Generalized hypergeometric arithmetic D-modules under a p-adic non-Liouvilleness condition

关于Hesse三次曲线的syntomic regulator与$p$-adic超几何函数

On the syntomic regulator of the Hesse cubic curves and $p$-adic hypergeometric functions

本文引入新型$p$-adic超几何函数，证明其满足Dwork型同余关系，并用于表达Hesse三次曲线的syntomic regulator，同时建立了与Asakura对数型$p$-adic超几何函数的变换公式。

与$p$-进多重多对数相关的$p$-进超几何函数

$p$-adic hypergeometric function related with $p$-adic multiple polylogarithms

本文受Ohno-Zagier公式启发，构造了高斯超几何函数的$p$-进类比，并借助与$p$-进多重多对数的关系建立严格解析延拓框架，证明其$p$-进高斯超几何定理。

p-adic Generalized Hypergeometric Equations from the Viewpoint of Arithmetic D-modules

We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\mathscr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.

The $p$-adic Gelfand-Kapranov-Zelevinsky hypergeometric complex

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $\mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points …

$A$-hypergeometric series and a $p$-adic refinement of the Hasse-Witt matrix

We identify the $p$-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic $p$ as the eigenvalues of a product of special values of a certain matrix of $p$-adic series. That matrix is a product $F(Λ^p)^{-1}F(Λ)$, where the entries in the matrix $F(Λ)$ are $A$-hypergeometric series with integral coefficients and $F(Λ)$ is independent of $p$.

关于对数型广义 $p$-adic 超几何函数的特殊值

On special values of generalized $p$-adic hypergeometric functions of logarithmic type

本文引入对数型广义 $p$-adic 超几何函数，证明其满足类似 Asakura 的同余关系，并通过数值计算和理论证明得出其在 $t=1$ 处特殊值在特定条件下为零。

$p$-adic formulas and unit root $F$-subcrystals of the hypergeometric system

We define the notion of {\it Dwork family of logarithmic $F$-crystals}, a typical example of which is the family of Gauss hypergeometricdifferential systems, viewed as parametrized by their exponents of algebraic monodromy. The $p$-adic analytic dependence of the Frobenius operation upon those exponents, is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the $p$-adic analytic continuation of the unit root $F$-subcrystal in the open tube of a singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of the Koblitz-Diamond formula $p$-adically analog to Gauss' evalu…

Dwork cohomology, de Rham cohomology, and hypergeometric functions

In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. N. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher cod…

超几何D-模与reductive群的指数和

Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups

本文定义了与有限域上reductive群表示家族相关的超几何指数和，引入超几何l-adic层和D-模，证明其是紧致的并估计其秩，利用Wang发展的向量丛一般基上的傅里叶变换理论，展示超几何D-模如何控制超几何层的一般行为，并应用结果估计超几何指数和。

Frobenius structures on hypergeometric equations

We give an exposition of Dwork's construction of Frobenius structures associated to generalized hypergeometric equations via the interpretation of the latter due to Gelfand-Kapranov-Zelevinsky in the language of A-hypergeometric systems. As a consequence, we extract some explicit formulas for the degeneration at 0 in terms of the Morita p-adic gamma function.

关于KZ方程的$p$-进解、寻常晶体与$p^s$-超几何解

On $p$-adic solutions to KZ equations, ordinary crystals, and $p^s$-hypergeometric solutions

本文研究$p$-进整数环上亏格$g$超椭圆曲线族对应的KZ联络，证明其局部平坦截面零化Gauss-Manin联络的秩$g$单位根$F$-子晶体且构成秩$g$自由$\boldsymbol{Z}_p$-模，还证明单位根子晶体模$p^s$的零化子由$p^s$-超几何平坦截面生成。

超几何周期 $p^s$-近似的幽灵与同余

Ghosts and congruences for $p^s$-approximations of hypergeometric periods

本文证明洛朗多项式常数项的 Dwork 型同余，并借此研究超几何与 KZ 方程模 $p^s$ 多项式解的算术和 $p$-进解析性质，进而展示 $p$-进 KZ 联络存在不变线子丛而复数情形没有。

Distinguished-root formulas for generalized Calabi-Yau hypersurfaces

By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in ${\mathbb P}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal $p$-divisibility. We study the $p$-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of $p$ times a product of special values of a certain $p$-adic analytic function ${\mathcal F}$. That function ${\mathcal F}$ is the $p$-adic analytic continuation of the ratio $F(Λ)/F(Λ^p)$, where $F(Λ)$ is a solution of the $A$-hypergeome…

Frobenius 交织子与 q-差分方程

Frobenius intertwiners for q-difference equations

本文研究射影空间余切丛上的量子差分方程，证明其具有 Frobenius 作用，给出交织子常数项的显式公式，并推广到 Dwork 的 p-adic 超几何与 Bessel 方程。