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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.01242 |
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| _version_ | 1866909980962062336 |
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| author | Ellenberg, Jordan Shusterman, Mark |
| author_facet | Ellenberg, Jordan Shusterman, Mark |
| contents | For a finite group $G$ and a sufficiently large (but fixed) prime power $q$ coprime to $G$ we obtain asymptotics for the number of regular Galois extensions $L/ \mathbb{F}_q(t)$, with $\mathrm{Gal}(L/\mathbb{F}_q(t)) \cong G$, ramified at a single place of $\mathbb{F}_q(t)$, thus making progress on a positive characteristic analog of the Boston--Markin conjecture. We also obtain similar results for other arithmetic functions of the product of places of $\mathbb{F}_q(t)$ ramified in $L$, and for more general one-variable function fields over $\mathbb{F}_q$ in place of $\mathbb{F}_q(t)$.
Some of our proofs make crucial use of a series of recent breakthroughs by Landesman--Levy, as well as a new `vanishing of stable homology in a given direction' result for representations of braid groups arising from braided vector spaces. Other inputs include a study of (rings of coinvariants of) braided vector spaces associated to racks with $2$-cocycles, a connection between convolution of arithmetic functions and direct sums of braided vector spaces, and a Goursat lemma for racks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Averages of Arithmetic Functions over Conductors of Function Fields Ellenberg, Jordan Shusterman, Mark Number Theory Algebraic Geometry Algebraic Topology Quantum Algebra 20F36 11G20 57K12 14F20 For a finite group $G$ and a sufficiently large (but fixed) prime power $q$ coprime to $G$ we obtain asymptotics for the number of regular Galois extensions $L/ \mathbb{F}_q(t)$, with $\mathrm{Gal}(L/\mathbb{F}_q(t)) \cong G$, ramified at a single place of $\mathbb{F}_q(t)$, thus making progress on a positive characteristic analog of the Boston--Markin conjecture. We also obtain similar results for other arithmetic functions of the product of places of $\mathbb{F}_q(t)$ ramified in $L$, and for more general one-variable function fields over $\mathbb{F}_q$ in place of $\mathbb{F}_q(t)$. Some of our proofs make crucial use of a series of recent breakthroughs by Landesman--Levy, as well as a new `vanishing of stable homology in a given direction' result for representations of braid groups arising from braided vector spaces. Other inputs include a study of (rings of coinvariants of) braided vector spaces associated to racks with $2$-cocycles, a connection between convolution of arithmetic functions and direct sums of braided vector spaces, and a Goursat lemma for racks. |
| title | Averages of Arithmetic Functions over Conductors of Function Fields |
| topic | Number Theory Algebraic Geometry Algebraic Topology Quantum Algebra 20F36 11G20 57K12 14F20 |
| url | https://arxiv.org/abs/2601.01242 |