Quasimodular solutions of a differential equation of hypergeometric type

Quasimodular forms and modular differential equations which are not apparent at cusps: I

In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Γ=Γ_0^+(N)$ generated by $Γ_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($Γ_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups $Γ_0^…

利用拟模形式计算双曲函数的倒数和

Evaluation of reciprocal sums of hyperbolic functions using quasimodular forms

本文研究了八族双曲函数无穷级数，提出利用复乘理论、模形式与拟模形式环及其微分算子计算其在 CM 点值的系统方法，并据此导出了 Fibonacci 数广义倒数和的代数无关性结果。

高阶 Schwarzian、拟模形式与等变函数

Higher Schwarzian, quasimodular forms and equivariant functions

本文研究高阶 Schwarzian 导数与拟模形式及等变函数的联系，证明亚纯函数等变当且仅当其高阶 Schwarzian 为指定权与深度的拟模形式，推广了经典结果。

Modular Linear Differential Operators and Generalized Rankin-Cohen Brackets

The aim in this paper is to give expressions for modular linear differential operators of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on …

Modular Ordinary Differential Equations on ${\rm SL}(2,\mathbb{Z})$ of Third Order and Applications

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form $y'''+Q_2(z)y'+Q_3(z)y=0$, $z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,\operatorname{Im}z>0 \}$, where $Q_2(z)$ and $Q_3(z)-\frac12 Q_2'(z)$ are meromorphic modular forms on ${\rm SL}(2,\mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on ${\rm SL}(2,\mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $\hatρ\colon {\rm SL}(2,\mathbb{Z})\to{\rm SL}(3,\mathbb{C})$ for this MODE and give the necessary …

Hypergeometric series, modular linear differential equations, and vector-valued modular forms

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.

On the modularity of solutions of certain differential equations of hypergeometric type

The purpose of this paper is to provide answers to some questions raised in a paper by Kaneko and Koike about the modularity of the solutions of a differential equations of hypergeometric type. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in other cases. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

Hypergeometric Functions and Carlitz Differential Equations over Function Fields

The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur's hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations with Carlitz derivatives (including a counterpart of regular singularity), umbral calculus, holonomic modules over the Weyl-Carlitz ring.

超几何微分方程的不规则 Hodge 滤过

Irregular Hodge filtration of hypergeometric differential equations

本文研究有理参数超几何联络的不规则Hodge滤过，借助几何解释重新证明Fedorov与Sabbah–Yu的Hodge数公式，并证明相关超几何和在乘法群上处处通常。

Geometry and arithmetic associated to Appell hypergeometric partial differential equations

In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives have the remarkable properties. We find that Appell hypergeometric partial differential equations can be reduced to four nonlinear partial differential equations by the use of our four derivatives. We investigate these four nonlinear partial differential equations. We are interested in some particular cases. In these cases, the system of linear partial differential equations has the remarkable propert…

A non-hypergeometric E-function

We answer in the negative Siegel's question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on André's theory of E-operators and Katz's computation of the Galois group of hypergeometric differential equations.

Solutions modulo $p$ of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz

We consider the Gauss-Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallelly to themselves. We reduce these equations modulo a prime integer $p$ and construct polynomial solutions of the new differential equations as $p$-analogs of the initial hypergeometric integrals. In some cases we interpret the $p$-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field $F_p$. That interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the number of po…

Hypergeometric decomposition of symmetric K3 quartic pencils

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard--Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.

Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.

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 …

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…

Duality relations for hypergeometric series

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for q-hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by N. M. Bailey in the 1930's on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.

Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations

It is well-known that Klein's lectures on the icosahedron and the solution of equations of fifth degree is one of the most important and influential books of 19th-century mathematics. In the present paper, we will give the complex counterpart of Klein's book, i.e., a story about complex regular polyhedra. We will show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equ…

Hypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $\mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these poly…

Determinant of $\mathbb F_p$-hypergeometric solutions under ample reduction

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work joint with V.\,Schechtman and called the $\mathbb F_p$-hypergeometric solutions. The dimension of the space of $\mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimen…

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。

The structure of bivariate rational hypergeometric functions

We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.

On the number of $p$-hypergeometric solutions of KZ equations

It is known that solutions of the KZ equations can be written in the form of multidimensional hypergeometric integrals. In 2017 in a joint paper of the author with V. Schechtman the construction of hypergeometric solutions was modified, and solutions of the KZ equations modulo a prime number $p$ were constructed. These solutions modulo $p$, called the $p$-hypergeometric solutions, are polynomials with integer coefficients. A general problem is to determine the number of independent $p$-hypergeometric solutions and understand the meaning of that number. In this paper we consider the KZ equation…

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form can be often read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of $p$ and from Weil's general bounds $|a(p)|\le2p^{(m-1)/2}$, where $m$ is the weight of the form. Furthermore, the critical $L$-values of the modular form are predicted to be $\mathbb Q$-proportional to the values of a related basis of solutions to the hypergeometri…

Solutions of KZ differential equations modulo $p$

We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions.