论文标题
Kloosterman集和$ \ Mathrm {SL} _3 $ -Kloosterman sums的参数化
Parametrization of Kloosterman sets and $\mathrm{SL}_3$-Kloosterman sums
论文作者
论文摘要
我们使用Bott-Samelson分解的启发，使用Weyl Group元素的减少单词分解来对$ \ mathrm {SL} _3 $大细胞Kloosterman设置进行分类。因此，$ \ mathrm {sl} _3 $ long word kloosterman和分解为更细的部分，我们将其作为两个经典kloosterman总和的乘积的有限总和。优秀的Kloosterman总和最终是在Bruggeman-Kuznetsov trace公式中要考虑的正确片段。另一个应用程序是一个新的显式公式，它根据双dirichlet系列的指数和概括了Ramanujan的公式，表达了三重除法总和函数。
We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $Γ_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.
