Orbifolds with signature (0;k,kⁿ⁻¹,kⁿ,kⁿ)

Angel Carocca, Rubén A. Hidalgo and Rubí E. Rodríguez

Vol. 263 (2013), No. 1, 53–85

DOI: 10.2140/pjm.2013.263.53

Abstract

Two interesting problems that arise in the theory of closed Riemann surfaces are (i) computing algebraic curves representing the surface and (ii) deciding if the field of moduli is a field of definition.

In this paper we consider pairs (S,H), where S is a closed Riemann surface and H is a subgroup of Aut(S), the group of automorphisms of S, so that S∕H is an orbifold with signature (0;k,kⁿ⁻¹,kⁿ,kⁿ) where k, n ≥ 2 are integers.

In the case that S is the highest abelian branched cover of S∕H we provide explicit algebraic curves representing S. In the case that k is an odd prime, we also describe algebraic curves for some intermediate abelian covers.

For k = p ≥ 3 a prime and H a p-group, we prove that H is a p-Sylow subgroup of Aut(S), and if p ≥ 7 we prove that H is normal in Aut(S). Also, when n≠3 we prove that the field of moduli in such cases is a field of definition. If, moreover, S is the highest abelian branched cover of S∕H, then we compute explicitly the field of moduli.

Keywords

algebraic curves, Riemann Surfaces, automorphisms, field of moduli

Mathematical Subject Classification 2000

Primary: 30F10, 30F40

Secondary: 14H37

Milestones

Received: 3 April 2012

Revised: 3 February 2013

Accepted: 26 February 2013

Published: 14 May 2013