In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperber's construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dwork's deformation equation for our family. Furthermore, with the help of Dwork's dual theory and deformation theory, the strong Frobenius structure of this equation is established. Our work adds some new evidences for Dwork's conjecture.

中文翻译：

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四分之一平面上广义非线性薛定谔方程的黎曼-希尔伯特方法

在本文中，我们关注环面上的一系列广义 Kloosterman 和。通过对 Haessig 和 Sperber 的构造进行一些更改，我们导出了一些与 $L$-函数相对应的 $p$-adic 上同调。我们提出了顶维上同调空间的基的显式形式，从而获得计算$L$-函数的牛顿多边形下界的具体方法。使用 GKZ 系统的理论，我们推导出我们家庭的 Dwork 变形方程。进而借助德沃克的对偶理论和形变理论，建立了该方程的强弗罗贝尼乌斯结构。我们的工作为德沃克的猜想增加了一些新的证据。