Enumeration of branched coverings of closed orientable surfaces whose branch orders coincide with multiplicity

Abstract

The number N-n,N- g,N- r of nonisomorphic n-fold branched coverings of a given closed orientable surface S of genus g with r >= 1 branch points of order n is determined. The result is given in terms of the Euler characteristic of the surface S with r points removed and the von Sterneck-Ramanujan function Phi(k, n) = Sigma((d, n) =1) exp (2 pi ikd/n). More precisely, if N-u,N-g,N-r = Sigma(l vertical bar n, l m = n) (m!lm)(v) Sigma(l)(k=1) (Phi(k, l)/n)(r) Sigma(m - 1)(s = 0) (-1)(sr) (s/m - 1)(-1).