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Branched coverings of Riemann surfaces whose branch orders coincide with the multiplicity. (English) Zbl 0701.30042
Suppose that S and T are compact Riemann surfaces and \(\pi\) : \(T\to S\) is a branched covering of multiplicity n. We consider a finite subset \(Y=\{y_ 1,...,y_ r\}\) from S such that \(\pi\) : \(T\setminus \pi^{- 1}(Y)\to S\setminus Y\) is an unbranched covering, and denote by \(S^ p_ k\) the number of branch points of order k of the mapping \(\pi\) in the preimage \(\pi^{-1}(y_ p)\), where \(k=1,...,n\) and \(p=1,...,r\). We call the matrix \(\sigma =(S^ p_ k)^{p=1,...,r}_{k=1,...,n}\) the ramification type of the covering \(\pi\). The coverings \(\pi\) : \(T\to S\) and \(\pi ': T'\to S\) are called equivalent if there exists a homeomorphism h: \(T\to T'\) such that \(\pi =\pi '\circ h.\)
The Hurwitz enumeration problem is to determine the number \(N_{n,g,\sigma}\) of nonequivalent coverings of the multiplicity n over a compact Riemann surface S of genus g with a given ramification type \(\sigma\). A. Hurwitz [Math. Ann. 39, 1-61 (1891), Math. Ann. 55, 53-66 (1902)] constructed a generating function for the number of nonequivalent coverings over the Riemann sphere having only simple branch points (of order 2) and proved that the number of such coverings can be expressed in terms of the characters of irreducible representations of symmetric groups. H. Röhrl [Trans. Am. Math. Soc. 107, 320-343 (1963; Zbl 0115.067)] obtained rough estimates for the number of nonequivalent coverings with given ramification type. The full description of the Hurwitz enumeration problem is contained in the author’s paper [Sib. Mat. Zh. 25, No.4(146), 120-142 (1984; Zbl 0598.30058)]. In the present paper the last result for coverings in the title is reduced to a form convenient for practical use.
Reviewer: A.D.Mednykh

MSC:
30F10 Compact Riemann surfaces and uniformization
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