Generalized modular forms including the weak Maass forms,the Ramanujan's Theta functions and the Tau function

TL;DR

This paper offers a geometric and algebraic reinterpretation of modular forms, linking weak Maass forms, Ramanujan's Theta functions, and the Tau function within a unified framework connected to the Langlands program.

Contribution

It introduces a generalized framework for modular forms that encompasses weak Maass forms, mock theta functions, and cusp forms, highlighting their interrelations.

Findings

Weak Maass forms are connected to Ramanujan's mock theta functions.

A geometric interpretation of modular forms is developed.

The framework relates modular forms to the Langlands program.

Abstract

The modular forms are revisited from a geometric and an algebraic point of view leading to a geometric interpretation of the weak Maass forms connecting them to the Ramanujan Mock Theta functions and to the cusp forms generated from the Langlands global program.