# Lifting of Modular Forms

## TL;DR

This paper demonstrates how to explicitly construct vector-valued modular forms for any Fuchsian group and finite-image representation by lifting scalar-valued forms from a subgroup, partially answering a longstanding question.

## Contribution

It provides a method to explicitly construct vector-valued modular forms for any admissible representation of a Fuchsian group via lifting scalar forms.

## Key findings

- Explicit constructions for vector-valued modular forms for any admissible multiplier.
- Partial answer to the existence question for nonzero component vvmf.
- Framework applicable to arbitrary Fuchsian groups and finite-image representations.

## Abstract

The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $\mathrm{G}$, for any representation $\rho:\mathrm{G} \longrightarrow \mathrm{GL}_{d}(\mathbb{C})$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $Ker(\rho)$ of $\mathrm{G}$. In this article vvmf are explicitly constructed for any admissible multiplier (representation) $\rho$, see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $\rho$ of a given $\mathrm{G}$, is there a vvmf with at least one nonzero component ?