Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 21st 2013
   • (edited Nov 21st 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexcreated stub for * [[p-adic homotopy]] with just some references and * [[p-adic geometry]] with just some pointers, cross-linked with * [[p-adic cohomology]]. * [[p-adic physics]] To be expanded...

   created stub for

   • p-adic homotopy

   with just some references and

   • p-adic geometry

   with just some pointers, cross-linked with

   • p-adic cohomology.

   • p-adic physics

   To be expanded…

2. • CommentRowNumber2.
   • CommentAuthorhilbertthm90
   • CommentTimeNov 21st 2013
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexWhen I use the term "p-adic cohomology," I usually mean $p=char(k)=\ell$. This usually involves switching to fppf or fpqc site if you want the same types of theorems, but even more typically it means using the crystalline cohomology (or Monsky-Washnitzer or rigid ...). I'm not sure how picky we want to be about this terminology or if it even matters when thinking in terms of the pro-etale version.

   When I use the term “p-adic cohomology,” I usually mean $p=char(k)=\ell$. This usually involves switching to fppf or fpqc site if you want the same types of theorems, but even more typically it means using the crystalline cohomology (or Monsky-Washnitzer or rigid …). I’m not sure how picky we want to be about this terminology or if it even matters when thinking in terms of the pro-etale version.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 21st 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexWait, so if you were to be picky, then which statement in which entry would you want to refine?

   Wait, so if you were to be picky, then which statement in which entry would you want to refine?

4. • CommentRowNumber4.
   • CommentAuthorhilbertthm90
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexI guess I wouldn't redirect to $\ell$-adic cohomology because it is fundamentally different, i.e. $p$-adic cohomology (as I use the term) is not just a limit over etale cohomology with coefficients in $\mathbb{Z}/p^n\mathbb{Z}$. I'd make it it's own page and talk about crystalline cohomology.

   I guess I wouldn’t redirect to $\ell$-adic cohomology because it is fundamentally different, i.e. $p$-adic cohomology (as I use the term) is not just a limit over etale cohomology with coefficients in $\mathbb{Z}/p^n\mathbb{Z}$. I’d make it it’s own page and talk about crystalline cohomology.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • (edited Nov 22nd 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, I gave [[p-adic cohomology]] its own page (super-stubby for the moment). I would be delighted if you could put into there some sentences on how you see the subject. But let me see if I understand the issue: so as we just discussed yesterday here, $\ell$-adic cohomology has genuine $\ell$-adic coefficients if regarded over the [[pro-etale site]]. So I suppose the remaining ambiguity in saying "p-adic cohomology" is that either the base of the geometry could be p-adic, or the coefficients of the cohomology could be (or both). Is that about right? (Feel free to set me straight here, you are the expert and I haven't looked much into adic stuff until now at all.) While I was at it, I also added to the references at [[crystalline cohomology]] a pointer to the [[comparison theorem (crystalline cohomology)]], which I also gave a brief entry of its own. You might want to look over this and maybe adjust a bit. I would be grateful.

   Okay, I gave p-adic cohomology its own page (super-stubby for the moment). I would be delighted if you could put into there some sentences on how you see the subject.

   But let me see if I understand the issue: so as we just discussed yesterday here, $\ell$-adic cohomology has genuine $\ell$-adic coefficients if regarded over the pro-etale site. So I suppose the remaining ambiguity in saying “p-adic cohomology” is that either the base of the geometry could be p-adic, or the coefficients of the cohomology could be (or both). Is that about right? (Feel free to set me straight here, you are the expert and I haven’t looked much into adic stuff until now at all.)

   While I was at it, I also added to the references at crystalline cohomology a pointer to the comparison theorem (crystalline cohomology), which I also gave a brief entry of its own. You might want to look over this and maybe adjust a bit. I would be grateful.

