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nLab > Latest Changes: adeles and ideles

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2013

    started stubs for ring of adeles and group of ideles – but they are not good yet

    • CommentRowNumber2.
    • CommentAuthorjim_stasheff
    • CommentTimeNov 22nd 2013
    Does anyone have a copy of the funeral of Bourbaki notice? in which figured the family including les Maamoiselles Adele et Idele? Soory - I couldn't resit the opportunity ;-)
    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 13th 2014

    I added more foundational material to ring of adeles.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014

    added a paragraph mentioning that one may regard the ring of adeles as the ring of functions on all formal disks over an arithmetic curve which have a pole at the origin for at most finitely many disks.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014

    added essentially the same paragraph also here to group of ideles.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014
    • (edited Jul 19th 2014)

    While Jim’s comment in #2 was maybe a bit lame as a joke, I keep being reminded of it whenever googling for “ring of adeles” and getting pictures of, indeed, the rings of Adele…

    Of course that’s nothing compared to gooling for “string theory” and “HoTT”…

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 19th 2014

    It’s not Jim’s joke. He’s referring to the joke announcement in 1968 by French mathematician Jacques Roubaud of the death of Bourbaki, see here. Adele and Idele are two of those “regretting to inform you”.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014
    • (edited Jul 19th 2014)

    All right, I see.

    Meanwhile, I tried to beautify the typesetting at ring of adeles a little. But it seems I am out of steam now and didn’t get that far. I made it to where it said (here)

    It may be shown that

    𝒪^≅∏ p∉S𝒪 v\widehat{\mathcal{O}} \cong \prod_{p \notin S} \mathcal{O}_v

    and I have fixed that to

    It may be shown that

    𝒪^≅∏ v∉S𝒪 v\widehat{\mathcal{O}} \cong \prod_{v \notin S} \mathcal{O}_v
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014

    Have added to ring of adeles the following items:

    The idea-section used to take quite a while to finally say what an adele is. That historical review is good, but I have prefaced it now by one sentence which says right away what the ring of adeles actually is:

    The ring of adeles 𝔸 k\mathbb{A}_k of any global field kk – in particular of the rational numbers ℚ\mathbb{Q} – is the restricted product of all formal completions k vk_v of kk at all its places vv, where the restriction is such that only a finite number of components have norm greater than 1.

    That style of definition is the one that lends itself most directly to the function field analogy and so I think this is what carries the best “Idea” in it.

    This style of definition is nicely reviewed in

    and I have added a pointer to that to the nLab entry.

    In particular the definition makes sense not just for number fields but for any global field. I have therefore added a section For global fields.

    Now, of course the definition there is just as well the definition for number fields, and hence there is now some overlap. Maybe this could be merged better into the existing material. But for the time being I think better to have some redundancy in the entry than to have it miss this important point. At least I need to go offline now.

    In fact I should have gone offline 15 minutes ago. Will be in trouble…

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 19th 2014

    Probably in that case the nLab page on restricted product ought to be beefed up categorically. The section on global fields in ring of adeles looks fine to me (thanks!), at least as far as it goes, but there ought to be more at restricted product about how restricted products should be interpreted appropriately as colimits in the relevant category. I think.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014
    • (edited Jul 19th 2014)

    Sure. As I said, I explicitly created an unsophisticated stub only. Am quasi-offline now. If or when you have energy to look into thus, I’d much appreciate it.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 20th 2014
    • (edited Jul 20th 2014)

    The restricted product description of the adeles is of course traditional, as well as being snappy. But the question is: what motivates it? Who ordered that?

    Remarks by Neil Strickland and Paul Garrett in this MO thread seem relevant in this regard. As noted by Neil, the restricted product description essentially falls out from tensoring the profinite completion of 𝒪\mathcal{O} with the field kk. Are we sure that some similar description can’t be worked out for the characteristic pp case? It would be nice to have both descriptions at hand, if possible.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2014
    • (edited Jul 20th 2014)

    Regarding what motivates it: as I said, I find this has a perfectly clear motivation from the function field analogy: the adeles are the ring of functions on all punctured formal disks in an arithmetic curve, all except finitely many extend to the unpunctured disk.

    This is well-motivated, because in the complex analytic case the analog function ring is that which gives the Cech cocycles for vector bundles on the curve, with respect to any cover obtained by removing finitely many points and patching these with formal disks –which are the standard “good” covers on curves.

    This perspective allows us to find plenty of incarnations of adeles outside number theory, notably in complex analytic geometry. This is contrary to the sentiment voiced in some of those MO comments that the adeles are an isolated phenomenon and a general abstract discussion via restricted products not of much use.

