Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)
Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)
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结果的拓扑和算术，II：结果 $=1$ 超曲面（附录由 C. Cazanave 编写）
DOI：
10.14231/ag-2017-019
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Wolfson
中科院分区：
文献类型：
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作者：
B. Farb;J. Wolfson
我们考虑一元n次多项式对的模空间$\数学{R}n$，其结果等于$1$。我们将这些代数簇的拓扑与它们的几何和算术联系起来。特别地，我们计算了它们的上同调，Frobenius的相关本征值，以及它们的点集的基数。当$q$和$n$互质时，我们证明了$数学{R}{n/\bar{mathbb{F}}_q}$的上同调是纯的，当且仅当$q\等价v1$mod$n$，我们还推导出了$SU(2)$n$的带电单极子的有限域对应的这些不变量的值，以及相应的强中心单极子的模空间$X_n$.
Cazanave的附录给出了点数的另一种基本计算方法。
We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$ polynomials whose resultant equals $1$. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their \'{e}tale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ are coprime, we show that the \'etale cohomology of $\mathcal{R}_{n/\bar{\mathbb{F}}_q}$ is pure, and of Tate type if and only if $q\equiv 1$ mod $n$. We also deduce the values of these invariants for the finite field counterparts of the moduli spaces $\mathcal{M}_n$ of $SU(2)$ monopoles of charge $n$ in $\mathbb{R}^3$, and the associated moduli space $X_n$ of strongly centered monopoles.
An appendix by Cazanave gives an alternative and elementary computation of the point counts.
DOI：
10.1515/9781400859306
发表时间：
1988
期刊：
--
影响因子：
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作者：
M. Atiyah;N. Hitchin
通讯作者：
M. Atiyah;N. Hitchin