Various aspects of hypergeometric functions

欧拉发现的超几何积分被许多作者用现代语言重新表述：扭曲同调和扭曲上同调的对偶。M.Kita和Yoshida为同调建立了期望交集理论，K.Cho和松本为上同调建立了期望交集理论。进一步的发展正在进行中，特别是松本的融合病例。这些可以被认为是扭曲版本的黎曼等式的周期积分。花村和吉田通过扭曲霍奇理论发现了黎曼不等式的扭曲形式。设X（k，n）是k-1维射影空间中n点集的位形空间。有几种构形空间可以表示为对称空间在不连续群下的商空间;最初的构形空间是X（2，4）* H/<$MA（2），其中H是上半空间，<$MA（2）是椭圆模群。Yoshida与松本和T.Sasaki一起发现了空间X（3，6）通过（3，6）型的超几何函数的模解释，它可以被概括为X（3，6）{z * M2（C）I（z-z *）/2 i> O}/Gamma，其中Gamma是作用于IV型埃尔米特对称域的算术子群。Yoshida写了两本关于这种解释的书。Kaneko发现，与D.Zagier，自守形式连接超几何函数和超奇异椭圆曲线。Kaneko Kaneko发现了j（gamma）的Fourier系数的一个新的算术公式，Watanabe利用Takano相空间的构造建立了一种新的非常透明的方法来寻找Painlv_函数的Okamoto变换，Kato雄心勃勃地试图寻找由Drinfeld对称空间和一致化微分方程p-向一致化的代数簇的例子;他已经发现，与M石田，新的假投影平面，并研究了他们的uniformizations复杂分析以及p-adically。

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.