Quantum Pick's Theorem in two dimensions

Abstract

Consider an $n$-dimensional sum of the form $S_k=\sum q^{f(\mathbf{z})}$ over lattice points $\mathbf{z}$ which lie in $K\mathcal{P}$, the $K$ scaling of an $n$-dimensional rational polytope $\mathcal{P}$. We formulate a conjecture on the structure of $S_K$ considered for $K$ prime, as a power series in $h=q-1$ modulo the $K^{th}$ cyclotomic polynomial, and in particular the existence of a decomposition in powers of the Gauss sum $\mathbb{G}$, up to the $n^{th}$, whose coefficients are almost independent of $K$, where the leading coefficient is related to the volume of $\mathcal{P}$ and the coefficient of ${\mathbb{G}}^r$ has contributions from the codimension $r$ skeleton of $\mathcal{P}$. The conjecture is proved for $n=1,2$.

arXiv comment35 pages