Various aspects of hypergeometric functions

由欧拉发现的超几何积分被许多作者用一种现代的语言重新表述：扭曲同调和扭曲上同调的对偶协调。期望交集理论由M.Kita和Yoshida为同调建立，K.Cho和Matsumoto为上同调建立。进一步的发展正在进行中，特别是松本的合流案例。这些可以被认为是周期积分的黎曼等式的扭曲版本。黎曼不等式的扭曲版本是由花村和吉田通过扭曲霍奇理论发现的。构形空间的模解释。设X（k, n）为k-1维射影空间中n个点集的位形空间。在不连续群下，若干组态空间可以表示为对称空间的商空间；原始矩阵为X(2,4) * H/GAMMA(2)，其中H为上半空间，GAMMA(2)为椭圆模群。吉田与Matsumoto和T.Sasaki一起，通过类型为（3,6）)的几何函数，发现了空间X（3,6）的模解释，可以概括为X(3,6) {z * M2(C) I (z -z*)/2i> O}/GAMMA，其中GAMMA是作用于类型为iv的埃尔米对称定义域上的算术子群。吉田就此解释写了两本书。Kaneko和D.Zagier一起发现了连接超几何函数和超奇异椭圆曲线的自同构形式。Kaneko发现了j（）的傅里叶系数的一个新的算术公式。Watanabe利用相空间的Takano构造建立了一种新的非常透明的方法来求Painlv_函数的Okamoto变换。f.k atto雄心勃勃地试图找到由Drinfeld对称空间和均匀化微分方程p- ady均匀化的代数变体的例子；他已经和石田先生一起发现了新的假投影平面，并对它们的均化复分析和p-adical进行了研究。

Hypergeometric integrals found by Euler was re-formulated in terms of a modern language by many authors : the dualﾟCpairing of twisted homologies and twisted cohomologies. Expected intersection theories were established by M.Kita and Yoshida for homologies, and by K.Cho and Matsumoto for cohomologies. Further developments are in progress, especially those for confluent case by Matsumoto. These can be considered to be twisted versions of Riemann's equality for period integrals. Twisted versions of Riemann's inequality were found, via twisted Hodge theory, by Hanamura and Yoshida.Modular interpretations of configuration spaces. Let X(k, n) be the configuration space of n-point-sets in the k-1-dimensional projective space. Several configuration spaces can be presented as quotient spaces of symmetric spaces under discontinuous groups ; the original one is X(2, 4) * H/GAMMA(2), where H is the upper half space and GAMMA(2) is an elliptic modular group. Yoshida found, with Matsumoto and T.Sasaki, a modular interpretation of the space X(3, 6) through hepergeometric function of type (3, 6)), which can be summerized as X(3,6) {z * M2(C) I (z -z*)/2i> O}/GAMMA, where GAMMA is an arithmetic subgroup acting on the hermitian symmetric domain of type IV.Yoshida wrote two books about this interpretation.Kaneko found, with D.Zagier, automorphic forms which connect hypergeometric functions and supersingular elliptic curves. Kaneko found a new arithmetic formulae for the Fourier coefficients of j(gamma).F.Watanabe established a new very transparent way to find Okamoto transformations for Painlv_ functions by using the Takano's construction of the phase spaces.F.Kato is ambitiously trying to find examples of algebraic varieties which are p-adically uniformized by Drinfeld symmetric spaces and the uniformizing differential equations ; he already found, with M.Ishida, new fake projective planes, and studied their uniformizations complex anlytically as well as p-adically.