Modular forms via invariant theory

TL;DR

This paper presents methods to construct modular forms from existing forms using invariant theory, including generalizations of the Rankin-Cohen bracket and approaches for vector-valued forms of higher genus.

Contribution

It introduces two new invariant-theoretic techniques for constructing modular forms, extending classical methods to more complex cases.

Findings

Generalizes the Rankin-Cohen bracket for elliptic modular forms

Provides methods for vector-valued modular forms of higher genus

Offers practical tools for modular form construction

Abstract

We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen bracket, while the second one deals with vector-valued modular forms of genus greater than one.