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On generalized superelliptic Riemann surfaces

Rubén A. Hidalgo, Saúl Quispe, Tanush Shaska

I-AIMS paper 2016-6 · Published in Transformation Groups (2025) · 2025

A conformal automorphism τ, of order n ≥ 2, of a closed Riemann surface 𝒳, of genus g ≥ 2, which is central in Aut(𝒳) and such that 𝒳/⟨τ⟩ has genus zero, is called a superelliptic automorphism of level n. If n = 2, then τ is the hyperelliptic involution and it is known to be unique. In this paper, for the case n ≥ 3, we investigate the uniqueness of the cyclic group ⟨τ⟩. Let τ₁ and τ₂ be two superelliptic automorphisms of level n of 𝒳. If n ≥ 3 is odd, then ⟨τ₁⟩ = ⟨τ₂⟩. If n ≥ 2 is even, the same uniqueness result holds, up to some explicit exceptional cases. We also provide conditions for these surfaces to be definable over their field of moduli.

MSC classes: Primary 14H37; Secondary 14H45, 30F10

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