6. • CommentRowNumber6.
   • CommentAuthorhilbertthm90
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexI think the main point when people make this distinction is that they want (I'm going to switch letters to try to be less confusing) $a$-adic cohomology to be a Weil cohomology theory that gives the "correct" vector space dimension over $\mathbb{Q}_a$. If $\ell\neq char(k)$, then the etale cohomology setup works. For example, for an elliptic curve if you pick some model for it over a characteristic 0 field and then base change to $\mathbb{C}$ it is a torus, so $b_1(E)=2$. This means we expect that $H^1_{et}(E_{\overline{k}}, \mathbb{Q}_\ell)$ to have dimension $2$. This is the case, independent of $\ell$ as long as $\ell$ is not the characteristic. If $p$ is the characteristic, then depending on whether or not $E$ is ordinary or supersingular you could even get that $H^1_{et}(E_\overline{k}, \mathbb{Q}_p)=0$. In this case we wouldn't want "p-adic cohomology" to mean use the $\ell$-adic cohomology setup with $\ell =p$ (besides giving the "wrong" answer it isn't a Weil cohomology theory either). At least for smooth things, if the term $p$-adic cohomology means crystalline cohomology, then $H^1_{crys}(E/W)\otimes_W Frac(W)$ is a 2 dimensional $\mathbb{Q}_p$-vector space and despite switching how it is calculated when $\ell=p$ we have "independence of $\ell$" when the terms get used in this way.

   I think the main point when people make this distinction is that they want (I’m going to switch letters to try to be less confusing) $a$-adic cohomology to be a Weil cohomology theory that gives the “correct” vector space dimension over $\mathbb{Q}_a$.

   If $\ell\neq char(k)$, then the etale cohomology setup works. For example, for an elliptic curve if you pick some model for it over a characteristic 0 field and then base change to $\mathbb{C}$ it is a torus, so $b_1(E)=2$. This means we expect that $H^1_{et}(E_{\overline{k}}, \mathbb{Q}_\ell)$ to have dimension $2$. This is the case, independent of $\ell$ as long as $\ell$ is not the characteristic. If $p$ is the characteristic, then depending on whether or not $E$ is ordinary or supersingular you could even get that $H^1_{et}(E_\overline{k}, \mathbb{Q}_p)=0$.

   In this case we wouldn’t want “p-adic cohomology” to mean use the $\ell$-adic cohomology setup with $\ell =p$ (besides giving the “wrong” answer it isn’t a Weil cohomology theory either). At least for smooth things, if the term $p$-adic cohomology means crystalline cohomology, then $H^1_{crys}(E/W)\otimes_W Frac(W)$ is a 2 dimensional $\mathbb{Q}_p$-vector space and despite switching how it is calculated when $\ell=p$ we have “independence of $\ell$” when the terms get used in this way.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • (edited Nov 22nd 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAll right, but how do you suggest to say it? From what you write I could deduce that you would want to say "p-Adic cohomology is what computes the right cohomology groups over the p-adics." But that seems a bit too vague to be useful. Or I could deduce that you would want to say "p-Adic cohomology is crystalline cohomology in positive characteristic." But that in turn seems to be too restrictive. (But this is your suggestion, right?)

   All right, but how do you suggest to say it? From what you write I could deduce that you would want to say “p-Adic cohomology is what computes the right cohomology groups over the p-adics.” But that seems a bit too vague to be useful. Or I could deduce that you would want to say “p-Adic cohomology is crystalline cohomology in positive characteristic.” But that in turn seems to be too restrictive. (But this is your suggestion, right?)

8. • CommentRowNumber8.
   • CommentAuthorhilbertthm90
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexIt might not be too restrictive to say that it is crystalline cohomology, because it will then just be a theorem that for a smooth variety in characteristic $p$ the other theories that people might refer to as $p$-adic cohomology are isomorphic to this. I think you can get different answers or extra structure from other theories. I'll check Dix Expose later, because I can't remember if this is in there. I think there is probably a useful way to phrase the vague version. Something like: p-adic cohomology is a cohomology theory for varieties in characteristic p taking values the p-adics that satisfies Grothendieck's six operations and [insert reasonable axioms] ... Then in the same way that the "Weil conjectures follow formally for any Weil cohomology theory" there are "theorems that follow formally from the axioms for any p-adic cohomology theory." All of this is just from a feeling I've developed from looking at lots of papers with "p-adic cohomology" in the title. Usually people will then define that they mean crystalline or Monsky-Washnitzer or rigid or Hodge-Witt, etc in the first paragraph.

   It might not be too restrictive to say that it is crystalline cohomology, because it will then just be a theorem that for a smooth variety in characteristic $p$ the other theories that people might refer to as $p$-adic cohomology are isomorphic to this. I think you can get different answers or extra structure from other theories.

