Masanori Asakura, Noriyuki Otsubo
We study the $H^2$ of certain surfaces with complex multiplication by a cyclotomic field. The periods are written in terms of values of the gamma function and the conjecture of Gross-Deligne is verified. The regulators of certain $K_1$-elements are written in terms of values of hypergeometric functions ${}_3F_2$, and we prove their non-vanishing.
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On the regulator of Fermat motives and generalized hypergeometric functions
我们计算了费马曲线的动力Beilinson调节器，并用广义超几何函数的特殊值表示。结果证明了该调节器的满射性，支持了关于L-函数特殊值的Beilinson猜想。
We study periods and regulators of a certain class of fibrations of varieties whose relative $H^1$ has multiplication by a number field. Both are written in terms of values of hypergeometric functions ${}_3F_2$ and the former reduces to values of the gamma function, which provide examples of the conjecture of Gross-Deligne.
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。
Regulator of the Hesse cubic curves and hypergeometric functions
该研究构造黑塞三次曲线动机上同调的积分元素，用两类超几何函数表示其调节子，得到了Bloch-Beilinson猜想关于L函数特殊值的数值例子。
Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points
本文研究了超椭圆曲线的动机上同调中的调节器构造，并通过超几何函数验证了其非平凡性及整数性，同时探讨了 $L$-函数特殊值的贝利松猜想。
In this paper we construct certain higher Chow cycles in the $K_{1}$ of the Jacobian of Fermat curves, generalising a construction of Collino. We further compute the regulator of these elements in terms of special values of hypergeometric functions. Otsubo has computed the regulator of certain elements of $K_0$ and $K_2$ of Fermat varieties and this paper is along the same lines.
The Ross symbol is defined to be an element {1-z,1-w\} in K_2 of a Fermat curve z^n+w^m=1. Ross showed that it is non-torsion by computing the Beilinson regulator. In this paper, we introduce a generalization of the Ross symbols in K_{d+1} of a variety (1-x_0^{n_0})\cdots(1-x_d^{n_d})=t. The main result is that the Beilinson regulator is described by the hypergeometric functions {}_{d+3}F_{d+2}'s.
For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.
We consider certain CM elliptic curves which are related to Fermat curves, and express the values of $L$-functions at $s=2$ in terms of special values of generalized hypergeometric functions. We compare them and a similar result of Rogers-Zudilin with Otsubo's regulator formulas, and give a new proof of the Beilinson conjectures originally due to Bloch.
This is a sequel of the paper "A generalization of the Ross symbols in higher K-groups and hypergeometric functions I" where we introduced higher Ross symbols in higher $K$-groups of the hypergeometric schemes, and discussed the Beilinson regulators. In this paper we give its p-adic counterpart and an application to the $p$-adic Beilinson conjecture for K3 surfaces of Picard number 20.
We use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by differentiating the Mahler measures and using hypergeometric identities. Since our proofs involve the regulator, they yield light into the expected relation of each Mahler measure to special values of $L$-functions of certain elliptic curves.
$_3F_2$ hypergeometric series and periods of elliptic curves
我们用经典的超几何级数表示椭圆曲线族的实周期。这个表达式是类似的结果小野其中涉及的跟踪Frobenius同一家庭的椭圆曲线的高斯超几何级数。这个类比进一步证明了经典和高斯超几何级数之间的相互作用。
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.
Periods of Hodge cycles and special values of the Gauss' hypergeometric function
本文通过计算 Fermat 畴扰动的周期，研究了由算术条件定义的 Hodge 圈子空间，给出了其维数上界，并推导出在单变量有理函数域上为代数的 Gauss 超几何函数的显式表达式。
Three vignettes on hypergeometric normal functions
本文利用Hodge理论方法，解释数论恒等式、描述Abel-Jacobi周期函数的Frobenius对偶，并提供Golyshev猜想的新证明。
Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.
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。
We consider the fourteen families $W$ of Calabi-Yau threefolds with one complex structure parameter and Picard-Fuchs equation of hypergeometric type, like the mirror of the quintic in $\mathbb{P}^4$. Mirror symmetry identifies the masses of even--dimensional D--branes of the mirror Calabi-Yau $M$ with four periods of the holomorphic $(3,0)$-form over a symplectic basis of $H_3(W,\mathbb{Z})$. It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under $Γ_0(25)$, whose Hecke eigenvalues are determined by the Hasse-…
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 …
Inspired by a result of Manin, we study the relationship between certain period integrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show that both of these properties can be computed in terms of "matching" classical and finite field hypergeometric functions, a phenomenon that has also been observed in elliptic curves and many higher dimensional varieties.
We give an expression for number of points for the family of Dwork K3 surfaces $$X_λ^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4λx_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy's $p$-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.
