Invariant rational forms for correspondences of curves

Text: We study the invariant rational 1-forms which correspondences of curves admit. We show that in the case of unequal degrees of the morphisms, the number of poles and zeroes of an invariant rational 1-forms is uniformly bounded. This bound is explicitly given in terms of the degrees of the morphisms and the genus of the curves and this bound is independent of the characteristic of the base field. This result is however not true in the case of equal degrees of the morphisms. In the case of correspondences of the projective line P1, when both the morphisms are conjugates to polynomial maps, under an assumption of sufficient separateness of the degrees, we prove that there are only two kinds of invariant rational 1-forms that they can admit. In one of the above two cases, the morphisms arise from the multiplicative maps. Video: For a video summary of this paper, please visit http://youtu.be/Ee9NRWL0_TM.