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nLab > Latest Changes: Global analytic geometry and analytic Langlands program

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  1.  
    • CommentRowNumber1.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)
    After reading some thread on mathoverflow, i decided to create the global analytic geometry and analytic Langlands program entries. The first one is now well started by Berkovich and Poineau mainly.

    The analytic Langlands program is an idea of mine that is not very precise but i am convinced would help beginners in arithmetic geometry to understand quickly important aspects of arithmetic problems, and develop interest for global analytic geometry. The point is to understand better why Langlands program is so hard using schemes and would be easier using analytic geometry. This is also true with the use of ind-schemes in local geometric langlands and Chiral algebras that should be replaced by t-adic analytic tools.

    Creating an entry on this would help put all references on the subject and make them better available.
    • CommentRowNumber2.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    This entry can also be related to the zeta function entry.
    • CommentRowNumber3.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 18th 2011

    Frédéric, this sounds like a very good idea – I hope you continue adding to it.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011

    Frédéric, there is an old stub rigid analytic geometry, which seem to be essentially about the same subject. I am familiar with rigid terminology, but not with “global” terminology. is this restriction to the case of global fields or what ? It is unfortunate terminology as in differential geometry global differential geometry is meants as a specific area of differential geometry.

    • CommentRowNumber6.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    Zoran, global analytic geometry contains usual analytic and rigid analytic geometry. It is geometry over the base banach ring Z, not over Q_p or C. Here, global is meant to be related to global field. It is a convenient terminology for an arithmetic geometer, and i think this article is addressed more to such a public than to a differential geometer.
    • CommentRowNumber7.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)
    Actually, global analytic geometry is more an extension of usual and rigid analytic methods to the case of global fields, than a restriction. The methods used in rigid analytic geometry a la Tate are very specific to non-archimedean fields and Berkovich's approach gives a very nice way to combine these methods more easily with usual complex analytic geometry (and actually put all of them together with algebraic geometry in the sense of scheme theory).

    It is very stricking to me that this subject has not attracted more attention in spite of the fact that it seems much better adapted to automorphic and other arithmetic stuff like zeta functions, than algebraic geometry (that is too rigid for say, Langlands program).

    At least, if one hopes one day to have a functorial study of zeta functions, it will be hard for this to avoid to pass through something like global analytic geometry to treat all local factors (associated to archimedean and non-archimedean varieties) on equal footing. In this sense, global analytic geometry seems to me well suited to the nlab spirit, because it goes in the direction of a functorial study of arithmetic geometry, direction that has been left alone so long because of the inertia of the arithmetic geometry Grothendieck school around scheme theory... Hum Hum... Am i doing politics here?
    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011

    Well, I do not know what do you mean by “usual rigid analytic geometry”. People whom I know classify under this name plethora of modern variants including e.g. by Huber and I think Berkovich spectra are also included.

    • CommentRowNumber9.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)
    By usual rigid analytic geometry, i mean the one invented by Tate in his papers on this. Huber and Berkovich use exactly the same building blocs, but with different corresponding spectra. Huber uses as points continuous higher rank seminorms, that are the natural point of Tate's rigid topos, and Berkovich uses R-valued seminorms, that have the advantage of allowing a combination with usual complex analytic geometry called analytic geometry over Z (or also global analytic geometry).

    Huber only works with non-archimedean geometry. Berkovich also has said a few words in his book about analytic spaces associated to general banach rings and in particular to Z with the archimedean norm. Poineau has developped this further in his thesis. You can take a look at the introduction of Poineau for large audience cited in the short description that he wrote at my demand for the EMS newsletter (this reference can be found at the global analytic geometry entry).
    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)

    Come on, rarely anybody works with the first version due Tate. If one restricts the name “rigid geometry” to his version then one may come in conflict with the practice of most active practioners. OK, I am happy with saying classical analytic spaces for the Tate version, but I continue to call the whole wide subject rigid analytic geometry. The name of Poineau is completely new to me. Thanks for pointing out. Somebody participating in a cycle of seminars at IHES explained to me 7 years ago about rigid geometry and they were working Huber version.

    When mentioning places at global analytic geometry, would you be so kind to write an entry for place ? I wanted to write one but felt incompetent.

