%0 Journal Article
%J Int. Math. Res. Not. 2004, no. 32, 1619-1683
%D 2004
%T Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation
%A Victor Z. Enolski
%A Tamara Grava
%X We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the Szego kernel of certain Riemann surfaces branched over the given 2m+2 points. These Riemann surfaces are constructed from a permutation representation of the symmetric group S_N to which the quasi-permutation monodromy representation has been reduced. The permutation representation of our problem generates the cyclic subgroup Z_N. For this reason the corresponding Riemann surfaces of genus N(m-1) have Z_N symmetry. This fact enables us to write the matrix entries of the solution of the NxN Riemann-Hilbert problem as a product of an algebraic function and theta-function quotients. The algebraic function turns out to be related to the Szego kernel with zero characteristics. From the solution of the Riemann- Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The tau-function of the Schlesinger system is computed explicitly. The rank 3 problem with four singular points (0,t,1,\\\\infty) is studied in detail. The corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi\\\'s theta-function with modulus T=T(t), Im(T)>0. The function T=T(t) is invertible if it belongs to the Siegel upper half space modulo the subgroup \\\\Gamma_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system.
%B Int. Math. Res. Not. 2004, no. 32, 1619-1683
%G en_US
%U http://hdl.handle.net/1963/2540
%1 1579
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-21T10:08:03Z\\nNo. of bitstreams: 1\\n0306050v2.pdf: 587970 bytes, checksum: 3d350364af41be2317176944f1d85868 (MD5)
%R 10.1155/S1073792804132625