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 \((\begin{smallmatrix}a&0\\ 0&b\end{smallmatrix})\) and \((\begin{smallmatrix} 0&b\\ a&0\end{smallmatrix})\). 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:
\[ \begin{array}{|c|c|} \hline H&\text{structure of $S$}\\ \hline =\{1_2\}&\text{abelian surface}\\ \hline \neq\{1_2\}&\text{K3 structure}\\ \hline \neq H_1 &\begin{array}{c|c} H=\langle[1,e_n]\rangle &\begin{array}{c|c} F(G)=\emptyset&\text{hyperelliptic surface}\\ \hline F(G)\not=\emptyset &\text{elliptic ruled surface}\\ \end{array}\\ \hline H=\langle[-1,1],[1,-1]\rangle\text{ or } \langle[-1,1]^*,[1,-1]\rangle &\begin{array}{c|c} F(G)=\text{ finite}&\text{Enriques surface}\\ \hline F(G)\supset\text{ curve}&\text{rational surface}\\ \end{array}\\ \hline \text{except the above}&\text{rational surface} \end{array}\\ \hline \end{array} \]
Especially we have that \(S\) is a rational surface if \(^\sharp H>24\) or the degree of the eigenvalues of \(M\) is 4.