    • CommentRowNumber11.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)
    My opinion is a bit different about the situation: everyone uses the basic acyclicity and other theorems of Tate, and (essentially) the same algebras of converging power series as him. It seems to me (perhaps mistaken) that Tate introduced rigid analytic space as built upon a site, whose building blocs are his affinoid algebras (essentially the same as Huber's and Berkovich) and with a special Grothendieck topology. Rigid analytic geometry is really this version of the theory, to my sense, and in any case, it is purely non-archimedean. The theory of coherent sheaves did not really change after Tate and it is the same in Huber and Berkovich. What improved with Berkovich is etale cohomology and the relation with trees and buildings, and with Huber is constructible theory (because he has the optimal model theoretic setting for that, with the points of Tate's rigid topos).

    I would call non-archimedean analytic geometry the whole subject you are talking about and global analytic geometry is a more general thing, that contains it but contains also all usual complex analytic geometry, and (normally) also algebraic geometry. My personal opinion is that rigid analytic geometry is, among specialist, the part of non-archimedean analytic geometry that was developped by Tate (coherent sheaves theory in particular, and the rigid site).
    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)

    I agree that the subject is unique; for example it draws much to the classical material in

    • Hans Grauert, Reinhold Remmert, Coherent analytic sheaves, Springer 1984

    The classical results like Weierstrass division, Weierstrass preparation and so on, hold over many fields; and one sees already there the role of Banach algebras etc. Now the Tate’s affinoid picture and usage of Grothendieck gluing are a different specific, but this is a technical matter of finding a more general and flexible setup, like the scheme theory is a natural extension of variety theory. On the other hand, your claim about unifying analytic and algebraic geometry confuses me a bit. I mean it is not only to define the objects – the crucial thing is to control some finiteness conditions. Now it is like approximating a sphere with polyhedra – for algebraic geometry the good objects and morphisms are finitely presented, proper and so on. The best objects in analytic world are far from satisfying such finiteness conditions. This is observed for example in the difference between classical Arakelov geometry and Durov’s algebraic version in his thesis.

    However, we show that our models X/SpecZ^X/\widehat{Spec \mathbf{Z}} (Z. Š.: typography – widehat should be there) of algebraic varieties X/QX/\mathbf{Q} define both a model 𝒳/SpecZ\mathcal{X} / Spec \mathbf{Z} in the usual sense and a (possibly singular) Banach (co)metric on (the smooth locus of) the complex analytic variety X(C)X(\mathbf{C}). This metric cannot be chosen arbitrarily; however, some classical metrics like the Fubini–Study metric on P n\mathbf{P}^n do arise in this way. It is interesting to note that “good” models from the algebraic point of view (e.g. finitely presented) usually give rise to not very nice metrics on X(C)X(\mathbf{C}), and conversely, nice smooth metrics like Fubini–Study correspond to models with “bad” algebraic properties (e.g. not finitely presented).

    Soibelman had some enthusiasm in developing a noncommutative version of Berkovich spectra and was added at the beginning with collaborating Alexander Rosenberg’s spectral experience (and formal methods in descent theory, likewise) in this project and one preprint appeared:

    • Yan Soibelman, On non-commutative analytic spaces over non-archimedean fields, preprint IHES, pdf

    Rosenberg speculated (personal communication) that the existence of Berkovich construction with all its properties in commutative case somewhat accidental from the point of view of a more general spectral theory he studied. Though some examples come from homological mirror symmetry, another impeding problem with nc Berkovich spectra is the lack of guiding examples.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeAug 18th 2011
    • (edited Aug 18th 2011)
    • R. Cluckers, L. Lipshitz, Fields with analytic structure, J. Eur. Math. Soc. 13, 1147–1223, pdf

    Edit: I put some references at rigid analytic geometry.

    • CommentRowNumber14.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    About Durov: i am not sure that working with finitely presented algebraic theories is the right thing to do when one wants to do analysis (analytic number theory), but perhaps i am not open enough...

