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The theory of automorphic forms is stated in modern language as a function on the adelic points of some reductive group satisfying certain properties (see for instance, Chapter 7 of An Introduction to the Langlands Program by Bump et al). This includes modular forms as a special case, whose classical theory well preceded the concept of adeles. When was this "adelic" viewpoint on automorphic forms and modular forms first developed?

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    $\begingroup$ By the time of Proc. Symp. Pure Math. IX (1966), about algebraic groups and discontinuous groups, this point of view was known. See the references in Godement’s papers there to earlier work by subsets of Gelfand, Graev, and Piatetski-Shapiro. $\endgroup$
    – KCd
    Aug 19, 2022 at 23:29
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    $\begingroup$ The first "published" appearance of adeles in the study of algebraic groups was Weil's 1961 IAS lecture notes Adeles and algebraic groups: see albert.ias.edu/bitstream/handle/20.500.12111/8024/… and this was published much later as a book by Birkhauser in 1982. $\endgroup$
    – KCd
    Aug 20, 2022 at 0:17
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    $\begingroup$ The answer by Emerton to the MO question mathoverflow.net/questions/75709/… should interest you. $\endgroup$
    – KCd
    Aug 23, 2022 at 5:56

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Already in 1950, Iwasawa and Tate's modernization of Hecke's theory "for GL(1)" was aggressively "adelized".

My recollection of many peoples' attitudes about that, in the 1970's, was that "the adelic viewpoint" was "an option", maybe a stylistic choice, and not at all necessarily advantageous.

Similarly, in the late 1940's and early 1950's, Godement and others were hoping for a very general representation theory of topological groups... but/and Harish-Chandra's pick-up of Wigner's and Bargmann's and others' physics-motivated explicit repn theory for small classical groups was a big change in attitude about the potential of generalization.

Meanwhile, already in the late 1950's, Gelfand and collaborators were indeed working to develop an understanding of automorphic forms on adele groups, emphasizing $GL(2)$. My recollection is that, in the mid 1970's, people still did view this as "an option". After all, R. Langlands' SLN 544, apparently composed in the mid-to-late 1960s) is not at all adelic.

S. Gelbart's 1974 orange Princeton-series book was in many ways aggressively forward-looking, talking about Langlands' ideas interpreted on adele groups, even if it was needlessly coy about various modern ideas.

(Yes, indeed, in the early 1960s, A. Weil's papers about Siegel-Weil formulas, oscillator repn, and so on, certainly did embrace "adelic" viewpoints. To my recollection, this perspective was not hugely popular. Similarly, T. Tamagawa's work that emphasized "an adelic viewpoint", was considered avante-garde...)

In terms of "fashion", to say the least, there seemed to have been no compulsion to "do things adelically" until after the 1977 Corvallis conference. :)

It does indeed seem to be true that there is not usually any effort exerted to persuade or explain why it's advantageous to "take an adelic viewpoint". As with many "fashions", a critical-thinking person could suspect that there is actually not good reason to adopt the style-of-the-day. In fact, to the contrary, as I eventually realized, there are many good reasons to "use an adelic viewpoint" when possible.

On another hand, some relatively recent work of Winnie Li (and others) about arithmetic of modular forms for non-congruence subgroups shows that "the adelic viewpoint" does not actually cover everything.

In summary, although the "adelic viewpoint" was well-documented by the early 1960's (Gelfand et al), people didn't necessarily believe it was an essential idea until maybe 1970. For example, the Godement-Jacquet treatment of standard L-functions for $GL(n)$, following Iwasawa-Tate.

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  • $\begingroup$ Hopefully it's alright to ask this semi-related follow-up - when did the theory of modular forms first become part of representation theory? $\endgroup$ Aug 20, 2022 at 15:25
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    $\begingroup$ @AntonHilado, I think it's not so much "become a part of repn theory", but, rather, when did people realize that ideas of repn theory were very useful in studying afms. Especially the big theorems about admissibility and factoring into (restricted) tensor products of local repns. By the early 1970s, though some experts could see it coming a few years before. $\endgroup$ Aug 20, 2022 at 21:58
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    $\begingroup$ @AntonHilado you might want to see Gelfand's 1962 ICM address "Automorphic Functions and the Theory of Representations, on pp. 74-85 of mathunion.org/fileadmin/ICM/Proceedings/ICM1962.1/…. $\endgroup$
    – KCd
    Aug 21, 2022 at 13:59

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