What Does Spec Z Look Like?

This week Laurent Fargues has started a series of lectures here at Columbia on Some new geometric structures in the Langlands program. Videos are available here, but unfortunately there is a problem with the camera in that room, making the blackboard illegible (maybe we can get it fixed…). Fargues however is writing up detailed lecture notes, available here, so you can follow along with those.

Fargues is covering the story of the Fargues-Fontaine curve and the relationship between geometric Langlands on this curve and arithmetic local Langlands that he worked out with Scholze recently. On Monday Scholze gave a survey talk in Bonn entitled What Does Spec Z Look Like?, video available here. Scholze’s talk gave a speculative picture of how to think about the global arithmetic story, with Spec Z as a sort of three-dimensional space. One thing new to me was his picture of the real place as a puncture, with boundary the twistor projective line. He then went on to motivate the course he will be teaching this fall with Dustin Clausen on Analytic Stacks. Here at Columbia we have an ongoing seminar on some of the background for this, run by Juan Rodriguez Camargo and John Morgan.

Update: Peter Scholze next week at the Rapoport conference will be giving a talk on new ideas about the twistor $$\mathbf{P}^1$ and real local Langlands. His abstract is

Towards a formulation of the real local Langlands correspondence as geometric Langlands on the twistor-$\mathbf P^1$

We will propose a formulation of the local Langlands correspondence for complex representations of real groups in terms of a(n everywhere unramified) geometric Langlands correspondence on the twistor-$\mathbf P^1$, analogous to our work with Fargues in the case of p-adic groups. This is motivated by discussions with Rodriguez Camargo, Pan, le Bras and Anschütz on the analogous case of locally analytic p-adic representations, and is different from the previous work of Ben-Zvi and Nadler in a similar direction. In particular, on the geometric side we get representations of the real group, encoded in terms of liquid quasicoherent sheaves on $[*/G(\mathbf R)^{la} ]$; and on the spectral side, we get representations of the real Weil group $W_R$, or rather vector bundles on $[(\mathbf A^2\backslash\{0\})/W_R^{la} ]$.

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24 Responses to What Does Spec Z Look Like?

  1. John Baez says:

    Peter wrote:

    “Scholze’s talk gave a speculative picture of how to think about the global arithmetic story, with Spec(ℤ) as a sort of three-dimensional space.”

    While there’s speculation in Scholze’s talk, I want to let beginners know that the 3-dimensional nature of Spec(ℤ) is not a speculative thing. In 1973 Barry Mazur worked out the etale cohomology of Spec(ℤ) ∪ {∞} and showed it has nontrivial cohomology up to and including dimension 3:


    (Apparently he also showed its etale cohomology vanishes in higher dimensions ‘up to 2-torsion’. Has anyone later computed the 2-torsion?)

    There has subsequently been a lot of work on how Spec(ℤ) ∪ {∞} resembles a 3-sphere and primes act like knots in this 3-sphere. Here are some relatively easy ways to learn more:

    Chao Li and Charmaine Sia, Knots and primes, https://www.julianlyczak.nl/seminar/knots2016-files/knots_and_primes.pdf

    Barry Mazur, Thoughts about primes and knots, https://www.youtube.com/watch?v=KTVEFwRbuzU

    Masanori Morishita, Analogies between knots and primes, 3-manifolds and number rings, https://arxiv.org/abs/0904.3399

    For people who know a fair amount of math but are in a rush, I recommend just checking out this analogy table from Li and Sia’s course notes:


  2. jack morava says:

    what John said.

  3. Peter Woit says:

    Thanks for writing up that background!

    The word “speculative” might not have been the right one. For the things John refers to it’s important to keep in mind that these are just analogies. Spec Z can’t be an actual 3-manifold.

    In the talk Scholze explained how his work with Fargues on the relation to geometric Langlands fits in with these analogies. My understanding now is that near the puncture corresponding to the infinite prime the “space” should have an action of the positive reals, looking like the twistor P^1 for a fixed value. Near a finite prime it should again have such an action, looking like a torus at a fixed value. Fargues has promised me he’ll explain this in detail in his lectures. This is all supposed to fit in with work of Deninger in which there’s a foliated space with a flow acting on it analogous to the Frobenius. Note that Deninger’s program is aimed at the Riemann hypothesis, so these are deep waters…

    Perhaps what is “speculative” is the idea that there is some sort of global version of the Fargues-Scholze story (which is local at each prime), and that this global version would give not just an analogy, but an actual space corresponding to Spec Z (for some exotic definition of what a “space” is).

  4. John Baez says:

    Hmm, Scholze actually makes a cryptic reference to the 2-torsion in the etale cohomology of Spec(ℤ) starting at around 19:15 in his talk, and leading up to “if you think deeply about the etale cohomology of Spec(ℤ) then there is some funny 2-torsion phenomenon having exactly to do with Gal(ℂ/ℝ) having infinite cohomological dimension, and if you just use ℝP² instead you get a much better answer”.

