Title:

Decomposition results for rings of modular forms

Abstract: 

Modular forms can be either seen as highly symmetric functions on the upper half-plane or as invariants of elliptic curves (with possible extra structure). To each symmetry group, we get a corresponding ring of modular forms. These rings are central objects in number theory and are in general difficult to compute. 

After an introduction to modular forms and elliptic curves, we will demonstrate that the additive structure on these rings is much easier to understand and provide in many cases decompositions into a few simple pieces.