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To a number theorist automorphic forms appear to be adelic point-counting generating functions for arithmetic schemes. This is what the conjectured equality of their $L$ functions tells us.

The fact that these generating functions are highly symmetric (Hecke action) seems to correspond to the fact that arithmetic schemes are highly symmetric (Galois action). That is why we try to match up Hecke symmetries with Galois symmetries when proving their relationship.

If we think of algebraic groups together with their arithmetic subgroups as comprising all the arithmetic symmetries there are, it seems reasonable that whenever these symmetries show up in nature, for example in the form of Galois action on an arithmetic scheme, we want to look up this comprehensive dictionary of arithmetic symmetries to see where it occurs in order to understand it.

Is this a clue to the awesome mystery that is the arithemetic Langlands correspondence? Does this count as a moral justification for the conjecture, Langlands' achievement being not only intuiting that auto-morphic actions of adelic algebraic groups are all the arithmetic symmetries there are, but also providing an indexing for the dictionary for this purpose by boldly extrapolating from class field theory after recasting it as the $GL_1$ case of a vast conjecture based on scant evidence at the time?

Even if the intuition is valid as far as it goes, it leaves big questions to think about, starting with "why only reductive groups?" and "what are the qualifiers that need to go in front of arithmetic symmetries to make it more precise?", which I'd love to understand.

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    $\begingroup$ This seems to me to be a classic example of "begging the question" (in the precise sense of the phrase). You are using the term "arithmetic symmetries" for two completely different things (actions of Galois on varieties, and Hecke actions on auto forms). Giving two things the same name doesn't make them the same! Saying that "they should be related because they are both examples of 'arithmetic symmetries'" is only natural in the light of the Langlands program itself which gives an objective justification for regarding these concepts as related. $\endgroup$ Feb 7 '21 at 17:17
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    $\begingroup$ @DavidLoeffler, Perhaps it is not clear, but I am saying - or really wondering - that algebraic groups+arithmetic subgroups provide all the "arithmetic symmetries" there are (sort of by definition?) prior to any Langlands program. Langlands program is then seen as saying that if you have any case of arithmetic symmetry arising in some other guise, e.g., Galois action on arithmetic schemes, it must be found among them. How precisely to locate them is, of course, his genius. Isn't a good chunk of math about taking two things that appear in different guises and showing them to be the same? $\endgroup$
    – P.H.
    Feb 7 '21 at 17:42
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    $\begingroup$ @DavidLoeffler, In math as in so many other areas of inquiry intuition often precedes demonstration. A conjecture is a precisely formulated intuition. A Langlands may intuit that seen in the right way automorphic representation (if thought of as constituents of $G(\mathbb{A}_K)$ acting on $G(K)$ \ $G(\mathbb{A}_K)$ for number field $K$ are examples of arithmetic symmetry as much as Galois actions are, so two should be related. As symmetries are encoded in group actions, arithmetic symmetries should be encoded in arithmetic actions of algebraic groups. Self (automorphic) actions are universal. $\endgroup$
    – P.H.
    Feb 7 '21 at 18:21
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    $\begingroup$ @P.H. I don't pretend to truly understand the content of your question, but I really like the way you seem to conceive mathematics. $\endgroup$ Feb 7 '21 at 20:59
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    $\begingroup$ @Heavensfall, It is not just algebraic groups, but together with their arithmetic subgroups. The precise way to put it is 𝐺(𝔸_𝐾) acting on 𝐺(𝐾) \ 𝐺(𝔸_𝐾) for number fields $K$, which takes arithmetic of $K$ into account. Sorry for using impressionistic language in the question. My aim was to emphasize intuition, while hopefully not leading it astray. The purpose of the question is in part to ask for more clarity from the experts. If you like, algebraic groups are algebraic symmetries, restricted here to arithmetic contexts, namely 𝔸_𝐾. Perhaps I should call them arithmetic-algebraic. $\endgroup$
    – P.H.
    Feb 8 '21 at 2:02
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One way to interpret this question:

Should we be able to guess or morally justify the philosophy behind the Langlands conjecture without analyzing important examples and special cases?

I'm going to answer this one "no, we shouldn't". The Langlands conjectures emerged from Langlands' construction of the $L$-function of an automorphic form and his prior knowledge of Artin $L$-functions of a Galois representation, class field theory, and other topics. Wiping this away, I don't see how one could have guessed it.

It's true that both sides can be described as "arithmetic symmetry", sort of (maybe one would better be described as "symmetric arithmetic"), but lots of things can be described as "arithmetic symmetry" at this level of detail, and they're not all in correspondence. Some examples are automorphisms of arithmetic schemes, fundamental groups of arithmetic schemes, and actions of fundamental groups on things other than vector spaces.

