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Multiple prime covers of the Riemann sphere. (English) Zbl 1108.30030

A cyclic \(q\)-gonal surface is a compact Riemann surface \(X\) of genus \(\geq 2\) which admits a cyclic group of automorphisms \(C_q\) of prime order \(q\) such that \(X/C_q\) has genus \(0\). The group \(C_q\) is called the \(q\)-gonal group for \(X\). If a \(q\)-gonal surface \(X\) is also a \(p\)-gonal surface for some prime \(p\not=q\), then \(X\) is called a multiple prime surface. The aim of this paper is to classify all multiple prime surfaces, that is, to find the automorphism group and the signature for the normalizer of a surface group for each such surface. As an application, the author proves the following:
Theorem: A cyclic \(q\)-gonal surface can be \(p\)-gonal for at most one other prime \(p\).
Theorem: If \(X\) is a multiple prime surface which is cyclic \(q\)-gonal and cyclic \(p\)-gonal, then any element from a \(q\)-gonal group commutes with any element from a \(p\)-gonal group.

MSC:

30F10 Compact Riemann surfaces and uniformization
14H37 Automorphisms of curves
14H30 Coverings of curves, fundamental group
30F60 Teichmüller theory for Riemann surfaces
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References:

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