论文标题
从其Zeta函数的伸出的椭圆曲线的betti和hodge数字
Betti and Hodge numbers of configuration spaces of a punctured elliptic curve from its zeta functions
论文作者
论文摘要
给定椭圆曲线$ e $在$ \ mathbb {c} $上定义的，令$ e^{\ times} $是通过删除点获得的$ e $的打开子集。在本文中，我们表明，$ i $ -th betti的无序配置空间的$ \ mathrm {conf}^{n}（e^{\ times}）$ $ e^{\ times} $上的$ n $点的$ n $点显示为在两个变量中的明显合理函数的系数。我们还将其Hodge数量计算为四个变量中另一个显式合理函数的系数。我们的结果很有趣，因为这些有理函数类似于$ \ mathbb {f} _ {q} $ - $ \ mathrm {conf}^{n}（e^{\ times}）$的点计数，可以从$ e $ $ e $的Zeta函数中获得$ e $ $ \ mathbb的Zeta函数。我们表明，$ i $ - th单数的共同体学组$ h^{i}（\ mathrm {conf}^{n}（e^{\ times}）$具有复杂系数的纯度是纯度$ w（i^{\ times}）$，是重量$ w（i）$，我们在本文中提供了明确的integer。这种纯度语句意味着我们关于贝蒂数字和霍奇号码的主要结果。我们的证明使用Totaro的光谱序列计算，该计算描述了$ h^{i}（\ Mathrm {conf}^{n}（e^{\ times}））$ h^{i}上混合霍奇结构的重量过滤。
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 $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$.
