Joint effective equidistribution of partial lattices in positive characteristic
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911292869050368 |
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| author | Horesh, Tal Paulin, Frédéric |
| author_facet | Horesh, Tal Paulin, Frédéric |
| contents | Let $ν$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_ν$ and completion $K_ν$, and let $1 \leq \mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_ν$-lattices $Λ$ of rank $\mathfrak{m}$ in $K_ν^{\;\textbf{d}}$ as their covolume tends to infinity: of their $K_ν$-linear span $V_Λ$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_ν^{\;\textbf{d}}$; of their shape in the modular quotient by $\operatorname{PGL}_\mathfrak{m}(R_ν)$ of the Bruhat-Tits buildings of $\operatorname{PGL}_\mathfrak{m}(K_ν)$; and of the shape of $Λ^\perp$ in the similar quotient for $\operatorname{PGL}_{\textbf{d}-\mathfrak{m}}(K_ν)$, where $Λ^\perp$ is the orthogonal partial $R_ν$-lattice of rank $\textbf{d}-\mathfrak{m}$ in the dual space of $K_ν^{\;\textbf{d}}$. The main tools are a new refined $\text{LU}$ decomposition by blocks of elements of $\operatorname{SL}_\textbf{d}(K_ν)$, techniques of Gorodnik and Nevo for counting integral points in well-rounded families of subsets of algebraic groups, and computations of volumes of various homogeneous spaces associated with partial $R_ν$-lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Joint effective equidistribution of partial lattices in positive characteristic Horesh, Tal Paulin, Frédéric Number Theory 11N45, 37A44, 20G30, 22F30, 14G17, 11H99, 28C10, 11G35 Let $ν$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_ν$ and completion $K_ν$, and let $1 \leq \mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_ν$-lattices $Λ$ of rank $\mathfrak{m}$ in $K_ν^{\;\textbf{d}}$ as their covolume tends to infinity: of their $K_ν$-linear span $V_Λ$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_ν^{\;\textbf{d}}$; of their shape in the modular quotient by $\operatorname{PGL}_\mathfrak{m}(R_ν)$ of the Bruhat-Tits buildings of $\operatorname{PGL}_\mathfrak{m}(K_ν)$; and of the shape of $Λ^\perp$ in the similar quotient for $\operatorname{PGL}_{\textbf{d}-\mathfrak{m}}(K_ν)$, where $Λ^\perp$ is the orthogonal partial $R_ν$-lattice of rank $\textbf{d}-\mathfrak{m}$ in the dual space of $K_ν^{\;\textbf{d}}$. The main tools are a new refined $\text{LU}$ decomposition by blocks of elements of $\operatorname{SL}_\textbf{d}(K_ν)$, techniques of Gorodnik and Nevo for counting integral points in well-rounded families of subsets of algebraic groups, and computations of volumes of various homogeneous spaces associated with partial $R_ν$-lattices. |
| title | Joint effective equidistribution of partial lattices in positive characteristic |
| topic | Number Theory 11N45, 37A44, 20G30, 22F30, 14G17, 11H99, 28C10, 11G35 |
| url | https://arxiv.org/abs/2404.04368 |