From the Carlitz exponential to Drinfeld modular forms

TL;DR

This paper introduces the elementary theory of Drinfeld modular forms for GL 2 over finite fields, connecting classical Carlitz exponential concepts to modern modular form theory, and explores future directions including vector modular forms.

Contribution

It provides an accessible introduction to Drinfeld modular forms and discusses potential developments in the theory of vector modular forms with ultrametric Banach algebra values.

Findings

Establishes foundational understanding of Drinfeld modular forms for GL 2.

Explores connections between Carlitz exponential and Drinfeld modular forms.

Suggests future research directions in vector modular forms.

Abstract

This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular group' GL 2 (Fq[$\theta$]) also providing some perspectives of development, notably in the direction of the theory of vector modular forms with values in certain ultrametric Banach algebras.