TY - JOUR
T1 - Thomae type formulae for singular Z_N curves
JF - Lett. Math. Phys. 76 (2006) 187-214
Y1 - 2006
A1 - Victor Z. Enolski
A1 - Tamara Grava
AB - We give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szego kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki\\\'s proof [31], obtained for non-singular ZN curves.
UR - http://hdl.handle.net/1963/2125
U1 - 2118
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation
JF - Int. Math. Res. Not. 2004, no. 32, 1619-1683
Y1 - 2004
A1 - Victor Z. Enolski
A1 - Tamara Grava
AB - 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.
UR - http://hdl.handle.net/1963/2540
U1 - 1579
U2 - Mathematics
U3 - Mathematical Physics
ER -