Title: Differential equations for Riemann and Prym theta-functions
Abstract: Recall that any theta-function constructed by means of the symmetric matrix {Bij} E GL (2,C), such that Re B is negative definite, has the form O(w I B) = ~ exp 2 ~ Bqnini + ~ njw , ni,...,nj=--oo i,j=l j=l where w = (wl, 9 .., wa) E C g, and has the periodic properties O(z + 2rilk) = O(z), O(z + f~) = exp ( _l~Bkk --z~)O(z), where lk = (0,...,0, 1,0...,0), f~ = Blk.The method of algebro-geometric integration of differential equations discovered by S. P. Novikov, B. A. Dubrovin, I. M. Krichever, P. Lax, V. B. Matveev and others demonstrates that, if the matrix B is connected with an algebraic curve, then its theta-functions (the Riemann and Prym theta-functions) satisfy some physical differential equations.Following [1, 2] in this note, we describe infinite systems of independent differential equations for Riemann and Prym theta-functions.