超幾何的黒写像

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1 * 9 : $-\cdot$ $F_{1}$ Appell, Lauricella $F_{D}$ $(3, 6)$ $E(3,6;1/2)$ $*2013$

2 : $\sum x^{n}$ $(1-x)^{-1}$ $arrow\sum\frac{(a)_{n}}{n!}x^{2}$ $(1-a)^{-a}$ $(a)_{n}=\gamma(a+n)/\gamma(a)$, $arrow\sum\frac{(a)_{n}(b)_{n}}{(c)_{n}n!}x^{n}$ $arrow$... $F(a, b, c;x)$, 1.2 : $\int_{0}^{1}t^{1-p}(1-t)^{1-q}dt$ $arrow\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b}dt$ $arrow$... Beta 1.3 $x(1-x)u^{\prime/}+\{c-(a+b+1)x\}u -abu=0.$ ( ) $\mathscr{u}$ $0,1,$ $\infty$ 2

3 $\circ$ ( H.A.Schwarz ) Sch: $X;=C-\{O, 1\}\ni x\mapsto u_{1}(x):u_{2}(x)\in P^{1}=CU\{\infty\}$ $\{u_{1}, u_{2}\}$ $a,$ $x$- $b,$ $c$ Sch : $a=b=1/2,$ $c=1$ $0,1,1/x,$ $\infty$ $C_{x}:s^{2}=t(1-t)(1-xt)$ ( ) ( ) ( ) Gauss-Manin ( 1 ) ; ( ) ( ) $H$ $\Gamma(2)$ $X=C-\{O, 1\}arrow H/\Gamma(2)$ 1 $=$ Hodge, Picard, $\circ=$ $=$ Fuchs, $=$ lambda 3

4 120 ( ) Pl ( $X$ $(2, 4)$ ) : (Istanbul) 4

5 $(b)_{n}$ $\sum\frac{(a_{1})_{n}.\cdot\cdot(a_{p})}{(b_{1})_{n}\cdot\cdot(b_{p-1})n!}x^{n}={}_{p}f_{p-1}(a, b;x)$ ( ) $50km$ $P$ $p=2$ $p=3$ 3.2 Appell $F_{1}$, Lauricella $F_{D}$ Beta $(1-xt)^{-b}$ $\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-x_{1}t)^{-b_{1}}\cdots(1-x_{n}t)^{-b_{n}}dt$ $n=1$ $n=2$ Appell $F_{1},$ $n\geq 3$ Lauricella ( 1 $F_{D}$ $D$ ) $P^{1}$ 5

6 122 $n+3$ $X(2, n+3)$ $n+1$ $n$- $\{z_{0}:z_{1} :... :z_{n}\in P^{n} z_{0} ^{2}- z_{1} ^{2}-\cdots- z_{n} ^{2}>0\}$ $\circ$ 3.3 $t$ ) $P^{k}$ $t_{1},$ $\ldots,t_{k}$ ( $n$ $(k,n)=(2, n)$ $(k,n)=(3,6)$ $(3, 5)$ $(2,5)$ : $P^{k-1}$ $n(\geq k+1)$ ( ) $X(k, n)=gl_{k}\backslash \{(\begin{array}{lll}x_{11} \cdots x_{1n}x_{k1} \cdots x_{kn}\end{array}) k\cross k$ $C\cross$ $\}$ /( )n ( $)$ $k$ $k$ $0$ $k=2$ $3\cross 5-$ ( 3 ) 6

7 :5-3 2 $=$ $2\cross 5$- $3\cross 5$- $2\cross 5$- ( ) $3\cross 5$- ( ) $3\cross 3$ 2 $x5$- $2\cross 2$ $3arrow k,$ $5arrow n,$ $2arrow n-k$ $X(k, n)\cong X(n-k, n)$ $X(2,4),$ $X(3,6),$ $\ldots$ $2k=n$ $X(3,6)$ $X(2,4)$ $x_{1},$ $x_{2},$ $x_{3},$ $x_{4}$ $X(2,4)$ $x_{1}=x_{4},$ $x_{2}=x_{3}$ ; $x_{1}=x_{3},$ $x_{2}=x_{4}$ ; $x_{1}=x_{2},$ $x_{3}=x_{4}$ $X(2,4)$ compact ; ; $X(2,4)$ $x_{1}=x_{2}$ $x_{3}=x_{4}$ $x_{1}=x_{3}$ $x_{2}=x_{4}$ $x_{1}=x_{4}$ $x_{2}=x_{3}$ ( ) ( ) $\lambda(x_{1}, \ldots, x_{4})=\frac{(x_{1}-x_{4})(x_{2}-x_{3})}{(x_{2}-x_{4})(x_{1}-x_{3})}$ $P^{1}$ $x_{1}$ $0$ $x_{2}$ $\infty$ $x_{3}$ 1 ( ) $x_{4}$ $(\begin{array}{llll}x_{01} x_{02} x_{03} x_{04}x_{11} x_{12} x_{13} x_{14}\end{array})$ 7