    I wish I could think deeply about the etale cohomology of Spec(ℤ). Alas I can’t, but I see that Gal(ℂ/ℝ) = ℤ/2 has as its classifying space ℝP^∞, whose cohomology has 2-torsion (namely, it’s ℤ/2) in every dimension. Is this what Mazur found for the etale cohomology of Spec(ℤ) ∪ {∞}?

    This is pretty technical but maybe someone reading this will know.

  5. John Baez says:

    Sorry, I meant “in every even dimension”.

  6. jsm says:

    Actually, it was Artin and Verdier who first worked out the etale cohomology of things like Spec(Z) (Artin-Verdier duality), but it was Mazur who popularized it. Let U=Spec(Z). It is possible to define a cohomology of U “with compact support” that fits into a long exact sequence
    H_c^r(U.F) -> H^r(U,F) -> H^r(Spec R,F)->
    The cohomology with compact support behaves more or less as it should, but the etale cohomology of the real numbers is just Galois cohomology with Galois group Z/2Z which has periodic cohomology, and so goes on for ever. Artin used to say that “Spec Z is singular at infinity”.

  7. Thomas says:

    The lectures by Gross which Fargues mentions in his talk seem to be available as text at: http://www.math.columbia.edu/~chaoli/docs/EilenbergLectures.html

  8. jack morava says:

    IIRC the mod two stuff is in the original Artin-Tate seminar notes where they calculate the Galois cohomology of the idele-class group and wrestle with the solenoid?

  9. DrDave says:

    Re: Black (Green) board legibility:
    The YT video is super low resolution, below what most phones and cameras record,
    so there is a good chance there is an original video with 720p or higher resolution. If so, it would be simple to boost the sharpness and contrast.
    However, if the presenter has slides or notes, then that is even easier.

  10. yyb says:

    I don’t know if this is related to analytic stacks or not, but 4 months ago someone asked on MO about Dustin Clausen’s talk about the “modified Hodge Conjecture” here https://mathoverflow.net/questions/446978/clausens-modified-hodge-conjecture/446992#446992 and Clausen’s answer is that the modified Hodge conjecture is related to “analytic k-theory” that’s possibly related to analytic stacks that they’re going to talk about this fall.

  11. Peter Woit says:

    Yes, the problem with the video is the very low resolution, which is what Columbia’s zoom configuration gives for cloud-stored recordings. Currently discussing this with the Columbia zoom people.

  12. Will Sawin says:

    If I can add some detail to what jsm said, the compactly-supported cohomology behaves as it should in that it has a form of Poincare duality which in particular means it vanishes in degrees above 3. So together with the fact that the cohomology of Spec R with coefficients in Z/2 is equal to Z/2 in every nonnegative degree, that immediately tells you that the ordinary cohomology of Spec Z with coefficients in Z/2 is equal to Z/2 in every degree >3.

    Note that this kind of situation is exactly what you see for noncompact manifolds – a long exact sequence between the ordinary cohomology, the compactly-supported cohomology, and the cohomology of a neigborhood of infinity. So from this perspective Spec Z behaves like a non-compact 3-manifold, except that a neighborhood of infinity is homotopic to the classifying space of Z/2Z, i.e. to infinite-dimensional real projective space. So somehow we have a 3-manifold which near infinity looks infinite-dimensional.

    Somehow the twistor P^1 must be related to this picture by the close relationship between the twistor P^1 and two-dimensional real projective space and some kind of approximation between infinite-dimensional real projective space and two-dimensional real projective space.

    Let me also point out that the notion of a 3-manifold with an action of R should not be very mysterious. A very classical way of producing a 3-manifold is to start with a surface and an automorphism of the surface, take the surface cross the interval, and then glue the two sides using the automorphism, producing a 3-manifold fibered over the circle. Such 3-manifolds naturally carry an action of R (locally acting by translation by the interval), whose periodic orbits correspond to fixed points of powers of the automorphism.

    The relevance of this to Spec Z (at least in a way that makes sense to me) goes something like this: An algebraic curve over a finite field is fibered over the spectrum of the finite field with fiber an algebraic curve over an algebraically closed field. Algebraic curves over C are Riemann surfaces, so the fiber behaves like a surface, and finite fields have Galois group Z-hat, so the spectrum of the finite field behaves like the classifying space of Z (fields behaving like classifying spaces of their Galois group), and thus like the circle. So the curve behaves like a fibered 3-manifold, with the automorphism being the generator of the Galois group, i.e. Frobenius. Periodic orbits correspond to fixed points of powers of Frobenius which are the same thing as prime ideals of the ring of functions on the curve.

