论文标题
计算Hecke运算符的一般线性组算术亚组
Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups
论文作者
论文摘要
我们提出了一种算法来计算Hecke运算符的算术算法的共同体，用于一般线性组$ \ mathrm {gl} _n $的算术亚组$γ$。这包括$ \ mathrm {gl} _n $上的数字字段或有限维划分代数。作为系数，我们可以使用任何有限的局部系数系统。与较早的方法不同，该算法适用于所有学位$ i $的同种学$ h^i $。它从全面的缩回$ \ tilde {w} $，一个$γ$ -Invariant细胞复合体，该复合物计算同时组。它使用称为“气质”的真实参数，将$ \ tilde {w} $扩展到一个较高的一个较高实际维度的新的脾气暴躁的复杂$ \ tilde {w}^+$。该算法已被编码为$ \ mathrm {sl} _n（\ mathbb {z}）$，用于$ n = 2,3,4 $;我们为$ \ Mathrm {Sl} _3（\ Mathbb {Z}）$的一致性子组提供了一些结果。
We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $Γ$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional division algebra. As coefficients, we may use any finite-dimensional local coefficient system. Unlike earlier methods, the algorithm works for the cohomology $H^i$ in all degrees $i$. It starts from the well-rounded retract $\tilde{W}$, a $Γ$-invariant cell complex which computes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex $\tilde{W}^+$ of one higher real dimension, using a real parameter called the temperament. The algorithm has been coded up for $\mathrm{SL}_n(\mathbb{Z})$ for $n=2,3,4$; we present some results for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$.
