Betti and Hodge numbers of configuration spaces of a punctured elliptic curve from its zeta functions

TL;DR

This paper computes Betti and Hodge numbers of configuration spaces on a punctured elliptic curve using zeta functions, revealing explicit rational functions and purity of mixed Hodge structures.

Contribution

It provides explicit formulas for Betti and Hodge numbers of configuration spaces on a punctured elliptic curve, linking them to zeta functions and demonstrating purity of the mixed Hodge structure.

Findings

Betti numbers as coefficients of rational functions

Hodge numbers as coefficients of rational functions

Mixed Hodge structures are pure of explicit weights

Abstract

Given an elliptic curve $E$ defined over $\mathbb{C}$, let $E^{\times}$ be an open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th Betti number of the unordered configuration space $\mathrm{Conf}^{n}(E^{\times})$ of $n$ points on $E^{\times}$ appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the $\mathbb{F}_{q}$-point counts of $\mathrm{Conf}^{n}(E^{\times})$, which can be obtained from the zeta function of $E$ over a finite field $\mathbb{F}_{q}$. We show that the mixed Hodge structure of the $i$-th singular cohomology group $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$ with complex coefficients is pure of weight $w(i)$, an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro's spectral sequence computation that describes the weight filtration of the mixed Hodge structure on $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$.