ag.algebraic geometry - Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?

Here is a vague suggestion as to why there might not be a "canonical" choice of basis. I will presume that you already have in mind a choice of basis for the homology of your elliptic curve. Write Gamma_{1,2} for the group of isotopy classes of oriented diffeomorphisms of the twice-punctured elliptic curve (i.e. the mapping class group) and Gamma_1 = SL_2(Z) for the mapping class group of the unpunctured elliptic curve. Then the natural surjection Gamma_{1,2} -> SL_2(Z) has a kernel G, which can be thought of as the braid group on two strands on the torus (I'm sure Tom can say more than I can, offhand, about what this group is.)

Now G acts on H_1(X,Z), where X is your spectral curve, and I think for your basis to be "canonical" it would want to be fixed by this. But I don't immediately see why this action would be trivial. (Of course, by construction it acts trivially on the natural quotient H_1(E,Z).)