    About algebraic geometry: i mean that the global sections of a global analytic space are algebraic usual algebraic functions. This means that you have all the information about a ring in its global analytic space, so global analytic spaces should contain fully faithfully schemes, non-archimedean analytic spaces and complex analytic spaces: it is a common generalization of these three theories.
    • CommentRowNumber15.
    • CommentAuthorfpaugam
    • CommentTimeAug 18th 2011
    Thanks for the reference about analytic fields.
    • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeAug 19th 2011

    14/Durov: that is what I was saying: Finitely presented things approximate all things (the latter being ind-objects, like sphere approximated by a sequence of polyhedra).

    I am not sure that I would accept the argument in second part as containing a scheme theory, by just having rings. The basic thing of scheme theory is the study of properties of morphism and how to get to there it is not clear to me from your description. For example, the basic thing how and when do you know that a given morphism in your geometry induces one in algebraic world (specially in nonaffine and relative case); and again questions of finiteness of induced morphisms. In rigid geometry there is a usual thing about the fiber relation with formal schemes. How is this in global geometry ?

    • CommentRowNumber17.
    • CommentAuthorfpaugam
    • CommentTimeAug 19th 2011
    Ok, i see for ind-things. But i prefer to work with analytic things directly.

    Schemes are locally representable sheaf on commutative rings for the Zariski topology, as you know. With this functor of point viewpoint, i can say most of what i needs about schemes. So one needs a category of analytic spaces that contains fully faithfully
    the category of rings to be more precise about what i said. It is not completely clear for now what a global analytic space should be. At least in Berkovich's book, it is a locally ringed spaces that is locally isomorphic to the support of a coherent ideal sheaf of the sheaf of functions on an open subset of the affine line. There is probably a better notion, related to overconvergent functions.
    I would tend to think that if one works with the global analytic affine spaces over Z, one gets global analytic spaces that contain
    fully faithfully rings, and also schemes of course, as is clear from the definition i gave above. So one only has to check rings, this is essentially what i was saying.

    Actually, i would prefer a functor of point approach to global analytic geometry (locally representable sheaves on affinoid subspaces of the affine line, the topology being well chosen and also the notion of affinoid...),
    exactly as Grothendieck's for schemes.

    Global geometry also contains trivial norms, that are very useful to talk about the variety over the residue field (a field with trivial norm). Over the p-adic norm, you should get non-archimedean (you say rigid) geometry.
    One works over the integers and not over Q, this is much better because it gives the missing trivial norms that are not
    visible over Q.

    Your question about special fiber of formal schemes is interesting. I don't know if it has an answer for the archimedean norm. Of course, you have this relation also in the part of global geometry that is given by rigid geometry... But global geometry is something really non-local (funny?). I don't know anything clever to say about your question, but this is certainly a good one. For example, the archimedean factor of L-functions seems to be related to a kind of specialisation procedure (explained in Deninger, Gamma factor II, at the end, about Simpson's construction using deformation to the normal bundle). This construction is also used in microlocalization... Humhum... A lot of mysteries and not many answers...
    • CommentRowNumber18.
    • CommentAuthorzskoda
    • CommentTimeAug 19th 2011
    • (edited Aug 19th 2011)

    Very interesting. Very lively field ! I hope more of it gets into nnLab.

    It is a bit puzzling to me that schemes are there fully faithfully, I mean faithfully that is OK, but fully…You know one expects to have more analytic morphisms than algebraic ones, unless in some special circumstances, like the projective case.

    Also for relative schemes one may need e.g. sheaves over the site of affine schemes over a base scheme (base not necessarily), not just for algebras above a ring (or above integers).

    • CommentRowNumber19.
    • CommentAuthorfpaugam
    • CommentTimeAug 21st 2011
    • (edited Aug 21st 2011)
    Of course, if one works on a local field, there are more analytic morphisms than algebraic, but over the ring of integers, i don't see how you can do that, because imposing analycity with respect to the trivial norm is very restrictive.

    Another way to say it is the following: the global sections of the sheaf of analytic functions on the global analytic spectrum of a ring A are exactly given by the ring A itself. A morphism of locally ringed spaces induces a morphism on global sections, and thus a morphism of rings. But this morphism determines everything in the case of a spectrum.

