Automorphisms group of generalized Fermat curves of type

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• 作者：Yolanda Fuertes^a ; ^{yolanda.fuertes@uam.es} ; Gabino Gonzá ; lez-Diez^a ; ^{gabino.gonzalez@uam.es} ; Rubé ; n A. Hidalgo^b ; ^{ruben.hidalgo@usm.cl} ; Maximiliano Leyton^c ; ^{leyton@inst-mat.utalca.cl}
• 关键词：30F10 ; 30F40 ; 14H37 ; 14H45
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2013
• 出版时间：October, 2013
• 年：2013
• 卷：217
• 期：10
• 页码：1791-1806
• 全文大小：538 K

文摘

The determination of the full group of automorphisms of a closed Riemann surface is in general a very complicated task. For hyperelliptic curves, the uniqueness of the hyperelliptic involution permits one to compute these groups in a very simple manner. Similarly, as classical Fermat curves of degree admit a unique subgroup of automorphisms isomorphic to , the determination of the group of automorphisms is not difficult.

In this paper we consider a family of non-hyperelliptic Riemann surfaces, obtained as the fibre product of two classical Fermat curves of the same degree , which exhibit behaviors of both elliptic and hyperelliptic curves. These curves, called generalized Fermat curves of type , are the highest regular abelian branched covers of orbifolds of genus zero with four cone points, all of the same order . More precisely, a generalized Fermat curve of type is a closed Riemann surface admitting a group , called a generalized Fermat group of type , so that and is an orbifold with signature . In this paper we prove the uniqueness of generalized Fermat groups of type . In particular, this allows the explicit computation of the full group of automorphisms of .