   I’ll check Dix Expose later, because I can’t remember if this is in there. I think there is probably a useful way to phrase the vague version. Something like: p-adic cohomology is a cohomology theory for varieties in characteristic p taking values the p-adics that satisfies Grothendieck’s six operations and [insert reasonable axioms] … Then in the same way that the “Weil conjectures follow formally for any Weil cohomology theory” there are “theorems that follow formally from the axioms for any p-adic cohomology theory.”

   All of this is just from a feeling I’ve developed from looking at lots of papers with “p-adic cohomology” in the title. Usually people will then define that they mean crystalline or Monsky-Washnitzer or rigid or Hodge-Witt, etc in the first paragraph.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeNov 22nd 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, thanks for the details. I have slightly edited the entry _[[p-adic cohomology]]_ now, but I would be really grateful if you could go to that entry and add some lines, the way you think about it. (Feel free to just remove what I have there, if it stands in the way.) Thanks.

   Okay, thanks for the details. I have slightly edited the entry p-adic cohomology now, but I would be really grateful if you could go to that entry and add some lines, the way you think about it. (Feel free to just remove what I have there, if it stands in the way.) Thanks.

10. • CommentRowNumber10.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 23rd 2013
    • PermaLink
    Author: Marc Hoyois
    Format: MarkdownItexBy the way, the fact that (pro-)étale cohomology does not give the correct $\ell$-adic cohomology when $\ell=p$ is related to the failure of [[strongly connected topos|strong connectivity]] of the (pro-)étale topos (discussed [here](http://nforum.mathforge.org/discussion/5473/etale-site/#Item_21)): the Künneth formula is one of the axioms that Weil cohomology theories must satisfy.

    By the way, the fact that (pro-)étale cohomology does not give the correct $\ell$-adic cohomology when $\ell=p$ is related to the failure of strong connectivity of the (pro-)étale topos (discussed here): the Künneth formula is one of the axioms that Weil cohomology theories must satisfy.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexI gave _[[p-adic homotopy theory]]_ a brief Idea-section and cross-linked a bit.

    I gave p-adic homotopy theory a brief Idea-section and cross-linked a bit.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexI never quite knew what to make of the existence of p-adic string theory... but in view of the topological refinement of the Witten genus by the string orientation of tmf it becomes rather compelling, doesn't it: the construction of tmf via those arithmetic squares explicitly says that and how the superstring correlator over elliptic curves over complex numbers is built from that over rational numbers and over p-adic integers. Does anyone know of a discussion of p-adic string theory from such a perspective?

    I never quite knew what to make of the existence of p-adic string theory… but in view of the topological refinement of the Witten genus by the string orientation of tmf it becomes rather compelling, doesn’t it: the construction of tmf via those arithmetic squares explicitly says that and how the superstring correlator over elliptic curves over complex numbers is built from that over rational numbers and over p-adic integers.

    Does anyone know of a discussion of p-adic string theory from such a perspective?

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 7th 2014
    • (edited Apr 7th 2014)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThe suggestion is that the relationship between $p$-adic string theory and the full theory reflects the completion $tmf_p$ within $tmf$?

    The suggestion is that the relationship between $p$-adic string theory and the full theory reflects the completion $tmf_p$ within $tmf$?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2014
    • (edited Apr 7th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for the question! Actually, I meant something more than that. I meant that in refining from the ordinary Witten genus $\Omega^{String}_\bullet \to MF_\bullet$ to the "topological Witten genus" $M String \to tmf$ one also generalizes from elliptic curves just over the complex numbers to elliptic curves over any ring. The Witten genus is constructed and argued for "actual"/"traditional" string worldsheets, which are Riemann surfaces (of genus 1 in the case at hand). But when one speaks of the Witten genus then as the [[string orientation of tmf]] then one has passed to the "absolute" moduli stack of elliptic curves. Moreover, the "standard" construction of $tmf$ (I made a note about that now at [tmf -- Definition and construction -- Decomposition via arithmetic fracture squares](http://ncatlab.org/nlab/show/tmf#DecomopositionViaArithmeticSquares)) is explicitly via decomposing the moduli stack into that of rational elliptic curves and that of $p$-adic elliptic curves. So in a very concrete sense, the "topological" refinement of the Witten genus is built by considering contributions from $p$-adic strings. Notice that the $p$-adic string literature likes to emphasize that the only thing that is made $p$-adic is the string's worldsheet (not its target spacetime). And this is exactly what happens here in the context of $tmf$, too: the spectrum $M String$ classifies String-cobordism, hence ordinary manifolds (not $p$-adic spaces) but the spectrum $tmf$ regards the string's worldsheet in full generality as an object in algebraic geometry, and its canonical "building blocks" in this generality are the rational and the $p$-adic elliptic curves.

