Geometry and arithmetic associated to Appell hypergeometric partial differential equations

算术超几何$\boldsymbol{\textrm{D}}$-模与约化群上的指数和

Arithmetic hypergeometric $\mathcal {D}$-modules and exponential sums on reductive groups

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。

Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpre…

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 …

可约秩2超几何动机的算术

The Arithmetic of Reducible Rank 2 Hypergeometric Motives

本文研究数域上可约且不规则的${}_3F_2(1)$超几何动机的算术，利用动机技术证明新求值公式，并给出与模形式$L$-函数精确值相关的例子。

超几何函数在SageMath中的代数与算术属性

Algebraic and Arithmetic Attributes of Hypergeometric Functions in SageMath

本文介绍了在SageMath中实现超几何函数的代数与算术属性算法，包括代数性判断、赋值计算及最小多项式生成。

Commensurability and arithmetic equivalence for orthogonal hypergeometric monodromy groups

We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to each other.

超几何函数的代数与算术属性的算法

Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

本文提出算法用于计算超几何函数的p进估值及模素数约简的 annihilating多项式。

Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the fami…

Dyson's Ranks and Appell-Lerch Sums

Denote by $p(n)$ the number of partitions of $n$ and by $N(a,M;n)$ the number of partitions of $n$ with rank congruent to $a$ modulo $M$. We find and prove a general formula for Dyson's ranks by considering the deviation of the ranks from the average: \begin{equation*} D(a,M) := \sum_{n= 0}^{\infty}\left(N(a,M;n) - \frac{p(n)}{M}\right) q^n. \end{equation*} Using Appell--Lerch sum properties we decompose $D(a,M)$ into modular and mock modular parts so that the mock modular component is supported on certain arithmetic progressions, whose modulus we can control. Using our decomposition, we show …

超几何函数值与算术Gevrey级数的线性独立性

Linear independence of values of hypergeometric functions and arithmetic Gevrey series

证明广义超几何函数${}_pF_q$在不同代数点处值的线性独立性，利用II型帕德逼近统一构造及新型非零论证，推广加强前人成果。

某些重量为二和四的海尔克埃钦形式的显式超几何模性

Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms

本文基于超几何模性方法，构造了两个eta商函数家族，利用Jacobi theta函数理论及其立方类比，将某些重量为二和四的海尔克埃钦形式的傅里叶系数表示为有限域周期函数，并获得新的模形式傅里叶系数与有限域Appell级数特殊值的关系式。

关于超几何动机的变换性质与丢番图方程

On Transformation properties of hypergeometric motives and Diophantine equations

本文研究高斯超几何函数变换公式在超几何动机上的算术类比，并将其应用于某些丢番图方程的研究。

有限域上的分圆矩阵与超几何函数

Cyclotomic matrices and hypergeometric functions over finite fields

本文借助有限域上的超几何函数，研究了涉及特征与二元二次型的分圆矩阵的算术性质，并证实了Zhi-Wei Sun提出的相关猜想。

Hypergeometric rational approximations to $ζ(4)$

We give a new hypergeometric construction of rational approximations to $ζ(4)$, which absorbs the earlier one from 2003 based on Bailey's ${}_9F_8$ hypergeometric integrals. With the novel ingredients we are able to get a better control of arithmetic and produce a record irrationality measure for $ζ(4)$.

高斯算术几何平均（AGM）到三变量迭代格式的推广

An extension of Gauss's arithmetic-geometric mean (AGM) to three variables iteration scheme

本研究将高斯算术几何平均（AGM）推广为三变量迭代格式，证明其在特定初值约束下的收敛性，给出极限的Appell超几何函数表示，并得到两类超几何函数的新关系。

Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz derivatives. The above class of equations includes some equations of hypergeometric type. Building on the work of Thakur, we develop his notion of the hypergeometric function of the first kind (whose parameters belonged initially to $\mathbb Z$) in such a way that it becomes fully an object of the func…

Arithmetic of linear forms involving odd zeta values

A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $ζ(2)$ and $ζ(3)$, as well as to explain Rivoal's "infinitely-many" result (math.NT/0008051) and to prove that at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$, and $ζ(11)$ is irrational.