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Quadratically enriched binomial coefficients over a finite field

Chongyao Chen, Kirsten Wickelgren

TL;DR

The paper defines quadratically enriched binomial coefficients over a finite field via trace forms in the Grothendieck–Witt group and proves a precise closed formula for the untwisted case: ${L/k race j} = {n race j} - (1-u)inom{(n-2)/2}{(j-1)/2}$ in $ ext{GW}( frac{}{})$, together with a twisted case: ${L[Q]/k race j} = \tfrac{1}{2}{2j race j}(1+u)$. The proofs translate the problem into necklace-orbit enumerations, leveraging Möbius inversion, Lucas/Kummer parity results, and a $C_2$-flipping symmetry, with a separate treatment for odd/even $2$-adic valuations of $n$ and $j$. A twisted analogue is obtained by a swapping action and a partition-theoretic reinterpretation, yielding an explicit expression involving a half-weighted binomial term. These results feed into arithmetic refinements of curve-counting formulas over non-algebraically closed fields in the framework of $ ext{A}^1$-homotopy theory, enabling more refined enumerative invariants over finite 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 étale extension of degree $n$. We also compute a quadratic twist. These (twisted) enriched binomial coefficients are defined in joint work of Brugallé 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.

Quadratically enriched binomial coefficients over a finite field

TL;DR

The paper defines quadratically enriched binomial coefficients over a finite field via trace forms in the Grothendieck–Witt group and proves a precise closed formula for the untwisted case: in , together with a twisted case: . The proofs translate the problem into necklace-orbit enumerations, leveraging Möbius inversion, Lucas/Kummer parity results, and a -flipping symmetry, with a separate treatment for odd/even -adic valuations of and . A twisted analogue is obtained by a swapping action and a partition-theoretic reinterpretation, yielding an explicit expression involving a half-weighted binomial term. These results feed into arithmetic refinements of curve-counting formulas over non-algebraically closed fields in the framework of -homotopy theory, enabling more refined enumerative invariants over finite 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 ring homomorphisms into an algebraic closure from an étale extension of degree . We also compute a quadratic twist. These (twisted) enriched binomial coefficients are defined in joint work of Brugallé and the second-named author, building on work of Serre. Such binomial coefficients support curve counting results over non-algebraically closed fields, using -homotopy theory.

Paper Structure

This paper contains 23 sections, 35 theorems, 105 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Summary of the proof
  3. Untwisted case
  4. Twisted case
  5. Acknowledgements
  6. Background
  7. Lucas and Kummer's theorems
  8. Quadratically enriched binomial coefficients over a field
  9. Untwisted case - Proof of Theorem \ref{['thm:closedformula-1']}
  10. Necklace interpretation
  11. Möbius inversion
  12. Proof of Theorem \ref{['thm:closedformula-1']} when $\nu_2(n)\geq1$ and $\nu_2(j)=0$
  13. Combinatorics of symmetric orbits
  14. Combinatorics of type 1 symmetric orbits
  15. Proof of Theorem \ref{['thm:closedformula-1']} when $\nu_2(n)=1$ and $\nu_2(j)\geq1$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $q$ be an odd prime power and let $j$ be a non-negative integer. Let $L/k$ be the finite extension of $\mathbb F_q$ of degree $n$. Then where $u$ is the non-square class in $\mathrm{GW}(k)$ and our convention is that $\binom{a}{b}:=0$, if either $a$ or $b$ is not in $\mathbb Z$.

Figures (11)

  • Figure 1: An element in $\mathrm{Orb}(C_6,\mathrm{Neck}(6,4))^f$ with a unique symmetry axis.
  • Figure 2: An element in $\mathrm{Orb}(C_6,\mathrm{Neck}(6,4))^f$ with two symmetry axes. The dashed line is a symmetry axis of type 1 and the solid line is of type 2.
  • Figure 3: An element in $\mathrm{Orb}(C_6,\mathrm{Neck}(6,4))^f$ decomposes to a symmetric product in $\mathrm{Orb}(C_3,\mathrm{Neck}(3,2)^f$ under $\phi$.
  • Figure 4: An element in $\mathrm{Orb}(C_5,\mathrm{Neck}(5,2))^f$.
  • Figure 5: The three distinct elements in $\mathrm{Orb}(C_{10},\mathrm{Neck}(10,4))^f$ that can be generated by the same element in $\mathrm{Orb}(C_5,\mathrm{Neck}(5,2))^f$.
  • ...and 6 more figures

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Theorem 2.1
  • Theorem 2.2: Kummer's theorem Kummertheorem
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Definition 3.1
  • Remark 3.2
  • ...and 61 more