J. William Hoffman, Fang-Ting Tu
We consider algebraic transformations of hypergeometric functions from a geometric point of view. Hypergeometric functions are shown to arise from the deRham realization of a hypergeometric motive. The $\ell$-adic realization of the motive gives rise to hypergeometric characters sums over finite fields. This helps to unify and explain some recent results about transformations of hypergeometric character sums.
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On rank $2$ hypergeometric motives
本文研究秩2超几何动机的性质，证明其预期特性，强调其在可处理性上的优势。
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。
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。
Hypergeometric Motives from Euler Integral Representations
本文通过欧拉积分表示研究超几何函数的有理函数覆盖族，提出局部紧致化方法，并利用L函数和环面ζ函数的乘积形式，发展出计算希格拉特数的组合算法。
Survey of hypergeometric motives, with a focus on their source varieties, Hodge numbers, and L-functions.
We calculate the Beilinson regulators of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the Beilinson conjecture on special values of L-functions.
We study a family of hypergeometric motives $H(α,β|t)$ attached to a pair of tuples $α=(1/4,1/2,3/4)$, $β=(0,0,0)$. To each such motive we can attach a system of $\ell$--adic realisations with the trace of geometric Frobenius given by the evaluation of the finite field analogue of complex hypergeometric function. Geometry of elliptic fibrations makes it possible to to realise the motive $H(α,β|t)$ as a pure Chow motive attached to a suitable K3 surface $V_{t}$.
Hypergeometric motives and the generalized Fermat equation
本文提出超几何动机作为研究广义费马方程整数解更自然的工具，以替代传统代数模型方法。
We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai $L$-functions of Hilbert modular forms.
We study classical hypergeometric series as a p-adic function of its parameters inspired by a problem in the American Mathematical Monthly solved by D. Zagier. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.
We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.
Hypergeometric Motives from Toric Hypersurfaces
本文研究由 d 维环面中 d+2 个单项式线性组合定义的一般超曲面的两种紧化，证明其 F_q-点数量可由有限超几何和表示，并通过引入 gamma 三元组将有限超几何和推广到素数幂情形，推广了 Beukers、Cohen 和 Mellit 的工作。
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.
In 1995, Thakur invented and studied positive characteristic analogues of hypergeometric functions. In this paper, we interpret the special values of those functions as periods of a pre-$t$-motive. As a consequence, we show their transcendence and linear independence results by using Chang's refined version of the Anderson-Brownawell-Papanikolas criterion. Furthermore, as by-products, we show some linear/algebraic independence results among the special values of Kochubei multiple polylogarithms according to our period interpretation and the corresponding refined criterion.
We prove two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them to the $p$-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the $p$-adic Clausen's transformation. We also deduce special values of certain $p$-adic hypergeometric functions.
We prove two transformations for the $p$-adic hypergeometric series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic hypergeometric series to deduce the transformations. We also find another transformation for the $p$-adic hypergeometric series from which many special values of the $p$-adic hypergeometric series as well as finite field hypergeometric functions are obtained.
We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric 2F1 functions over a finite field F_q. We prove this conjecture and give an application. The proof depends on a new linear transformation formula for pseudo hypergeometric functions over F_q.
New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the $_5F_4$ hypergeometric function. We use these results to obtain two explicit $_5F_4$ evaluations, and several new formulas for $1/π$.
Transformation formula of Dwork's $p$-adic hypergeometric function
本文给出了Dwork的$p$-adic超几何函数在$t$与$t^{-1}$之间的变换公式，并在附录中引入蕴含其特殊情形的有限模拟。
In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using the Teichmüller character of finite fields and quotients of the $p$-adic gamma function. This function extends hypergeometric functions over finite fields to the $p$-adic setting. In this paper, we give certain transformation formulas for the function $_{n}G_{n}[\cdots]$ which are not implied from the analogous hypergeometric functions over finite fields.
We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function's relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.
Congruence relations for $p$-adic hypergeometric functions $\widehat{\mathscr{F}}_{a,...,a}^{(σ)}(t)$ and its transformation formula
本文引入了一类新的$p$-adic超几何函数，证明了它们满足同余关系从而收敛，并在特定情形下建立了与M. Asakura定义的对数型$p$-adic超几何函数之间的变换公式。
On transformation formulas of $p$-adic hypergeometric functions
本文回顾了三种$p$-adic超几何函数及其猜想变换公式，并证明了一个变换公式蕴含另一个。
We find summation identities and transformations for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$Z_λ: x_1^d+x_2^d=dλx_1x_2^{d-1}$$ over a finite field $\mathbb{F}_p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ on the family $Z_λ$ in terms of quotients of $p$-adic gamma function under the condition that $d|p-1$. In this paper, we first express the number of points over a finite field $\mathbb{F}_p$ on the family $Z_λ$ in terms of McCarthy's $p$-adic hypergeometric series for a…
We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions to establish a finite-field analogue of Koike and Shiga's cubic transformation for the Appell hypergeometric function $F_1$, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas …
We classify all the zeros and non-zero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic setting which can be described as $p$-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the $p$-adic setting as well as Gaussian hypergeometric series.
