Modular forms of weight one: Galois representations and dimension

TL;DR

This paper explores the properties of weight one modular forms, focusing on their analytic and arithmetic aspects, and discusses their connections to Galois representations, aiming to make these topics accessible to non-specialists.

Contribution

It provides an accessible introduction to weight one modular forms and their Galois representations, with a focus on clarity and motivation for non-analytically oriented number theorists.

Findings

Insights into the analytic properties of weight one modular forms

Connections established between modular forms and Galois representations

Motivations for conjectures relating to modular forms and Galois theory

Abstract

The present notes are the expanded and polished version of three lectures given in Stanford, concerning the analytic and arithmetic properties of weight one modular forms. The author tried to write them in a style accessible to non-analytically oriented number theoritists: in particular, some effort is made to be precise on statements involving uniformity in the parameters. On the other hand, another purpose was to provide an introduction, together with a set of references, consciously kept small, to the realm of Galois representations, for non-algebraists -- like the author. The proofs are sketched, at best, but we tried to motivate the results, and to relate them to interesting conjectures.