Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ehrhart polynomials, Hecke series, and affine buildings

Claudia Alfes, Joshua Maglione, Christopher Voll

Abstract

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.

Ehrhart polynomials, Hecke series, and affine buildings

Abstract

Given a lattice polytope and a prime , we define a function from the set of primitive symplectic -adic lattices to the rationals that extracts the th coefficient of the Ehrhart polynomial of relative to the given lattice. Inspired by work of Gunnells and Rodriguez-Villegas in type , 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 , their eigenvalues are independent of and expressed as polynomials in . We define local zeta functions that enumerate the values of these Hecke eigenfunctions on the vertices of the affine Bruhat--Tits buildings associated with -adic symplectic groups. We compute these zeta functions by enumerating -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.
Paper Structure (9 sections, 10 theorems, 54 equations, 1 figure, 1 table)

This paper contains 9 sections, 10 theorems, 54 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Zeta functions of Ehrhart coefficients
  3. Symplectic Hecke series
  4. The Hermite--Smith generating function
  5. Main results
  6. The type--A story
  7. Examples
  8. Hecke eigenfunctions
  9. Local Ehrhart--Hecke zeta functions

Key Result

Theorem 1.1

Let $n\in\mathbb{N}$, $\bm{a} = (1, 2, \dots, n)\in \mathbb{N}^n$, $\bm{d} = (n, n-1, \dots, 1)\in \mathbb{N}^n$, and let $\langle, \rangle$ be the usual dot product. Then

Figures (1)

  • Figure 3.1: Polytopes and some values of $\mathscr{E}_{2,2,1,P}$ displayed on lattices in the affine building of type $\widetilde{\mathsf{A}}_1$ associated with the group $\mathop{\mathrm{GSp}}\nolimits_2(\mathbb{Q}_p) \cong \mathop{\mathrm{GL}}\nolimits_2(\mathbb{Q}_p)$. The center vertex corresponds to the homothety class of the identity, and the values are the linear coefficients of the Ehrhart polynomials with respect to the corresponding lattices.

Theorems & Definitions (13)

  • Theorem 1.1: Andrianov
  • Example 1.2
  • Theorem A
  • Corollary B
  • Theorem C
  • Corollary D
  • Theorem E
  • Theorem F
  • Corollary 2.1
  • proof
  • ...and 3 more