# zbMATH — the first resource for mathematics

Die Automorphismengruppen hyperelliptischer Kurven. (The groups of automorphisms of hyperelliptic curves). (German) Zbl 0588.14022
Let $$\Gamma$$ be a hyperelliptic curve over an algebraically closed field of characteristic 0 and G a group of automorphisms of $$\Gamma$$ containing the canonical involution. The authors study the group theoretic structure of G and relate it to the ramification behavior of the morphism $$p:\quad \Gamma \to \Gamma /G.$$ Using the well known classification of finite subgroups of PGL(2,$${\mathbb{C}})$$, they first obtain a classification of the possible ramification types of p. The computation of the number of involutions in G then shows the main result (theorem 5.1): Up to isomorphism, G depends only on the ramification type of p.
Actually, the authors do much more: For each type, the corresponding group is determined, and a way to obtain normal forms is indicated. The proofs use complicated case by case considerations which are carried out only in typical special cases.
Reviewer: E.-U.Gekeler

##### MSC:
 14H45 Special algebraic curves and curves of low genus 14L30 Group actions on varieties or schemes (quotients) 14H30 Coverings of curves, fundamental group
Full Text:
##### References:
 [1] IWASAWA, K.: On solvable extensions of algebraic number fields. Ann. of Math.58, 548-572 (1953) · Zbl 0051.26602 · doi:10.2307/1969754 [2] ROTMAN, J.J.: An Introduction to homological Algebra, New York-San Francisco-London, Academic Press 1979 · Zbl 0441.18018 [3] TSUJI, R.: On conformal Mapping of a hyperelliptic Riemann Surface onto itself, Kodai Math. Sem. Reports10, 127-136 (1958) · Zbl 0085.06903 · doi:10.2996/kmj/1138844026 [4] WEBER, H.: Lehrbuch der Algebra, Bd.II, 2.Aufl. Braunschweig 1899 · JFM 30.0093.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.