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。
Elements in $K_4$ and regulator maps of Fermat curves
针对所有 N≥3 的 Fermat 曲线 x^N+y^N=1，构造了 K_4^{(3)} 群中的显式元素，并证明其 Beilinson 调节子像非零，给出了调节子积分的显式公式及渐近行为。
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…
Hypergeometric functions and a family of algebraic curves
设$λ\in \mathbb{Q}\setminus \{0, 1\}$和$l \geq 2$，用$C_{l,λ}$表示$\mathbb{Q}$上具有$$y^l=x(x-1)(x-λ).$$给出的仿射方程的非奇异射影代数曲线。本文定义了类似于椭圆曲线实周期的$Ω(C_{l, λ})$，并找到了它与一般超几何级数的关系。给出了有限域上$C_{l, λ}$上点的个数与高斯超几何级数的关系。最后给出了\cite{rouse}的一个结果的另一种证明.
Periods for rank 1 irregular singular connections on surfaces
我们定义了一个周期配对平坦的，不规则的奇异，秩一连接，满足一个技术条件，关于其平稳集，在复杂的表面之间的de Rham上同调的连接和修改后的奇异同调，快速衰减同调。我们证明了这给出了一个完美的对偶性。作为一个例子，构造了一个二维的合流超几何。
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…
The Explicit Hypergeometric-Modularity Method I
本文提出显式超几何-模性方法，将给定超几何数据关联到模形式，利用拉马努金椭圆函数理论、形式群律和超同余，构造eta商并给出应用。
Higher regulators, periods and special values of the degree four L-function of GSp(4)
我们考虑与辛群GSp（4）的一个自守表示相关的4次L函数.从Beilinson的Eisenstien符号出发，构造了GSp（4）的Shimura簇上的一些动机上同调类.我们表明，这些类的绝对霍奇调节器下的图像消失的Baily-Borel紧的志村品种的边界上。这允许将这些类的产品的阿基米德积分，哈里斯的隐匿期不变，德利涅期间和特殊价值的L函数预测贝林森的猜想。所考虑的表示被假定为具有关于各向同性对称矩阵的贝塞尔模型。
A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data $HD$ are primitive and self-dual. If the data are also defined over $\mathbb Q$, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of $\ell$-adic representations of the absolute Galois group of …
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…
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。
Homological periods and higher cycles
本文为代数概形构造同调调节子与高阶环类映射，并在代数数上给出度1周期同调的几何描述，证明Grothendieck周期猜想蕴含广义Roitman定理。
On special values of Jacobi-sum Hecke L-functions
对于与费马曲线相关的动机，动机上同调中有一些元素的调节子是用广义超几何函数的特殊值来表示的。利用它们，我们在某些情况下数值验证了Beilinson猜想，并找到了L函数在0.这些看起来类似于具有复乘的椭圆曲线周期的Chowla-Selberg公式，其通过Birch和Swinnerton-Dyer猜想与1处的L值相关。
Construction of Higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$
本研究针对$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖曲面族，构造了$(2, 1)$型高阶Chow环，通过计算超越调节映射的像，证明其在$\mathbb{Z}$上线性无关且生成$\operatorname{rank} \ge 36 \cdot φ(N)$的子群。
Exceptional Sets for Certain ${}_2F_1$ Hypergeometric Functions
研究针对单值群为竹内I类算术三角群的\({}_2F_1(a,b,c;z)\)，利用超几何 - 模恒等式、Wüstholz关于阿贝尔簇周期的超越性结果及Schneider关于椭圆曲线\(j\)不变量代数值的经典结果，明确确定其例外集\(E(a,b,c)\)。
q-Terms, singularities and the extended Bloch group
我们的论文起源于体积猜想到$q$ -超几何项的多重和的推广。2006年，Kontsevich在奥胡斯大学的一个问题列表中概述了这一概括; \cite{Ko}。我们引入了$q$ -超几何项（简称$q$ -项）的概念.后者是线性型的$q$ -因子在多个变量中的比率的乘积。在本文的第一部分中，我们展示了如何构造元素的布洛赫群（及其扩展版本）给定的\qterm。它们在布洛赫-维格纳映射或罗杰斯双对数下的像是一个权为2的周期的有限集，在孔采维奇-扎吉埃意义下。在本文的第二部分中，我们介绍了一个特殊的$q$ -项的概念，其相应的多项式序列，其生成系列。特殊$q$ -项的例子自然来自量子拓扑学，特别是来自结的平面投影。这两个部分是绑在一起的猜想，涉及到一个特殊的$q$ -长期与扩展布洛赫群的相应元素的周期生成系列的奇点。在某些情况下（如$4_1$结），猜想是已知的。
Coxeter Symmetry in Hypergeometric Functions and Elliptic Integral Moments
本文建立超几何框架，利用$W(D_5)$对称性统一推导椭圆积分矩、调和和及特殊值恒等式，揭示其共同对称性来源。
