Enumeration of the Branched Zp-Coverings of Closed Surfaces

A continuous function p : S̃ → S between two surfaces S̃ and S is called a branched covering if there exists a finite set B in S such that the restriction of p to S̃ − p−1(B), p | S̃−p−1(B) : S̃− p −1(B)→ S− B, is a covering projection in the usual sense. The smallest set B of S which has this property is called the branch set. A branched covering p : S̃ → S is regular if there exists a (finite) group A which acts pseudofreely on S̃ so that the surface S is homeomorphic to the quotient space S̃/A, say by h, and the quotient map S̃→ S̃/A is the composition h◦p of p and h. In this case, the groupA is the group of covering transformations of the branched covering p : S̃→ S. We call it a branched A-covering. Two branched coverings p : S̃→ S and q : S̃ → S are equivalent if there exists a homeomorphism h : S̃→ S̃ such that p = q ◦ h. Since A. Hurwitz showed how to classify the branched coverings of a given surface [6], this area has been studied in [1, 2, 5] and their references. Recently, Kwak et al. enumerated the number of equivalence classes of the branched A-coverings of surfaces, when A is the cyclic group Zp or the dihedral group Dp of order 2p, p prime [7, 9]. In this paper, we enumerate the equivalence classes of branched A-coverings p : Si → S with branch set B, when A is the group Zp ⊕ Zp ⊕ · · · ⊕ Zp ≡ mZp, the direct sum of m copies of Zp.