Quadratically enriched binomial coefficients over a finite field

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ring homomorphisms into an algebraic closure from an étale extension of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -homotopy theory.