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
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(A) General Public (at most high school background, with no particular interest in science)
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(B) Scientists (some undergraduate years or curiosity and interest toward science)
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(C) Mathematicians (think of the question as an analog to Sarnak's book "Some applications of modular forms", but for automorphic forms/representations)
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(D) Number Theorists (inner motivations, e.g. Langlands program, allowed)
Type of argument
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(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.)
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(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.)
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(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.
Best Answer
The specific issue of what automorphic forms on bigger groups than $GL(2)$ over $\mathbb Q$ (for example) may tell us about automorphic forms (and L-functions) for $GL(2,\mathbb Q)$ or $GL(1,k)$ for number fields $k$ does have at least a few good answers. First, about 1960 and a little before, Klingen's proof that zeta functions of totally real number fields $k$ (and L-functions of totally even characters on such fields) have good special values at positive even integers used the idea of pulling back holomorphic Hilbert modular Eisenstein series to elliptic modular forms. (I heard G. Shimura lecture on this c. 1975, and it was quite striking.)
Another example: already in the 1960s, J.-P. Serre and others saw that holomorphic-ness of symmetric-power L-functions for $GL(2)$ holomorphic modular forms would prove Ramanujan-conjecture-type results. How to prove that holomorphy? By finding an integral representation of such L-functions, and using that. This has met with varying degrees of limited success, e.g., in papers of H. Kim and F. Shahidi.
The previous example was grounded in the general pattern of Langlands-Shahidi treatment of L-functions in terms of constant terms of cuspidal-data Eisenstein series on (necessarily larger) reductive groups. The specifica cases where various Levi-Malcev components of parabolics were products of $GL2$'s or $SL2$'s produced several "higher" L-functions for $GL2$.
As variation on that, already in the Budapest conference in 1971, Piatetski-Shapiro observed that (what we often nowadays call) Gelfand pairs could produce Euler produces via integral representations (usually involving Eisenstein series). Various success-examples of this idea included work of PiatetskiShapiro-Rallis, Shimura, myself, M. Harris, S. Kudla, various collaborations among these people, and several others, beginning in the late 1970s. E.g., I was fortunate enough to stumble upon an integral representation for triple tensor product L-functions for $GL2$ in terms of an integral representation against Siegel Eisenstein series on $Sp(3)$ (or $Sp(6)$, if one prefers). M. Harris and S. Kudla found another such integral representation that covered special value results in the "other range" (in terms of P. Deligne's conjectures).
In yet other terms, Jacquet-Lapid-Rogawski (and several others) have demonstrated that a variety of L-functions appear as periods of Eisenstein series on "larger" reductive groups. (One novelty is using relative trace formulas to exhibit Euler products when the simpler "Gelfand pair" idea is not quite sufficient.)