AI 中文总结
本文定义约化群有限表示族关联的超几何指数和，引入带Frobenius结构的算术超几何$\boldsymbol{\textrm{D}}$-模研究该指数和，得到其估计并构造相关$F$-等晶体。
AI 中文摘要
对于约化群的有限表示族，我们在该群上定义一个劳伦多项式，与该劳伦多项式关联的指数和称为超几何指数和。我们引入算术超几何$\boldsymbol{\textrm{D}}$-模来研究超几何指数和，它是带Frobenius结构的超完整算术$\boldsymbol{\textrm{D}}$-模，使得有理点处的Frobenius迹即为该指数和。在劳伦多项式非退化的轨迹上，该算术超几何$\boldsymbol{\textrm{D}}$-模定义了沿退化轨迹上收敛的$F$-等晶体，作为应用我们得到了超几何指数和的一个估计。
英文摘要
For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric $\mathcal D$-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic $\mathcal D$-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. Over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric $\mathcal {D}$-module defines an $F$-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.