It's true that we are more likely to see a correspondence between two classes of highly symmetric, arithmetic objects than between one such class and one class of non-symmetric or non-arithmetic objects. For example, we are unlikely to find a deep correspondence between Galois representations and finite graphs.

Thus I think your approach is somewhat backwards. The way to guess the Langlands correspondence is to first notice that special cases of the two sides correspond, and that concrete features appear similar on both sides, using easier steps like the Satake isomorphism, class field theory, and so on, and then generalize. Trying say "I believe a priori there should be some correspondence between these two types of objects, now I just have to figure out which ones match up with which ones and what that means in concrete terms" is trying to make the hard parts easy and the easy parts hard.

The closest thing I can imagine to this is to take the physical perspective on geometric Langlands, derive it from physical ideas (electric-magnetic duality in $N=4$ supersymmetric Yang-Mills), and deduce that it should have some arithmetic analogue or consequence. This faces two challenges, one is that making the translation is very hard (even having access to both the physical and arithmetic-geometric sides, it took Kapustin and Witten a lot of effort to see how they match up) and the physical ideas are not "morally obvious" in themselves.

This perspective gets the closest to seeing the two sides as "fundamentally the same". We think of representations of the Galois group into $\hat{G}$ as $\hat{G}$-bundles with flat connection, and we think of points of $G(F)\backslash G(\mathbb A_F)/ G( \hat{\mathcal O}_F)$ as bundles without flat connection (where the flat connection reappears when we take the cotangent bundle of the moduli space of bundles), and we compare the two types of bundles. But the comparison is not anything simple - note that $G$ is not $\hat{G}$ - it's roughly supposed to be an advanced type of Fourier transform.

One reason it's hard to imagine understanding this without getting into the weeds is that any philosophical description of how the Galois group of $\mathbb Q$ relates to representations of arithmetic groups over $\mathbb Q$ seems like it would apply equally well to how the Galois group of $\mathbb Q(x)$ relates to representations of arithmetic groups over $\mathbb Q(x)$. However, the first step of the Langlands program, class field theory, fails in that setting. There are notions of higher class field theory, but they don't just involve comparing representations and representations, and they don't seem likely to generalize to non-abelian groups. So we're saying something special about number fields that doesn't hold for higher-dimensional fields. (From my arithmetic-geometric perspective, I would say class field theory is a manifestation of Poincare duality, and the statement of Poincare duality depends heavily on the dimension. From the physical perspective, Langlands comes from a duality in four-dimensional field theory, and you don't get the same duality in higher dimensions.)

Why reductive groups and not general groups?

The rule of thumb you should always use in these types of things is that before conjecturing something for reductive groups you should test it for $\mathbb G_m$, and before conjecturing something for unipotent groups, or non-reductive groups, you should test it for $\mathbb G_a$. Characters of $\mathbb G_m(\mathbb A_F)/\mathbb G_m(F)$ correspond to characters of the Galois group by class field theory. Characters of $\mathbb G_a(\mathbb A_F)/\mathbb G_a(F)$ correspond to elements of $F$, not anything Galois-theoretic. What more is there to say?

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  • $\begingroup$ Your answer starts with a gross mis-interpretation of my question. No wonder the rest of its sounds to me like a screed based on misguided premises. I am indeed looking for a high-level understanding of why “automorphic” should have anything to do with “Galois”. More detailed the better, but one begins somewhere. Indeed it is not a substitute for getting into the weeds and doing the dirty work - exceptional mathematicians have been doing for 50 odd years - but it is not contradictory to it either; rather it complements it. There is the forest to admire as well as the trees. $\endgroup$
    – P.H.
    Feb 9 '21 at 2:07
  • $\begingroup$ I did not say that that’s how Langlands came up with it; it was indeed in part extrapolation - extraordinarily bold extrapolation - from known cases, but also more than a dollop of imagination and intuition. I wish I could know what his thoughts and insights were when he formulated his conjecture, but his 1967 letter to Weil throws no light on the matter, nor do his subsequent writings or talks as far as I am aware. $\endgroup$
    – P.H.
    Feb 9 '21 at 2:08
  • $\begingroup$ I just want to understand for myself why the two should be related at all in intuitive terms because class field theory is a flimsy rationale for it, and makes for one only in its re-interpretation in light of Langlands’ extrapolation. I do wish historians of contemporary mathematics would weigh in on it with concrete evidence, but that is at one remove from the spirit of my question. $\endgroup$
    – P.H.
    Feb 9 '21 at 2:08
  • $\begingroup$ @P.H. I am sorry my answer was not to your liking. I hope the small bits of positive information I included (e.g. on unipotent groups, and on the connections to physics) were helpful. $\endgroup$
    – Will Sawin
    Feb 9 '21 at 2:13
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    $\begingroup$ @P.H. This might not be so helpful, but isn't the Langlands program itself an answer to that question? $\endgroup$
    – Will Sawin
    Feb 9 '21 at 22:08

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