    Relative things can be treated with functors on rings i think. Can you be more specific on the second point?
    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2012
    • (edited Jan 4th 2012)

    I am late to this discussion here (by about half a year :-) but anyway, here is a comment in reply to #17, where Frédéric said:

    Actually, i would prefer a functor of point approach to global analytic geometry (locally representable sheaves on affinoid subspaces of the affine line, the topology being well chosen and also the notion of affinoid…), exactly as Grothendieck’s for schemes.

    I would like to have this, too! Moreover, I would like to have the site consisting of affinoids which are contractible – or rather, what I really want is that the site is oo-cohesive.

    Of course, I can simply define the site this way. Therefore, what I would really like to understand is how “good” the topos over such a site is, meaning, how faithfully (global) analytic geometry is embeddable in the topos over a site of contractible affinoids.

    Berkovich’s result that everly “locally smooth” pp-adic analytic space is locally contractible suggests that such a site of contractible affinoids would be “good enough”.

    Any help in making this more precise would be very much appreciated.

    • CommentRowNumber21.
    • CommentAuthorzskoda
    • CommentTimeJan 4th 2012
    • (edited Jan 4th 2012)

    I am sure that many do via functor of points approach (not Berkovich himself). Ayoub has that approach but he looks predominantly on the analytic Nisnevitch site for the obvious reasons. How about Kedlaya’s work ?

    P.S. Brian Conrad’s work UNIVERSAL PROPERTY OF NON-ARCHIMEDEAN ANALYTIFICATION pdf is predominantly in the setup of ringed topoi.

    P.S.2 maybe of some use – Prop. 2.4 in pdf

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2012
    • (edited Jan 5th 2012)

    Thanks for the links.

    Let me maybe say what I think needs to be checked and done.

    According to fact 3.4.5. here, every locally smooth analytic space has a cover by contractible ind-objects of affinoids.

    So, for what I have in mind, it seems we want to be looking at the site (IndAff) contr(Ind Aff)_{contr} of contractible ind-objects of affinoids and check, first of all, whether it is a locally infinity-connected site.

    For this we need to check if every split hypercover of a contractible ind-affinoid by contractible ind-affinoids has the property that after contracting each component space to a point, the resulting simplicial set is weakly contractible (hence: whether every contractible ind-affinoid is also contractible in étale homotopy.)

    First of all, by the above fact we should have covers of any contractible ind-affinoid by contractible ind-affinoids. To promote this to a split hypercover, we need to iteratively cover the intersections of this by contractible ind-affinoids again.

    So first of all: are locally smooth analytic spaces stable under intersection/pullback?

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 5th 2012

    I know my last suggestion to pose an MO question led to no answers, but wouldn’t this be a good question to ask there?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2012
    • (edited Jan 5th 2012)

    I will eventually.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2012

    Okay, I have posted something to MO.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2012
    • (edited Jan 5th 2012)

    Zoran,

    concerning the terminology of “rigid” at analytic space:

    when looking at any review of the matter, I am getting away with the same impression that Frédéric seems to have expressed above: Berkovich-style analytic geometry is usually presented as going beyond Tate’s rigid analytic geometry. It seems to be misleading to me to speak of Berkovich spaces as “rigid analytic spaces”, since the main point that every review emphasizes is that they are different.

    Generally: where does the term “rigid” here come from, anyway? Which aspect of Tate’s spaces is it supposed to reflect? What is rigid about these spaces?

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 5th 2012

    Kedlaya on p. 3 of these slides asks and answers Why Rigid? And he contrasts with Berkovich’s approach.

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012

    Yes, that’s how I understand it:

    • CommentRowNumber29.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2012

    Urs, re your MO question. I wonder if you could have posed the question in such a way that analytic geometers knowing nothing about ∞\infty-sheaves over ∞\infty-cohesive sites could answer.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012

    Right, I’ll try again later.

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012

    On the other hand, it is true that some peope say “Berkovich’s theory of rigid analytic spaces”, for instance in the first sentences here.

