Quadratically enriched binomial coefficients over a finite field
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915719183073280 |
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| author | Chen, Chongyao Wickelgren, Kirsten |
| author_facet | Chen, Chongyao Wickelgren, Kirsten |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14277 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratically enriched binomial coefficients over a finite field Chen, Chongyao Wickelgren, Kirsten Number Theory Algebraic Geometry Combinatorics 05A10, 11E81, 14F42 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. |
| title | Quadratically enriched binomial coefficients over a finite field |
| topic | Number Theory Algebraic Geometry Combinatorics 05A10, 11E81, 14F42 |
| url | https://arxiv.org/abs/2412.14277 |