Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this aim has been erected as one of the topmost paradigm of research in number theory, reasons for formulating motivations for studying automorphic forms may have faded. I am seeking here for such motivations and facts leading to making so many efforts toward the theory of automorphic forms.
Why are we interested in automorphic forms/representations?
There are many different levels of reading of this question, and it might be of use to both have in mind the possibility of some of them -- for we often stay at one particular of them, depending on our interests and culture --and the different treatment they require.
Type of audience
(A) General Public (at most high school background, with no particular interest in science)
(B) Scientists (some undergraduate years or curiosity and interest toward science)
(C) Mathematicians (think of the question as an analog to Sarnak's book "Some applications of modular forms", but for automorphic forms/representations)
(D) Number Theorists (inner motivations, e.g. Langlands program, allowed)
Type of argument
(0) Meta (e.g. unification of different notions; "representations are a way to makes a group a group of motions, so endow it with a more geometrical structure, change the insight, or note that we know better many objects by their actions than by their elements", etc.)
(1) Outside motivations (e.g. PDE model many dynamics coming from physics or biology; high-dimensional spaces as efficient setting for robotics; formal logic leading to specification and verification, etc.)
(2) Math Facts (e.g. historical reasons of why automorphic forms are powerful; "such fact reduces the study of a certain kind of objects to a simpler one", how it intervenes in a classification as a relevant part of the field, etc.)
(3) Math Hopes (e.g. what a given result on automorphic forms would bring)
This is a proposition and every comment or suggestion towards those classification are welcome.
Some comments on the classification
- the "Audience" classification can be superfluous because it can be an issue concerning language and culture (replacing "PDE" and "representation" by "evolution law" or "way to move" can allow to make an argument work for both C and A as well) more than contents, but not always
- there is no necessary relation between the "Audience" tag and the "Argument" one. Even if some of those categories seems to be mutually exclusive (as A2), this is also part of the challenge to be able to present the whole diversity of motivations for automorphic forms to every audience.
The purpose of this thread if both to gather ideas to answer the natural questions of mathematicians as well as friends about the reason of spending so much time working on those topics, and also to reinvent our field in facing different aspects and motivations for the automorphic world as well as having to put words on motivations that are sometimes far away from our mind (and I do not claim they are necessary). Those could also lead to natural introductions and motivations for students.
Ideas, precise references, detailed answers or specific examples are all welcome.
Note: Even if many examples motivating modular forms or Maass forms arise in the literature, I fail for months to find something for automorphic forms without specific distinction.