Ehrhart polynomials, Hecke series, and affine buildings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914689761411072 |
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| author | Alfes, Claudia Maglione, Joshua Voll, Christopher |
| author_facet | Alfes, Claudia Maglione, Joshua Voll, Christopher |
| contents | Given a lattice polytope $P$ and a prime $p$, we define a function from the set of primitive symplectic $p$-adic lattices to the rationals that extracts the $\ell$th coefficient of the Ehrhart polynomial of $P$ relative to the given lattice. Inspired by work of Gunnells and Rodriguez-Villegas in type $\mathsf{A}$, we show that these functions are eigenfunctions of a suitably defined action of the spherical symplectic Hecke algebra. Although they depend significantly on the polytope $P$, their eigenvalues are independent of $P$ and expressed as polynomials in $p$. We define local zeta functions that enumerate the values of these Hecke eigenfunctions on the vertices of the affine Bruhat--Tits buildings associated with $p$-adic symplectic groups. We compute these zeta functions by enumerating $p$-adic lattices by their elementary divisors and, simultaneously, one Hermite parameter. We report on a general functional equation satisfied by these local zeta functions, confirming a conjecture of Vankov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ehrhart polynomials, Hecke series, and affine buildings Alfes, Claudia Maglione, Joshua Voll, Christopher Combinatorics Group Theory Number Theory Given a lattice polytope $P$ and a prime $p$, we define a function from the set of primitive symplectic $p$-adic lattices to the rationals that extracts the $\ell$th coefficient of the Ehrhart polynomial of $P$ relative to the given lattice. Inspired by work of Gunnells and Rodriguez-Villegas in type $\mathsf{A}$, we show that these functions are eigenfunctions of a suitably defined action of the spherical symplectic Hecke algebra. Although they depend significantly on the polytope $P$, their eigenvalues are independent of $P$ and expressed as polynomials in $p$. We define local zeta functions that enumerate the values of these Hecke eigenfunctions on the vertices of the affine Bruhat--Tits buildings associated with $p$-adic symplectic groups. We compute these zeta functions by enumerating $p$-adic lattices by their elementary divisors and, simultaneously, one Hermite parameter. We report on a general functional equation satisfied by these local zeta functions, confirming a conjecture of Vankov. |
| title | Ehrhart polynomials, Hecke series, and affine buildings |
| topic | Combinatorics Group Theory Number Theory |
| url | https://arxiv.org/abs/2402.15412 |