    For instance the starting point of geometric Langlands is that all automorphic adelic representation theory finds its “natural explanation” from this perspective.

    My favorite of the bunch of MO comments in that thread is hence the one that got accepted. This also provides what looks like clearly the right categorical perspective on restricted products in general.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 20th 2014

    Thanks Urs. What you wrote is a little too compressed for me to follow in detail, but I take it to mean both viewpoints then are valid and useful, and I suspect the treatment given for number fields can be extended in a natural way to cover all global fields, but I haven’t worked out how just yet.

    Of course restricted products, or more generally certain filtered colimits of systems built on products, recur frequently in mathematics. Direct sums and ultraproducts are just two.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2014

    Hi Todd, the story is discussed more in section 3.2 of the lectures by Frenkel which are linked to in the entry. (Sorry, I am still just on my phone.)

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 20th 2014

    Thanks. I added a little text to the section on global fields, which I should think isn’t too obtrusive.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2014
    • (edited Jul 22nd 2014)

    I have tried to expand the story for the rational numbers – here by making more explicit a bunch of basic but important facts. For instance I added explicitly the reason why the rationalization of the product over all ℤ p\mathbb{Z}_p is the restricted product over the ℚ p\mathbb{Q}_p, and I added the remark how the factor of ℝ\mathbb{R} becomes part of that restricted product if one takes it not over all primes, but over “all” places.

    One thing I need to clarify is whether I need to say “all real” places here or something. (?)

    • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 22nd 2014
    • (edited Jul 22nd 2014)

    Re #17: that set of remarks (or that story) is essentially recapitulated in the material on the situation for number fields. A key part of the story is the topology: that what emerges naturally is not the subspace topology where we view the restricted direct product as a subset of the full direct product, but something finer (given by a filtered colimit induced by a localization, given by inverting all nonzero elements in the ring of integers).

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2014

    Isn’t the topolgy that of the restricted product of topological tings? I thought that was the point.

    • CommentRowNumber20.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 22nd 2014

    Yes, the colimit topology is the restricted product topology, and that is the point. Part of that point is that it isn’t the subspace topology!

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2014

    Okay good.

    Re #18: Do you think I created too much overlap by my recent addition? Sorry if so. I just thought these basic facts are important to highlight right away for the simplest case over Q.

    • CommentRowNumber22.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 22nd 2014

    It’s fine. I’m actually glad, because I inwardly felt that the POV given in the section on number fields wasn’t getting much sympathy from you earlier. Your addition makes me feel a little differently.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2014

    As for symathy, I now do value the point of view of rationalization the integral adeles. Our recent discussion ofthe fracture square shows that this is indeed maybe the most fundamental perspective. Though the expression of that via the restricted product seems in turn to be the royal road to understanding the geometric meaning.

    • CommentRowNumber24.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 22nd 2014

    This makes me happy. The pushout statement for the fracture square meshes with the colimit topology statement touched upon in #18.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2014
    • (edited Jul 23rd 2014)

    Here is an into-the-blue question, before I call it quits for today: is the construction of the ring of adeles in any intesting way part of an adjunction?

    • CommentRowNumber26.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 25th 2014

    Not sure. But let me put a pin through the observation that for number fields F/ℚF/\mathbb{Q}, we have 𝔸 F≅F⊗ ℤ𝔸 ℤ\mathbb{A}_F \cong F \otimes_\mathbb{Z} \mathbb{A}_\mathbb{Z}.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeAug 14th 2014

    Added to this remark the definition of the “finite adeles” (omitting the factor ℝ=ℚ ∞\mathbb{R} = \mathbb{Q}_\infty).

    • CommentRowNumber28.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 2nd 2015

    Added to group of ideles some missing information on the topology.

    • CommentRowNumber29.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 13th 2015

    Added the proposition that the adeles 𝔸 ℚ\mathbb{A}_\mathbb{Q} form a locally compact Hausdorff commutative ring (in particular, complete w.r.t. its uniformity).

    • CommentRowNumber30.
    • CommentAuthorspitters
    • CommentTimeAug 17th 2015

    There are two recent papers by Connes and Consani using topos theory to treat aspects of adeles and ideles. I haven’t had the time to look at them yet.

    • CommentRowNumber31.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 16th 2017

    Added to global field the neat axiomatization due to Artin and Whaples, that they are precisely the fields which obey a product formula (the one which enters consideration of the idele class group).

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