Quotients of Abelian Surfaces

Let L be an abelian function field of two variables over C, and K be a Galois subfield of L, i.e., L is a finite algebraic Galois extension of K. We classify such K by a suitable complex representation of the Galois group G = Gal (L/K). Let A be the abelian surface with the function field L. Since g e G induces an automorphism of A, we have a complex representation gz = M(g)z + t(g\ where M(g) e GL2(C), z e C , and t(g) e C. Fixing the representation, we put G0 = {g eG\M(g) is the unit matrix}, H = {M(g)\g e G} and H1 = {M(g)e H\det M(g) = 1}. Then we have the following exact sequences of groups: