## 古典解析セミナー

担当者 大島 利雄, 坂井 秀隆

### 2011年06月24日(金)

15:00-16:30   数理科学研究科棟(駒場) 128号室

A Schwarz map of Appell's $F_2$ whose monodromy group is
related to the reflection group of type $D_4$ (JAPANESE)
[ 講演概要 ]
The system of differential equations for Appell's hypergeometric function $F_2(a,b,b',c,c';x,y)$ has four fundamental solutions.
Let $u_1,u_2,u_3,u_4$ be such solutions. If the monodromy group of the system is finite, the closure of the image of the Schwarz map $U(x,y)=(u_1(x,y),u_2(x,y),u_3(x,y),u_4(x,y))$
is a hypersurface $S$ of the 3-dimensional projective space ${\\bf P}^3$. Then $S$ is defined by $P(u_1,u_2,u_3,u_4)=0$ for a polynomial $P(t_1,t_2,t_3,t_4)$.
It is M. Kato (Univ. Ryukyus) who determined the parameter
$a,b,b',c,c'$ such that the monodromy group of the system for $F_2(a,b,b',c,c';x,y)$ is finite. It follows from his result that such a group is the semidirect product of an irreducible finite reflection group $G$ of rank four by an abelian group.
In this talk, we treat the system for $F_2(a,b,b',c,c';x,y)$ with
$(a,b,b',c,c')=(1/2,1/6,-1/6,1/3,2/3$. In this case, the monodromy group is the semidirect group of $G$ by $Z/3Z$, where $G$ is the reflection group of type $D_4$. The polynomial $P(t_1,t_2,t_3,t_4)$ in this case is of degree four. There are 16 ordinary singular points in the hypersurface $S$.
In the rest of my talk, I explain the background of the study.