Hypergeometric decomposition of Delsarte K3 pencils
研究五族项目ive四次Delsarte K3表面，通过超几何求和给出有限域上的点计数公式，并在复数域上匹配周期与超几何微分算子，分解L函数为超几何L函数和狄利克雷ζ函数。
Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions
本文提出一种与超几何积分相关的motives理论，证明散射振幅背景下Gauss超几何函数上的motivic Galois群作用猜想，并展示Lauricella超几何函数的局部与全局作用的兼容性。
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric functions is also studied.
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpre…
$L$-values of certain weight 3 Modular Forms and Transformations of Hypergeometric Series
本文基于有限Coxeter群框架，对可用超几何-模性方法获得$L$-值的权重3 Hecke特征形式进行了完整分类，并利用超几何函数视角确定了模形式因有限阶特征扭曲产生的差异，重新解释了经典超几何恒等式。
In this paper, we will obtain new algebraic transformations of the $_2F_1$-hypergeometric functions. The main novelty in our approach is the interpretation of identities among $_2F_1$-hypergeometric functions as identities among automorphic forms on different Shimura curves.
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some $q$ orthogonal polynomials, and various multi-sums that are expressible as infinite products.
If $k$ is set equal to $a q$ in the definition of a WP Bailey pair, \[ β_{n}(a,k) = \sum_{j=0}^{n} \frac{(k/a)_{n-j}(k)_{n+j}}{(q)_{n-j}(aq)_{n+j}}α_{j}(a,k), \] this equation reduces to $β_{n}=\sum_{j=0}^{n}α_{j}$. This seemingly trivial relation connecting the $α_n$'s with the $β_n$'s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for…
Supercongruences arising from a ${}_7F_6$ hypergeometric transformation formula
本文借助 ${}_7F_6$ 超几何变换公式证明两个超同余式，其中之一证实 Guo–Liu–Schlosser 猜想并推广 Long–Ramakrishna 的结果。
It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function ${}_2F_1$ arises from a relation between modular curves, namely the covering of $X_0(3)$ by $X_0(9)$. In general, when $2\le N\le 7$ the N-fold cover of $X_0(N)$ by $X_0(N^2)$ gives rise to an algebraic hypergeometric transformation. The N=2,3,4 transformations are arithmetic-geometric mean iterations, but the N=5,6,7 transformations are new. In the final two the change of variables is not parametrized by rational functions, since $X_0(6),X_0(7)$ are of genus 1. Since their quotients $X_0^+(6),X_0^+(7)$ unde…
Sum representations of Appell-Lauricella functions over finite fields using confluent hypergeometric functions and their applications
本文利用有限域上的合流超几何函数证明了 Appell-Lauricella 函数的求和表示，并以此推导出其在有限域上的变换、求和与化简公式。
Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include $q$-analogues of supercongruences (referring to $p$-adic identities remaining valid for some higher power of $p$) established by Long, by Long and Ramakrishna, and several other $q$-supercongruences. The six basic hypergeometric transformation formulas which ar…
Theta functions and transformations of bilateral basic hypergeometric series
本文通过theta函数和双侧基本超几何级数建立新变换公式，构造伴生q级数，证明McIntosh关于q级数渐近变换的猜想，并扩展了Ramanujan和McIntosh的一些恒等式。
Hypergeometric $L$-functions in average polynomial time, II
本文提出一种算法，能在拟线性时间内计算固定超几何动机在所有素数处的 Frobenius 迹的模 $p^e$ 约简，扩展了先前模 $p$ 工作，关键在于引入通用素数构造扩展以支持 $p$-adic 超越函数计算。
Arcsine kernels, inverse-binomial harmonic sums, and classical hypergeometric transformations
研究将反正弦幂展开相关系数族放入双曲反正弦核，通过有限傅里叶投影和梅林变形等方法，得到含特定参数的恒等式及相关结果，还记录了核间卷积等细节，指出直接四次核会导致不同的\({}_4F_3\)族。
Expansion formulas for elliptic hypergeometric series
本研究提出一种沿良配Bailey引理思路获取椭圆超几何级数展开公式的替代方法，复现已有结果并得到1个新展开公式，进而推导出19个新变换公式，这些公式在p=0的基本超几何情形下也属全新。
In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann $Ξ$-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann $Ξ$-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby genera…
On the adelic Gaussian hypergeometric function
通过超几何曲线塔定义特殊高斯型的阿代尔超几何函数，该函数取值于阿代尔完备群环，插值所有有限域上的同类型超几何函数，并证明经典变换公式与求和公式。
For motives associated with Fermat curves, there are elements in motivic cohomology whose regulators are written in terms of special values of generalized hypergeometric functions. Using them, we verify the Beilinson conjecture numerically for some cases and find formulae for the values of L-functions at 0. These appear analogous to the Chowla-Selberg formula for the periods of elliptic curves with complex multiplication, which are related with the L-values at 1 by the Birch and Swinnerton-Dyer conjecture.
We study the quotient of hypergeometric functions \begin{equation*} μ_{a}^*(r)=\fracπ{2\sin{(πa)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation for $a\in(0,1/2]$, find an infinite product formula for $μ_{1/3}^*(r)$ by use of the properties of $μ_{a}^*(r)$ and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.
Hypergeometric functions over finite fields
本文利用修改后的Gauss和定义了有限域上的广义超几何函数，证明其求和、变换与乘积公式，展现与复数域经典超几何函数的类比，并应用于K3曲面zeta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。
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