2020/09/16 by Gilyoung Cheong, Yifeng Huang, Cheong, Gilyoung +1 · 1 citation
Mathematics · #14F45 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2009.07976
openalex publication_date 2020/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an elliptic curve E defined over ℂ, let E× 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 Confn(E×) of n points on E× 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 \mathbbFq-point counts of Confn(E×), which can be obtained from the zeta function of E over a finite field \mathbbFq. We show that the mixed Hodge structure of the i-th singular cohomology group Hi(Confn(E×)) 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 Hi(Confn(E×)).