    • CommentRowNumber32.
    • CommentAuthorfpaugam
    • CommentTimeJan 6th 2012
    Hi all,

    The answer to your question, Urs, is hard: what you want, i guess, is to have a kind of p-adic Riemann-Hilbert, because contractible in analysis means that you can solve linear PDEs. This exists, by Kedlaya's devissage, only when one works with Frobenius modules
    (Christol-Mebkhout). I am convinced this is the right direction to look for, and i am presently working on this, on any analytic basis.
    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)

    re #26, on the meaning of “rigid”:

    in _p-Adic differential equations on p. 18 Kedlaya explains:

    The traditional method is Tate’s theory of rigid analytic spaces, so-called because one develops everything “rigidly” by imitating the theory of schemes in algebraic geometry, but using rings of convergent power series instead of polynomials.

    I have added that to the entry rigid analytic geometry.

    (And to share a personal opinion: I am glad to know now what “rigid” is supposed to allude to, but I think this is bad choice of terminology.)

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)

    Hi Frédéric,

    thanks for your reply! This sounds most interesting.

    I have the two obvious questions:

    1. could you point me to some references that would help me understand what you mean when you say “…by Kedlaya’s devissage, only when one works with Frobenius modules (Christol-Mebkhout).” ?

    2. is your work that you allude to top secret or would there be a chance to talk with you about it (here or maybe by private email)?

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012

    Ah, I guess the Christol-Mebkhout theorem in question is that discussed here.

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)

    I have found this here on étale homotopy of analytic spaces, but I don’t see yet that/if it helps with my question above:

    • de Jong, Étale fundamental groups of non-archimedean spaces (numdam)
    • CommentRowNumber37.
    • CommentAuthorfpaugam
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)
    Dear Urs,

    Yes, i was talking of p-adic monodromy theorem (Bourbaki seminar of Comez on his web page and course of Christol on his web page) and of the recent results of Kedlaya, that gives a devissage of overconvergent F-isocrystals, that are p-adic analogs
    of local systems. The complex analog is for holonomic D-modules:
    http://arxiv.org/abs/1001.0544
    and the F-isocrystal case is:
    http://arxiv.org/abs/0712.3400

    Yes, i know de Jong's paper. I think the best approach to local systems is through differential equations: these are the building blocs of differential equations... In the p-adic case, one needs a Frobenius structure to prove the monodromy theorem, as explained by Christol in his lecture notes, at the end.
    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)

    Okay, thanks a lot for the references! I am way behind, but maybe I can catch up a little.

    Just so that we don’t talk past each other, can you say what general statement it is that you have in mind when you say “I am working hard on this”? Is “this” related to constructing a site of analytic spaces contractible in étale homotopy?

    • CommentRowNumber39.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2012

    It’s a shame so much has to go on secretly. The dream of the Cafe was to have research open to view. I wonder what tends to happen with ’top secret’ work, such as this and this.

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)

    I wonder what tends to happen with ’top secret’ work, such as this and this.

    I agree with your attitude here.

    Just for the record, on the two specific examples that you link to:

    the first one referred to stuff that Alejandro Cabrera had told me about his upcoming thesis, back then. This has long appeared since then. I had seen it once, but right now I can’t find the link… (?)

    The second refers to an insight that has meanwhile been fully clarified in section 4.2 of Topological Quantum Field Theories from Compact Lie Groups.

    • CommentRowNumber41.
    • CommentAuthorfpaugam
    • CommentTimeJan 6th 2012
    I also agree with David, but not completely. I think the nlab is good for sharing ideas and documentation, to learn things, but we need to publish at some point, so we (maybe i should say I, here) can't do research continuously in front of everyone, because we don't only live of math, and we have to feed our family, at some point.

    I told you essentially all i know and all the serious things i can say on this. I will tell you the rest when it becomes serious (need to crash test my things, and i don't want to waste your time; i just don't want to say completely false things to you). My opinion is that the right notion of local system in analytic geometry is given by the local solution spaces of differential equations. You may also work with etale local systems, but i now prefer differential equations. Of course, solving differential equations involves etale coverings, at some point, as one can see in the p-adic monodromy theorem. But one needs some frobenius structure, in the p-adic case. I think the two Kedlaya references are exactly what one needs to define a good notion of local system in analytic geometry, because he does the devissage, and this is the hardest part.

