HOMOLOGY CLOSED RIEMANN SURFACES

作者：Hidalgo Ruben A^*

来源：The Quarterly Journal of Mathematics, 2012, 63(4): 931-952.

DOI：10.1093/qmath/har026

A closed Riemann surface of genus at least 2 can be described by many different objects, for instance, by algebraic curves and by torsion-free co-compact Fuchsian groups. If a torsion-free co-compact Fuchsian group is provided, then in general it is a difficult task to obtain an algebraic curve describing the surface uniformized by the given Fuchsian group. We consider those closed Riemann surfaces appearing as a maximal Abelian cover of an orbifold; called homology closed Riemann surfaces. These surfaces are uniformized by the (torsion-free) derived subgroup of certain co-compact Fuchsian groups of genus zero. We describe a general method to obtain algebraic curves for homology closed Riemann surfaces. We make this explicit for the case of (i) hyperelliptic homology closed Riemann surfaces and (ii) homology closed Riemann surfaces being the highest Abelian covers of orbifolds with triangular signature. The structure of the cover groups are also provided. As a simple application, we note that if S is a closed Riemann surface and A %26lt; Aut(S) is an Abelian group so that S/A has triangular signature, then S (and a Galois cover with A as its deck group) can be defined over Q. This says, in other words, that Abelian regular dessins d%26apos;enfants are definable over Q. We also prove that if two orbifolds with triangular signatures have conformally equivalent homology closed Riemann surfaces, then they are necessarily conformally equivalent as orbifolds.

• 出版日期2012-12