Multiplying Modular Forms

TL;DR

This paper explores the connection between the algebraic structure of modular forms and the representation theory of real semisimple Lie groups, providing explicit constructions of intertwining operators that reflect the ring structure.

Contribution

It explicitly links the ring structure on modular forms with branching problems in representation theory and constructs a family of intertwining operators illustrating this connection.

Findings

Constructed explicit intertwining operators for discrete series representations.

Connected the ring structure of modular forms with representation branching problems.

Discussed analytic subtleties and conditions like discrete decomposability.

Abstract

The space of elliptic modular forms of fixed weight and level can be identfied with a space of intertwining operators, from a holomorphic discrete series representation of SL2(R) to a space of automorphic forms. Moreover, multiplying elliptic modular forms corresponds to a branching problem involving tensor products of holomorphic discrete series representations. In this paper, we explicitly connect the ring structure on spaces of modular forms with branching problems in the representation theory of real semisimple Lie groups. Furthermore, we construct a family of intertwining operators from discrete series representations into tensor products of other discrete series representations. This collection of intertwining operators provides the well-known ring structure on spaces of holomorphic modular forms. An analytic subtlety prevents this collection of intertwining operators from directly yielding a ring structure on some spaces of modular forms. We discuss discrete decomposability, and its role in constructing ring structures.