Linear independence measures for Chowla--Selberg periods
通过同时Padé近似估计超几何函数，研究特定d和D下1、π√d、Ω_D/π和π/Ω_D的线性形式，揭示其线性独立性。
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。
Ramanujan-type identities for Shimura curves
在1914年，拉马努金给出了一个列表的17个身份表示$1/π$作为线性组合的价值观的超几何函数在某些有理数。从那时起，许多作者发现了类似性质的恒等式。如今，这种恒等式的标准方法之一是使用模曲线理论。本文考虑Shimura曲线的情形，得到了包含特殊值的超几何函数和Gamma值乘积的Ramanujan型公式. Gamma值的乘积与椭圆曲线的周期相关，其具有与Q（\sqrt{-3}）和Q（\sqrt{-4}）的复乘。
Let $E/\mathbb{Q}$ be a number field of degree $n$. We show that if $\operatorname{Reg}(E)\ll_n |\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg_{n,\varepsilon} |\operatorname{Disc}|^{-1/4-\varepsilon}$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf{PGL}_n(\mathbb{Z})\backslash\mathbf{PGL}_n(\mathbb{R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.
Counting matrix points on certain varieties over finite fields
本文将有限域超几何函数方法推广到矩阵点计数，构造矩阵椭圆曲线及 K3 曲面族的计数公式，并证明误差项的 Sato-Tate 分布。
We investigate arithmetic aspects of the middle degree cohomology of compactified Picard modular surfaces $X$ attached to the unitary similitude group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$. We construct new Beilinson--Flach classes on $X$ and compute their Archimedean regulator. We obtain a special value formula involving a non-critical $L$-value of the degree six standard $L$-function, a Whittaker period, and the regulator. This provides evidence for Beilinson's conjecture in this setting.
We give a generalization of $p$-adic congruences for truncated period functions, that were originally discovered for a class of hypergeometric functions by Bernard Dwork.
Hypergeometry and the AGM over Finite Fields
本文利用有限域超几何函数研究 AGM 的有限域类比，通过“水母群”图结构证明正定二元二次型高斯类数的新恒等式，并揭示水母大小与类群中 2 之上素数阶的关系。
Identities for the Ramanujan zeta function
我们证明公式的Ramanujan τ zeta函数的特殊值。我们的公式表明，$L(Δ, k)$是一个周期的意义上的Kontsevich和Zagier时，$k\ge12$。作为一个例子，当$L(Δ, k)$。
On the representability of sequences as constant terms
本文研究有理序列何时可表示为多元 Laurent 多项式幂的常数项，给出常系数线性递推序列的完整刻画，并分类一类简单超几何序列族中的常数项序列。
Apéry's theorem and problems for the values of Riemann's zeta function and their $q$-analogues
本专著旨在被视为我的history（D.Sc.）。论文;由于这一点，因为一切都已经出现在英语，它是专门在俄罗斯进行。该专着包括一个详细的介绍和七章，代表了我的工作的一部分影响Apéry的证明从1978年的非理性的$ζ(2)$和$ζ(3)$，价值的黎曼zeta函数。第1章是关于“$ζ(5)$、$ζ(7)$、$ζ(9)$和$ζ(11)$四个数中至少有一个是无理数”（部分基于arXiv：math.NT/0206176）。第2章解释了Beukers在他的Apéry结果的证明中引入的广义重积分和非常平衡的超几何级数之间的联系;它基于arXiv：math.CA/0206177。第3章综述了一些算术和超几何$q$ -类比，并建立了$ζ(2)$的$q$ -类比的无理性测度$μ(ζ_q(2))<3.518876$;它严格遵循Sb.Math.193（2002），1151- 1172中的内容，但也结合了Smet和Van Assche（arXiv：0809.2501 [math.CA]）对超几何构造的更尖锐的分析，以改进2002年的结果。第4章专门讨论$μ(ζ(2))<5.095412$度量，基于arXiv：1310.1526 [math.NT];第5章建立从$(3/2)^k$到最近整数的距离的估计值$||(3/2)^k||>0.5803^k$，英文版发表在J. Théor上。Nombres Bordea…
Representability of G-functions as rational functions in hypergeometric series
本文利用微分Galois理论证明，存在G-函数无法表示为具有有界奇点的线性微分方程解的有理函数，从而对G-函数能否表示为超几何级数有理函数的问题给出了无条件的否定回答。
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