    But Z behaves like the ring of functions on a curve since both are Dedekind domains of finite type, so its spectrum should behave like a 3-manifold with an action of R whose periodic orbits correspond to prime ideals, i.e. primes.

  13. Will Sawin says:

    After watching the video, indeed it seems to be RP^2, i.e. the quotient of the twistor P^1 by complex conjugation, rather than the twistor P^1 itself, that appears.

    The fact that it’s RP^2 / the twistor P^1 that appears seems to be a way of capturing Hodge theory, so I guess the reason we see RP^infinity from the perspective of the étale cohomology of Spec(Z) is that étale cohomology doesn’t really capture Hodge theory.

  14. John Baez says:

    Will Sawin wrote:

    “But Z behaves like the ring of functions on a curve since both are Dedekind domains of finite type, so its spectrum should behave like a 3-manifold with an action of R whose periodic orbits correspond to prime ideals, i.e. primes.”

    I haven’t had time to watch the whole video yet – just the first part. So, if anyone feels like it, I’d love to hear how close Scholze is to formalizing this hand-wavy but delightful intuitive picture using his new ideas.

    (I’ll watch the rest as soon as I have time.)

  15. Will Sawin says:

    @John Baez: The thing that most surprised me about the construction is that the action of R is completely literal – there is really a one-parameter continuous (in the appropriate sense) family of automorphisms (or endomorphisms – if I understand right one starts with a space with a semigroup action and can use it to construct a space with a group action by a general construction). I wouldn’t have expected such an action to literally exist on any reasonable variant of Spec Z.

    On the other hand, I don’t think Scholze made any claims in the talk about the space being like a manifold. Well, the space is defined by starting with an (suitably analytified) spectrum of a ring of power series, so I guess one can locally (in a certain sense) express functions on the space as power series, which makes it like a manifold, but this is somewhat indirect (and anyways these are 1-variable power series, not 3-variable). Since the space is closely related to Spec Z there perhaps exists some cohomology theory for it which recovers the cohomology of Spec Z and thus makes the space behave homologically like a 3-manifold.

    For the intuition about Spec Z locally looking like certain spaces at certain points, the talk sketches some local picture along these lines. This is localization in the algebraic geometry sense, i.e. potentially arbitrary maps from another space to your space, so it doesn’t completely fit the handwavy intuition of literal open sets.

  16. anon says:

    Presumably this 3-manifold is still very prickly – spine like, like the 1-skeleton it maps to. I mean, do the primes 3 and 5 (say) interact only at the generic point? Or do we expect the orbits around 3 wind around the orbits around 5 (and so forth), producing Massey products or some such, in an as yet unseen global theory? That would be rather shocking. But if not, what number theoretic information is this 3-manifold picture going to carry?

    (I’m still reading Bhatt’s wonderful prismatic gauge lectures, and havent absorbed the advances at p – neither in stacky language nor via TCH etc. So I’m somewhat behind.)

  17. jack morava says:

    … do the primes 3 and 5 (say) interact only at the generic point? … Or do we expect the orbits around 3 wind around the orbits around 5 (and so forth), producing Massey products or some such, in an as yet unseen global theory? That would be rather shocking….

    For such products see Morishita, Masanoi, Milnor invariants and Massey products for prime numbers, Compos. Math. 140 (2004) MR2004124, and John B’s citation for Morishita upthread.

  18. anon says:

    Thank you Jack! That’s very pretty. So “of course” primes p,q interact – quadratic reciprocity is such an interaction. Properties of global etale cohomology of Spec of a number field already allow one to see this kind of phenomena as a Massey product. Scholze’s 3 manifold `Spec Z’ would give a `cycle theoretic’ interpretation of this as an actual linking number between actual R-orbits. And presumably much more?

    Any predictions on what new `hard’ information such a picture of Spec Z would tell us?

    (I’m already blown away by the stacky/de Rham intepretation of d dimensional varieties over finite fields having etale cohomological dimension 2d+1; Bokstedt periodicity and Bott, R[u,v]/uv-p. And that’s just the local picture.)

    Also: is it trolling on this blog to ask whether there are `arithmetic’ versions of 3+d-dim TFTs, and what they assign to d-dimensional arithmetic varieties? And what an honest number theoretic punchline of having the evident analogous structure would be?

    The enormous gulf between the easy to predict analogies, and the often simple but unknown definitions needed to make them work never ceases to amaze.

  19. Another anon says:

    This is maybe a little bit off-topic, but there is a Simons Collaboration on Perfection in Algebra, Geometry, and Topology to tackle major open questions in number theory, algebraic geometry, p-adic geometry, commutative algebra, and algebraic topology here: https://scop.math.berkeley.edu/ with 13 people as principal investigators (including Scholze, Clausen, Bhatt, Lurie, Mathew, etc) with Martin Olsson as the collaboration director.

