Symbolic Computation for Rankin-Cohen Differential Algebras

Zagier pursues the study of the algebraic structure that this operation gives to the ring of modular forms viewed as a differential module, observing that it is “not clear how far we would have to go to get the first relation or how much further to ensure that all subsequent relations obtained would be consequences of ones already found”. Instead of determining the relations, he proposes the abstract concept of a Rankin-Cohen differential algebra and gives a “partial structure theorem”. In this work, we propose to use Symbolic Computation to detect minimal sets of relations for the case study of Γ(7), the modular group of the Klein curve, the only algebraic curve of genus three with the largest possible group of automorphisms, motivated by the first-named author’s Ph.D. Thesis [Farr], which uses techniques that allow us to deal explicitly with certain modular forms. We apply the theory of Gröbner bases (as in [EGÔP]) to control the weight of the relations, and then perform a search (implemented in Maple syntax) for complete, minimal sets ot relations weight-by-weight; in consequence, our results only reach a(ny) finite given weight, but these relations are of interest, given the large number of open problems that concern the Klein curve (more specifically stated below).