8 124 $GL_{2}$ $(C^{*})^{4}$ $(\begin{array}{llll} \lambda\end{array})$ 3.5 $(3, 6)$- $P^{2}$ $(3, 6)$- 6 $X(3,6)$ $f_{\delta}(x, \alpha)=\int\int_{\delta}\prod_{j=1}^{6}\ell_{j}(x, s, t)^{\alpha_{j}-1}dsdt.$ $(s, t)$ $P^{2}$ $\ell_{j}(x, s, t)=x_{0j}+x_{1j}s+x_{2j}t, j=1, \ldots, 6$ $L_{J}$ $x$ $X(3,6)$ $3\cross 6$- ( ) $x=(\begin{array}{llllll} x_{1} x_{2} x_{3} x_{4}\end{array})$ $\alpha_{j}$ $\triangle$ $\alpha_{1}+\cdots+\alpha_{6}=3$ $P^{2}-\cup L_{j}$ 6 $E(3,6;\alpha)$ $u_{1},$ $\ldots,$ $u_{6}$ $X(3,6)\ni x\mapsto u_{1}(x)$ :.. : $u_{6}(x)\in P^{5}$ $\alpha$ $\alpha j=1/2$ ( ) $\alpha j$ 1/6 ( $ST$ 34) $\alpha j$ 1/3 8

9 $E(3,6;1/2)$ $\alpha j$ 1/2 $P^{2}$ ( ) $KO$ $K3$ $(2, 4)$ $P^{5}$ IV $\Gamma$ $H_{2}=\{z\in M_{2\cross 2}(C) \frac{z-z^{*}}{2i}>0\}$ $X(3,6)arrow H_{2}/r$ $2\cross 2$- ( $X(2,6)$ ) $960$ $(4, 8)$ $(5, 10)$ $k\geq 4$ $X(k, 2k)arrow H_{k-1}/\Gamma,$ $H_{k-1}=\{z\in M_{(k-1)\cross(k-1)}(C) \frac{z-z^{*}}{2i}>0\}$ ( ) ( 2 $=Siegel$ 9

10 126 ) $(2013$ $)$ ; $(2,4)$ $(3,6)$ ( ) $C\ni x\mapsto(u_{0}(x), u_{1}(x))\in C^{2}$ (affine Schwarz map) 5 $a,$ $b,$ $c$ ( ) $x$ $1-x$ $SL$ ( $u $ $0$ ) ( ) $u +Q(x)u=0,$ $Q= \frac{1}{4}\{\frac{1-\mu_{0}^{2}}{x^{2}}+\frac{1-\mu_{1}^{2}}{(1-x)^{2}}+\frac{1+\mu_{\infty}^{2}-\mu_{0}^{2}-\mu_{1}^{2}}{x(1-x)}\}.$ $\mu_{0},$ $\mu_{1},$ $\mu_{\infty}$ $0,1,$ $\infty$ $1-\mathcal{C},$ $\mathcal{c}-a-b,$ $b-a$ 10

11 127 $\mu_{0},$ $\mu_{1},$ $\mu_{\infty}$ ( 2 ) ; 6 $C-\{O, 1\}$ $P^{1}$ $GL_{2}(C)$ $P^{1}$ $P^{1}$ : $SL_{2}(R)$ $SL_{2}(Z)$ $GL_{2}(Z[i])$ ( ) $P^{1}$ 6.1 graph ; ( ) 11

12 128 $H_{R}^{3}$ $2\cross 2$ Hermite ( ) $Her_{2}^{+}/R_{>0}$ ( ) $Her_{2}^{+}/R_{>0}\ni(\begin{array}{l}zs\overline{z}t\end{array})rightarrow(\frac{z}{t }\frac{\sqrt{st- z ^{2}}}{t})\in C\cross R_{>0}$ $Her_{2}^{+}$ $GL_{2}(C)$ $g$ $ghg^{*}$ $CU\{\infty\}\cong P^{1}$ 6.2 $A$ $2\cross 2$ $\frac{d(u,v)}{dx}=(u, v)a$ $U(x)$ Hermite $U(x)U(x)^{*}$ $hsch:x\mapsto U(x)U(x)^{*}\in H_{R}^{3}$ (hyperbolic Schwarz map) $u^{\prime/}+p(x)u +q(x)u=0$ $SL$ $(u, u )$ $2\cross 2$- ; $SL$ $u_{1}.u_{2}$ $U(x)=(\begin{array}{ll}u_{1} u_{1}^{/}u_{2} u_{2}\end{array})$ $SL$ $SL$ $(u, u )$ $SL$ ( ) ( ) $SL$ 12

13 $\bullet$ $\bullet$ 129 1: $C$ 6.3 $SL$ $u +Q(x)=0$ $\{u_{1}, u_{2}\}$ $hsch:x(2,4)\ni x\mapsto U(x)U(x)^{*}\in H_{R}^{3}$ $ Q(x) =1$ $x$ $C$ ( ) $C$ $hsch(c)$ $hsch(c)$ ( $)$ $(2, 3)$- $hsch(c)$ ( ) $(a=b=1/2, c=1)$ $C$ $ $ $\{0,1\}$ ( $\Re x=1/2$ ) ( 1 ) ( ) ( 2 ) $\{0,1, \infty\}$ $\{0,1\}$ 13

14 $\bullet$ $\bullet$ 130 2: ( ) (derived Schwarz map) : $dsch:x(2, x)\ni x\mapsto u_{1} (x):u_{2} (x)\in P^{1}$ $x$- ( ) $u -xu=0$ 14

15 131 3: ( $SL$ ) $Q=x$ $C=\{ x =1\}$ 1 ( $C$ ) ( 3 ) 6.4 $Q=x$ 15

16 132 7 $H_{R}^{3}$ 4: $(3,6)$ T. Hoffmann, W. Rossman ( ) $ $ (de Sitter) Rossman 16

一般演題（ポスター）

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