  20. David Ben-Zvi says:

    @anon: Arithmetic (Topological) Quantum Field Theory is in fact a thriving research area, initiated by Minhyong Kim. You can see more about the topic from the talks at this seminar https://homepages.warwick.ac.uk/~u1972347/AGQFT.html and there will be a program on Arithmetic QFT at Harvard’s Center of Mathematical Sciences and Applications Feb-March 2024, with a research conference March 26-29.

  21. Kirti Joshi says:

    @Dr. Woit: In the context of the current discussion here, let me point out my preprint, Constructions II(1/2), posted on the arXiv on May 17, 2023. My paper furthers the analogy between Riemann surfaces and Number Fields and establishes a Teichmuller Theory of Number Fields. Specifically in the context of this discussion here, let me say that in my work I construct a global topological object which resembles Fargues-Fontaine curves (of both Y and X types) at all primes of the number fields simultaneously.

    My treatment at archimedean primes is different from Fargues’s treatment, but also closer to the original Banach space approach of Fargues-Fontaine at finite primes. Notably, I work with a certain (infinite dimensional) complex Banach algebra which serves as the ring of functions on the archimedean Fargues-Fontaine curve (of my theory)–and this curve appears as the spectrum (of maximal ideals) of this Banach algebra of my theory–just as the closed maximal ideals of the Fargues-Fontaine algebra B are related to the p-adic Fargues-Fontaine curve.

    This global object which I construct is equipped with a natural action of the multiplicative group of the number field which at each prime operates through Frobenius morphism (at each finite prime) and through the multiplicative group at archimedean primes (I assume that the number field has no real embeddings for simplicity). [Further properties of this global object are established in my paper.] This global topological object is metrisable and locally i.e. at each prime of the number field, this object is equipped with the natural metric provided by Fargues-Fontaine theory.

    I show that the points of this global topological object parameterize distinct arithmetic avatars of the number field just as points of the Teichmuller space of a Riemann surface parameterize different complex function theories (i.e. distinct complex structures) on a fixed topological surface.

    Hence my work further deepens the analogy between Number Fields and Riemann surfaces.

    Each avatar of the Number Field (called an arithmeticoid in my paper) also provides holomorphic structures (at all primes) in the sense of Berkovich analytic spaces (at archimedean primes one obtains analytic spaces) and I show that deforming the avatar of the number field also deforms Berkovich analytic structures. This has consequences for anabelian geometry related to Mochizuki’s IUT. Mochizuki, in his IUT, has asserted the existence of deformations of arithmetic structure of a number field whose existence I have established in my work.

    Current versions of my preprints related to this can be found on the arXiv using this link or through my webpage. I will be happy to answer any questions.

  22. Kirti Joshi says:

    Let me further remark on one important aspect of my global arithmetic construction alluded to in my comment above. My (global) construction has certain automorphisms/symmetries (at all primes) which have a nice geometric counterpart which appears in the Virasoro Uniformization Theorem of Beilinson-Schechtman (also Kontsevich) via action of automorphisms of a punctured formal disc–detailed, for instance, in the Ben-Zvi–Frenkel book in 17.3.2). The Virasoro uniformization and related actions plays an important role in Beilinson-Drinfeld work on Geometric Langlands, and is also of significance in Conformal Field Theory.

    Mochizuki, in his IUTT, asserts that automorphisms of certain unit groups change arithmetic holomorphic structures (he calls this property Indeterminacy Ind2).

    In my Constructions I paper, I have proved a precise version of this important assertion. [Informally speaking, my approach to establishing this claim is that one should work with the automorphisms of the perfectoid punctured unit disc–precise statements are in my paper. In my Untilts paper, I provide a natural geometric construction of arithmetic holomorphic structures which also establishes their existence in Mochizuki’s context.]

    So the global object I construct is at once geometric and arithmetic in nature and allows me to prove a precise version of Mochizuki’s claim (alluded to above). My approach also agrees with Fargues-Fontaine “holomorphic functions of the prime number p” point of view (at all primes p) and automorphisms of the relevant unit group provides “holomorphic changes of the coordinate p” (p is any prime) in a manner entirely analogous to the Beilinson-Schechtman setting and also agrees with Mochizuki’s multiplicative monoidal approach to thinking about number fields and varieties over them. Minhyong Kim’s theory requires working with fundamental groups arising from distinct base-points (and paths connecting them)–and my theory (and Mochizuki’s IUTT) requires working with tempered fundamental groups arising from distinct base-points.

    Given the close parallel the above described object and its symmetries bear to its counterparts in Geometric Langlands (and Conformal Field Theory), it would be interesting to see if it plays an analogous role in the corresponding arithmetic contexts.

  23. Yonz says:

    Will the Rapoport conference talk be available online? I’m quite interested in watching that. Thanks!

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