Timeline for Dictionary of arithmetic symmetries and Langlands
Current License: CC BY-SA 4.0
13 events
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| Feb 8, 2021 at 14:48 | answer | added | Will Sawin | timeline score: 8 | |
| Feb 8, 2021 at 7:08 | comment | added | Sylvain JULIEN | Also, one dare say that every symmetry is by nature algebraic... | |
| Feb 8, 2021 at 2:02 | comment | added | P.H. | @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. | |
| Feb 8, 2021 at 1:52 | comment | added | Heavensfall | Why should symmetry came from algebraic groups be called arithmetic symmetry instead of algebraic symmetry? | |
| Feb 8, 2021 at 1:40 | history | edited | P.H. | CC BY-SA 4.0 |
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| Feb 7, 2021 at 20:59 | comment | added | Sylvain JULIEN | @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. | |
| Feb 7, 2021 at 18:21 | comment | added | P.H. | @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. | |
| Feb 7, 2021 at 17:42 | comment | added | P.H. | @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? | |
| Feb 7, 2021 at 17:27 | review | Close votes | |||
| Feb 13, 2021 at 15:29 | |||||
| Feb 7, 2021 at 17:17 | comment | added | David Loeffler | 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. | |
| Feb 7, 2021 at 16:43 | history | edited | P.H. | CC BY-SA 4.0 |
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| Feb 7, 2021 at 16:25 | review | First posts | |||
| Feb 7, 2021 at 16:39 | |||||
| Feb 7, 2021 at 16:22 | history | asked | P.H. | CC BY-SA 4.0 |