Lei Fu, Daqing Wan
We calculate the trivial factors of the L-functions for both finite and p-adic infinite symmetric products of the Kloosterman sheaf.
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Deligne constructed a remarkable local system on $\bP^1-\{0,\infty\}$ attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct l-adic Kloosterman sheaves for…
$L$-functions of Kloosterman sheaves
本研究将Kloosterman层对称幂相关motive的动机$L$-函数的亚纯延拓与函数方程结果从$n=1$推广到部分$n>1$情形及相关二维motive，并证明了联系Kloosterman层迹与模形式傅里叶系数的若干Evans型猜想。
A certain "condition (S)" on reductive algebraic groups was introduced in [GT2], in which a slightly stronger condition (S+) was shown to have very strong consequences. We show that a wide class of Kloosterman and hypergeometric sheaves satisfy (S+). For this class of sheaves, we determine possible structure of their monodromy groups.
Euler characteristics of the generalized Kloosterman sheaves for symplectic and orthogonal groups
本文研究了 Yun 构造的广义 Kloosterman 层相关局部系统的单值性，计算了其在辛群与正交群标准表示下的欧拉示性数，为 Reeder-Yu 关于 epipelagic 表示 Langlands 参数预测的 Swan 导子猜想提供了证据。
Kloosterman sheaves and Bessel functions for generic principal series of finite groups
针对有限域上拟分裂约化群的一般主系列，运用几何朗兰兹纲领技术，建立贝塞尔函数特殊值与克洛施曼层弗罗贝尼乌斯作用迹的关联，并给出含例外群在内的显式例子。
We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\ge 2$ on $\mathbb{G}_m/\mathbb{F}_q$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz's results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_q)$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maxim…
Bilinear forms with trace functions over arbitrary sets, and applications to Sato-Tate
本文证明了丰饶层迹函数在任意子集支撑下的一般双线性形式非平凡上界，结合加性组合与Kloosterman层乘积和结果突破了Pólya-Vinogradov障碍，并推导出稀疏族中Kloosterman和与椭圆曲线Frobenius迹的Sato-Tate分布。
Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums
本文通过双线性形式和广义克卢斯特曼和，证明了关于指数和的非平凡界限，并应用到三阶$L$-函数的第一矩分析中。
Kloosterman sums for a finite field arise as Frobenius trace functions of certain local systems defined over $\Gm$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic Galois representation that comes from geometry. We also give bounds on the ramification of these Galois representations. All this is done in the g…
Bilinear forms with Kloosterman and Gauss sums in function fields
针对有限域有理函数域中 Kloosterman 和与 Gauss 和双线性型缺乏研究的问题，基于模同余解数的新界限，提供了若干新的非平凡界。
Symplectic Kloosterman Sums for $\operatorname{Sp}(2n)$ with Powerful Moduli
本文研究了Sp(2n)的辛Kloosterman和的界，并将其应用于共 prime 对称对的等分布。
We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the Pólya-Vinogradov range. We then derive applications to the second moment of holomorphic cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\GL_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.
Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families
本文确定了线性约束 Kloosterman 族的顶权层的连通几何幺正群，通过四阶矩恒等式和射影射线论证，在无需有限群分类的情况下证明其为特殊线性群 SL_r(n)。
By a result of Biswas and Dos Santos, on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite, that is it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for non-proper non-smooth algebraic stacks over arbitrary fields of characteristic $p>0$. As by-product we have the following partial generalization of the Biswas-Dos Santos' result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over $k$ a vector bundl…
Bilinear forms with trace functions
本文提出了一种基于分层定理和鲁棒准则的新方法，用于在波利亚-文纳戈夫范围内获得双线性迹函数和的非平凡界。
On values of the Bessel function for generic representations of finite general linear groups
本文研究了有限一般线性群不可约一般表示的Bessel函数，给出了其递推表达式，发现其特殊值与exotic Kloosterman sums的L-函数系数相关，并以此证明了多项式根的性质及Shintani base change下Bessel函数值的关联。
Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Isolation of the residual spectrum via Poisson Summation
本文在GL(3,Q)上推广Aldug为GL(2,Q)发展的Poisson求和方法，通过假设猜想A分离平凡表示的贡献，并通过Poisson求和得到具体展开。
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