To let $ f: X rightarrow Y $ be an Abelian Galois coverage of non-singular complete curves over algebraic $ k $where the order of the Galois group $ A $ is coprime to the characteristic of $ k $, We can display the function field $ K (X) $ As a $ K (Y) $ Vector space, and through Galois theory we know its structure as $ k (A) $ Module is given by $ K (Y) (A) $,

Given our assumptions, we see that $ K (X) $ has a basis of eigenvectors for $ A $, corresponding to the various one-dimensional representations of $ A $ about $ k $, It has an explicit basis from $ f_ lambda in K (X) $ so that $ g.f_ lambda = lambda (g) .f $ to the $ lambda $ a character of $ A $ with value in $ k $, So the dividers shared this $ f_ lambda $ are canonical and we have one for each character of $ A $,

My question is, can you describe these dividers more geometrically?

I'm vaguely aware that all coverings of this form should come from Jacobian isogenies, but my knowledge of Abelian curve coverings is not great, so excuse me if this question is really simple.