Quotients of abelian surfaces
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\hss \vtop{\hsize=.6\hsize \centerline{\strut hyperelliptic surface}}} \hrule \line{\vtop{\hsize=.4\hsize \centerline{\strut \(F(G)\not=\emptyset\)}} \hss\vrule\hss \vtop{\hsize=.6\hsize \centerline{\strut elliptic ruled surface}}}}} \hrule \line{\vtop{\hsize=.4\hsize \centerline{\strut \(H=\langle[-1,1],[1,-1]\rangle\)} \centerline{\strut or \(\langle[-1,1]^*,[1,-1]\rangle\)}} \vrule \vtop{\hsize=.6\hsize \line{\vtop{\hsize=.4\hsize \centerline{\strut \(F(G)=\) finite}} \hss\vrule\hss \vtop{\hsize=.6\hsize \centerline{\strut Enriques surface}}} \hrule \line{\vtop{\hsize=.4\hsize \centerline{\strut \(F(G)\supset\) curve}} \hss\vrule\hss \vtop{\hsize=.6\hsize \centerline{\strut rational surface}}}}}\hrule \line{\vtop{\hsize=.4\hsize \centerline{\strut except the above}} \hss\vrule\hss \vtop{\hsize=.6\hsize \centerline{\strut rational surface}}}} \hss\vrule} \hrule}} Especially we have that \(S\) is a rational surface if \(^\sharp H>24\) or the degree of the eigenvalues of \(M\) is 4.
scientific article; zbMATH DE number 785370