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By Greco S., Strano R.

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Example text

Consider the elliptic integrals (of first kind) s z(s) = jdxlvx(X-1)(X-t) , tEC\{O,l}. So The integrand can be understood as the holomorphic differential form w = dx 1y on the elliptic curve E t : y2 = X(X -l)(X -t) and the integral can be taken along paths on E t joining points So and s on E. Since E t is not simply-connected, the 32 2 PICARD Curves value z( s) depends on the choice of paths. But it is unique modulo the lattice /\t of period integrals a E H1(Et,Z). Jw, a ABEL and JACOBI studied the inverse function s(z).

G6. They have been explicitly described already by PICARD [60] (with correction in [61]) and ALEZAIS. Their symplectic lifts Gi = *gi E §p(6, Z), i = 1, ... 50). 28 (i), (iii) for suitable holomorphic functions th on lB it is sufficient to check them for the generators of r( yC3). According to our claim th = Thba we have now only to look for holomorphic functions T h on H3 satisfying the six restricted functional equations Th 0 Gi = (detg;)2. jg; . Th on lB C H 3 , i = 1, ... ,6. 43) Step 2: RIEMANN'S Theorem.

22 in order to find the "typical period points" 0 by calculating IIi l . II = (E310). PICARD carried out this calculation in [60]. 64) -u, with u = A2/Ab V = pA3/AI (AI cannot be equal to 0). 65) 2Re(v) + lul 2 < 0 . 1 Ball Uniformization of Algebraic Surfaces Let X be a normal complex algebraic surface. We assume for a moment that X is compact. Then it supports only a finite number of singularities. Furthermore we assume that all these singularities are of quotient or ball cusp type. 1. A surface germ (U, P), U an open analytic surface (neighbourhood of P), P E U, is called a quotient singularity, if (U, P) is the finite quotient (V, O)/G of a smooth germ (V, 0).

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