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Dr. Katz DEq Homework Solutions 97

# Dr. Katz DEq Homework Solutions 97 - (x 1(x 2 with...

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Algebraic Solutions of Differential Equations 97 are conjugate (by a diagonal matrix) in GL(2, C). As already noted, Co(a,b,c)=Co(a+r,b+s,c+t), and as nl(P~-{0, 1, oo}; is the free group on 70 and 71, the representations in question are indeed conjugate. Q.E.D. 6.8.0. In this section, we study the relation between the hypergeometric equation and curves of the form y"= xa(x-1)b(x--2) c. This relation was known to Euler, in the form of his integral representation (cf. [1], p. 115, (6), or [45], p. 293) 9O F(~, fl,~; 2)= F(7) 5x=_~(x_l)~_r r(fl) r(7 - fl) It was reconsidere recently by Messing ([32]), from the conception of whose manuscript we have borrowed heavily. (6.8.1) Let n, a, b, c be strictly positive integers. We denote by X(n;a, b, c) the spectrum of the smooth C(2)-algebra (6.8.1.0) A(n; a, b, c)= C(2) Ix, y, 1/y]/(y" - x"(x - 1) b (x - 2)c) C(2) [x, y, 1/y], which is finite and &ale over (6.8.1.1) B = C (2) [x] [ 1/x
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Unformatted text preview: (x - 1)(x - 2)], with basis (6.8.1.2) 1, 1/7, l/y 2, . .., 1/y "-1 . To any n-th root of unity ~/~,, we assosciate the C(2)-automorphism of X(n; a, b, c) given by XF--~ X (6.8.1.3) y~--~ ~y. This defines an action of/~, on X(n; a, b, c). Let X denote the "identical" character of /~,, i.e., Z(~)=~, and for any integer L let X(f) denote the character of/4, given by (6.8.1.4) ;((~) (~)= ~e. This basis (6.8.1.2) of A(n; a, b, c) over B is just its "isotypique" decomposition according to the characters of/J,: (6.8.1.5) P(X(-E)) A(n; a, b, c)=y-eB = lim y-e-,, C(2)[x]. F~ 9O The de Rham cohomology group H~R(X(n; a, b, c)/C(2)) is the co- kernel of the exterior differention map (6.8.1.6) A(n; a, b, c) d , 12~(,;~,b,c)/c~ ~ because X(n; a, b, c) is affine and smooth of relative dimension one over C(2). The module f2]~,;,,h.~)/C~Z) is free over A(n; a, b, c) with basis dx. 7 Inventiones math., VoL 18...
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