Quotients of abelian surfaces
type: Item
instance of: scholarly article
title: Quotients of abelian surfaces
zbMATH Open document ID: 0852.14016
author: Hisao Yoshihara
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1996-12-08T00:00:00Z
review text: Let \(L\) be an abelian function field of two variables over \(\mathbb{C}\) and \(K\) a Galois subfield of \(L\), i.e., \(L\) is a finite algebraic Galois extension of \(K\). We classify such \(K\)'s by a suitable complex representation of the Galois group \(G= \text{Gal} (L/K)\), which turns out to be solvable. Let \(A\) be the abelian surface with the function field \(L\). For \(g\in G\), we have a complex representation \(gz= M(g)z+ t(g)\), where \(M(g)\in GL_2 (\mathbb{C})\), \(z\in \mathbb{C}^2\), and \(t(g)\in \mathbb{C}^2\). Fixing the representation, we put \(G_0= \{g\in G\mid M(g)\) is the unit matrix\}, \(H= \{M(g) \mid g\in G\}\) and \(H_1 = \{M(g)\in H\mid \text{det } M(g)= 1\}\). Let \(S\) be a relatively minimal model of \(A/G\) and \(F(G)\) denote the set of fixed points of \(G\). Let \([a, b]\) and \([a, b]^*\) denote the matrices \({{a\;0} \choose {0\;b}}\) and \({{0\;b} \choose {a\;0}}\). Let \(1_2\) denote the unit matrix. If a group is generated by \(g_1, \dots, g_m\), then it is denoted by \(\langle g_1, \dots, g_m \rangle\). Put \(e_n= \exp (2\pi \sqrt {-1} /n)\). Then the main result is stated as follows, where \(n= 2, 3, 4\) or 6: \centerline{\vtop{\hsize=.9\hsize \hrule \line{\vrule\hss \vtop{\hsize=.2\hsize\centerline{\strut\(H\)}} \hss\vrule\hss \vtop{\hsize=.8\hsize \centerline{\strut structure of \(S\)}} \hss\vrule} \hrule \line{\vrule\hss \vtop{\hsize=.2\hsize \centerline{\strut\(=\{1_2\}\)}} \hss\vrule\hss \vtop{\hsize=.8\hsize \centerline{\strut abelian surface}} \hss\vrule} \hrule \line{\vrule\hss \vtop{\hsize=.2\hsize \centerline{\strut \(\not=\{1_2\}\)}} \hss\vrule\hss \vtop{\hsize=.8\hsize \centerline{\strut K3 structure}} \hss\vrule} \hrule \line{\vrule\hss \vtop{\hsize=.2\hsize \centerline{\strut \(\not=H_1\)}} \hss\vrule\hss \vtop{\hsize=.8\hsize \line{\vtop{\hsize=.4\hsize \centerline{\strut \(H=\langle[1,e_n]\rangle\)}} \vrule \vtop{\hsize=.6\hsize \line{\vtop{\hsize=.4\hsize \centerline{\strut \(F(G)=\emptyset\)}} \hss\vrule…
Mathematics Subject Classification ID: 14K99
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 785370
zbMATH Keywords: quotient surface
zbMATH Keywords: abelian function field
zbMATH Keywords: abelian surface
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195164795
OpenAlex ID: W1988446232
cites work: Finite automorphism groups of complex tori of dimension two
cites work: Q3253828
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cites work: Normal Two-Dimensional Singularities. (AM-71)
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cites work: Degree of irrationality of abelian surfaces
cites work: Structure of complex tori with the automorphisms of maximal degree
cites work: Galois subfields of abelian function field of two variables
DOI: 10.2977/PRIMS/1195164795
description: scientific article; zbMATH DE number 785370