On branched covering of compact Riemann surfaces with automorphisms
In this work, we give an algorithm to count the different conformal equivalence classes of compact Riemann surfaces that admit a group of automorphisms isomorphic to Z/nZ, n ? N, and that are branched coverings ofthe Riemann sphere, with signature ((n, 0); m1 ,m2 ,m3 ), m1 ,m2,m3 ? N.
By using the previous result, we count the different conformal equivalence classes of compact Riemann surfaces in the cases of coverings with signature ((p, 0); p, p, p), p ? 5 and prime, and signature ((p2, 0); p2 , p2 , p), p ? 3 and prime.
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