    There is also another reference on this: it is Berkovich's article
    http://www.wisdom.weizmann.ac.il/~vova/Integration_2004.pdf
    where he generalizes the Coleman approach to integration to higher dimension.

    In general, the theory of local systems in p-adic analytic geometry is not completely satisfactory, up to my knowledge.

    That's all folks!
    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)

    Thanks again, Frédéric. I need to be concentrating on other things now anyway, so I’ll be happy to wait for what you will publish later.

    • CommentRowNumber43.
    • CommentAuthorfpaugam
    • CommentTimeJan 6th 2012
    • (edited Jan 6th 2012)
    I forgot to give another good reference on the various etale coverings in analytic geometry: it is chapter III of André's book:

    http://arxiv.org/abs/math/0203194
    • CommentRowNumber44.
    • CommentAuthorEric
    • CommentTimeJan 7th 2012

    Speaking of publications, you are probably aware of this:

    • CommentRowNumber45.
    • CommentAuthorzskoda
    • CommentTimeJan 9th 2012
    • (edited Jan 9th 2012)

    Urs 26: Surely Berkovich introduced new formalism into the wider subject of rigid analytic geometry, not contained in Tate’s Huber did introduce an alternative formalism which has some advantages etc. There are many directions, and it is standard to call them all rigid analytic geometry and all of them feature some variant of rigid analytic spaces. The rigid aspect is seen already at the level of algebraic geometry, let me mention standard phrases like, rigid spaces as the fiber for formal geometry, rigid GAFA, and rigid cohomology…

    If one wants to take strictly Berkovich spectrum or Berkovich analytic space and not say, Huber’s version one can call it by that name. In 12 I mentioned some problems with noncommutative version.

    • CommentRowNumber46.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2012

    let me mention standard phrases like, rigid spaces as the fiber for formal geometry, rigid GAFA, and rigid cohomology…

    Yes, and all these are made more flexible by Berkovich spaces. A Berkovich analytic space is not necessarily the fiber of a formal geometry. So it seems misleading to call it by the same name.

    In any case, the majority of publications that I have seen distinguishes Berkovich spaces from rigid spaces terminologically, so the entries should reflect that. We can add the emphasis that Berkovich spaces are “just a generalization” as extra information.

    • CommentRowNumber47.
    • CommentAuthorfpaugam
    • CommentTimeJan 18th 2012
    I was not aware of publications on the nlab.
    • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2012

    I was not aware of publications on the nlab.

    It’s only just getting started. So far there is one publication, Leinster2011 (publications), and one more submission is currently still with the referee. After a second publication is through, we are planning to try to formally instantiate the editorial board and then eventually try to establish a recognized journal.

    • CommentRowNumber49.
    • CommentAuthorfpaugam
    • CommentTimeJan 22nd 2012
    That's a good idea!
    • CommentRowNumber50.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2012

    I see in section 4.4 ’Derived Complex Analytic Geometry’ of Structured Spaces Lurie says

    With some effort, the ideas presented in this section can be carried over to the setting of rigid analytic geometry. We will return to this subject in [46].

    [46] Toric varieties, elliptic cohomology at infinity, and loop group representations. In preparation.

    • CommentRowNumber51.
    • CommentAuthorfpaugam
    • CommentTimeApr 29th 2012
    Thanks for the reference David. Yes, it's easy to define derived global analytic spaces, by using the locally finite etale analytic topos as building blocs, simplicial algebras on these (that commute to transversal fiber products) and then simplicial sheaves on the associated homotopical site, a la Dubuc-Lurie.

    What looks clear to me is that the analytic setting may be as useful in geometric Langlands program as it is in usual Langlands program (where you can't avoid it, because the p-adic moduli spaces are not algebraic and you need infinite etale coverings there). I tend to find it strange that people continue to work with schemes there, where analytic varieties seem better adapted to the study of D-modules, say, because you may look for analytic solutions, and algebraic solutions of PDEs are not really interesting. Also, it is useful to have balls in the analytic setting, and to study not only formal discs, but true analytic discs (and analytic factorization spaces and chiral algebras), to have finer information and the true tools of algebraic analysis and generalized functions.
    • CommentRowNumber52.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 10th 2013

    I see Frédéric’s habilitation memoire appeared.

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