    Thanks for the question!

    Actually, I meant something more than that. I meant that in refining from the ordinary Witten genus $\Omega^{String}_\bullet \to MF_\bullet$ to the “topological Witten genus” $M String \to tmf$ one also generalizes from elliptic curves just over the complex numbers to elliptic curves over any ring.

    The Witten genus is constructed and argued for “actual”/”traditional” string worldsheets, which are Riemann surfaces (of genus 1 in the case at hand). But when one speaks of the Witten genus then as the string orientation of tmf then one has passed to the “absolute” moduli stack of elliptic curves.

    Moreover, the “standard” construction of $tmf$ (I made a note about that now at tmf – Definition and construction – Decomposition via arithmetic fracture squares) is explicitly via decomposing the moduli stack into that of rational elliptic curves and that of $p$-adic elliptic curves. So in a very concrete sense, the “topological” refinement of the Witten genus is built by considering contributions from $p$-adic strings.

    Notice that the $p$-adic string literature likes to emphasize that the only thing that is made $p$-adic is the string’s worldsheet (not its target spacetime). And this is exactly what happens here in the context of $tmf$, too: the spectrum $M String$ classifies String-cobordism, hence ordinary manifolds (not $p$-adic spaces) but the spectrum $tmf$ regards the string’s worldsheet in full generality as an object in algebraic geometry, and its canonical “building blocks” in this generality are the rational and the $p$-adic elliptic curves.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have expanded a bit at _[[p-adic physics]]_ along the above lines and then also split off an entry _[[p-adic string theory]]_.

    I have expanded a bit at p-adic physics along the above lines and then also split off an entry p-adic string theory.

16. • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 7th 2014
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThanks! So there's a _rational_ string theory too?

    Thanks!

    So there’s a rational string theory too?

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2014
    • (edited Apr 7th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThat's a good question, but beware that there is some gap in the literature between what people traditionally call "[[p-adic string theory]]" and what happens in $tmf$. Traditional p-adic string theory considers the open string, sees the integrals over the real numbers appearing in its scattering amplitudes, where the real numbers parameterize the _boundary_ of the open string, and observes that these integrals still make sense and are interesting also over the $p$-adic numbers $\mathbb{Q}_p$. In fact all the standard literature on $p$-adic strings will say that it is unclear how to formulate p-adic closed strings, since it is not clear, they say, which adic version of the complex numbers (now parameterizing the interior of the string worldsheet) one should use. It is only me here suggesting (or so it seems, that was my original question if anyone has considered this before) that in some precise way the "topological Witten genus" with values in $tmf$ receives contributions from "p-adic worldsheets" in the sense of elliptic curves over the $p$-adic integers $\mathbb{Z}_p$. From looking around (but I may be missing something) this seems to not have occured to champions of what to date is called "$p$-adic string theory". But, yeah, to the extent that what I am getting at here makes sense, maybe one should also speak of string theory over $Spec(R)$ for any other ring $R$.

    That’s a good question, but beware that there is some gap in the literature between what people traditionally call “p-adic string theory” and what happens in $tmf$.

    Traditional p-adic string theory considers the open string, sees the integrals over the real numbers appearing in its scattering amplitudes, where the real numbers parameterize the boundary of the open string, and observes that these integrals still make sense and are interesting also over the $p$-adic numbers $\mathbb{Q}_p$.

    In fact all the standard literature on $p$-adic strings will say that it is unclear how to formulate p-adic closed strings, since it is not clear, they say, which adic version of the complex numbers (now parameterizing the interior of the string worldsheet) one should use.

    It is only me here suggesting (or so it seems, that was my original question if anyone has considered this before) that in some precise way the “topological Witten genus” with values in $tmf$ receives contributions from “p-adic worldsheets” in the sense of elliptic curves over the $p$-adic integers $\mathbb{Z}_p$.

    From looking around (but I may be missing something) this seems to not have occured to champions of what to date is called “$p$-adic string theory”.

    But, yeah, to the extent that what I am getting at here makes sense, maybe one should also speak of string theory over $Spec(R)$ for any other ring $R$.

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexAdded the original reference of Mandell to _[[p-adic homotopy theory]]_

    Added the original reference of Mandell to p-adic homotopy theory