao Chen·Kirsten Wickelgren
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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.

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Chongyao Chen, Kirsten Wickelgren. 2024-12-18. Quadratically enriched binomial coefficients over a finite field. https://arxiv.org/abs/2412.14277

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