## The stable cohomology of SL(F_p)

Back by popular demand: an actual mathematics post!

Today’s problem is the following: compute the cohomology of $\mathrm{SL}(\mathbf{F}_p)$ for a (mod-p) algebraic representation.

Step 0 is to say what this problem actually is. It makes sense to talk about certain algebraic representations of $\mathrm{SL}_n(\mathbf{F}_p)$ as n varies (for example, the standard representation or the adjoint representation, etc.). For such representations, one can prove stability phenomena for the corresponding cohomology groups. But my question is whether one can actually compute these groups concretely.

The simplest case is the representation $\mathbf{L} = \mathbf{F}_p$ and here one has a complete answer: these cohomology groups are all zero in higher degree, a computation first done by Quillen and which is closely related to the fact that the $K_n(\mathbf{F}_p) \otimes \mathbf{F}_p = 0.$

Most of the references I have found for cohomology computations of special linear groups in their natural characteristic consider the case were p is very large compared to n, but let me remind the reader that we are exactly the opposite situation. One of the few references is a paper of Evens and Friedlander from the ’80s which computes some very special cases in order to compute $K_3(\mathbf{Z}/p^2 \mathbf{Z}).$

Note, however, that p should still be thought of as “large” compared to the partition which defines the corresponding stable local system(s).

In order to get started, let us make the following assumptions:

ANZATZ: There exists a space X with a $\mathrm{SL}(\mathbf{Z}_p)$ pro-cover such that:

1. The corresponding completed cohomology groups with $\mathbf{F}_p$ coefficients are $\mathbf{F}_p$ for i = 0 and vanish otherwise.
2. If $\mathbf{L}$ is the mod-p reduction of (an appropriately chosen) lattice in an algebraic representation of $\mathrm{SL}(\mathbf{Z}_p),$ then $H^i(X,\mathbf{L}) = 0$ for i small enough compared to the weight of $\mathbf{L}.$

Some version of this is provable in some situations and it may be generally true, but let us ignore this for now. (One explicit example is given by the locally symmetric space for $\mathbf{SL}(\mathbf{Z}[\sqrt{-2}])$ and taking the cover corresponding to a prime $\mathfrak{p}$ of norm p satisfying certain global conditions.) The point is, this anzatz allows us to start making computations. From the first assumption, one deduces by Lazard that

$H^i(X(p),\mathbf{F}_p) = \wedge^i M,$

where M is the adjoint representation. But now one has a Hochschild-Serre spectral sequence:

$H^i(\mathrm{SL}(\mathbf{F}_p), \mathbf{L} \otimes \wedge^j M) \Rightarrow 0.$

The point is now that one can now start to unwind this (even knowing nothing about the differentials) and make some conclusions, for example:

1. $H^1(\mathrm{SL}(\mathbf{F}_p),\mathbf{L}) = 0.$
2. $H^2(\mathrm{SL}(\mathbf{F}_p),\mathbf{L}) = H^0(\mathrm{SL}(\mathbf{F}_p),\mathbf{L} \otimes M).$

In particular, the first cohomology always vanishes, and the second cohomology is non-zero only for the adjoint representation where it is one dimensional. (One can see the non-trivial class in H^2 in this case coming from the failure of the tautological representation to lift mod p^2.) Note of course I am not claiming that the first cohomology vanishes for all representations, but only the “algebraic” ones, and even then with p large enough (compared to the weight). Note also that one has to be careful about the choice of lattices, but that is somehow built into the stability — for n fixed, the dual of M is given by trace zero matrices in $M_n(\mathbf{F}_p)$ and so (from the cohomology side) “$\mathbf{L} = M$” is the correct object to consider rather than its dual since the dual is not stable even in degree zero. But I think you can secretly imagine that p is big enough and the weight small enough so that you can choose n so that all these representations are actually irreducible).

The first question is whether 1 & 2 are known results — I couldn’t find much literature on these sort of questions (they are certainly consistent with the very special cases considered by Evens and Friedlander).

The second question is what about degrees bigger than 2? For H^3 things start getting a little murkier, but it seems possible that H^3 always vanishes. Beyond that (well even before that) I am just guessing. But one might hope to even come up with a guess the the answer which is consistent with the spectral sequence above.

Posted in Mathematics, Uncategorized | | 3 Comments

## Harris versus Buzzard

Michael Harris has a new article at quanta. The piece is (uncharacteristically?) coy, referring to the laments of two logicians without divulging either their names or their precise objections, making oblique references to a cabal of 10 mathematicians meeting at the institute, and making no reference at all to his own significant contribution to the subject. But that aside, the piece relates to one of themes from Michael’s book, namely, what is mathematics to mathematicians? In this case, the point is made that mathematics is decidedly not — as it is often portrayed — merely a formal exercise of deducing consequences of the axioms by modus tollens and modus ponens. More controversial, perhaps, is the question of what number theorists stand to gain by a massive investment in the formalization of mathematical arguments in (say) Lean. (I “say” Lean but I don’t really know what I am talking about or indeed have any idea what “Lean” actually is.) As you know, here at Persiflage we like to put words in people’s mouths which may or may not be a true reflection of their actual beliefs. So let’s say that MH believes that any thing produced by such programs will never produce any insight — or possibly not in anyway that would count as meaningful insights for humans (if a computer could talk, we wouldn’t be able to understand it). KB believes that without the promised salvation of computer verified proofs, modern number theory is in danger of shredding itself before your eyes like that Banksy. What do you think? Since everything comes down to money, the correct way to answer this question is to say what percentage of the NSF budget are you willing to be spent on these projects. Nuanced answers are acceptable (e.g. “as long as some really smart people are committing to work on this the NSF should get ahead of the curve and make it a priority” is OK, “better this than some farcical 10 million pound grant to study applications of IUT” is probably a little cheeky but I would accept it if you put your real name to it).

Let the battle begin!

(Photo credit: I went to Carbondale to see the solar eclipse, but all I saw was this lousy sign. The other is just a random web search for “vintage crazy pants”.)

Posted in Mathematics | | 24 Comments

## Choices

Who is your preferred next prime minister? I guess it depends on what variety of politician you prefer.

Posted in Politics, Waffle | 1 Comment

## Referee Requests

###################################################

From: Mathematics Editorial Office
Subject: [Mathematics] Review Request
Date: December 26, 2018 at 7:43:31 AM CST

Dear Professor Calegari,

Happy new year. We have received the following manuscript to be considered for publication in Mathematics (http://www.mdpi.com/journal/mathematics/) and kindly invite you to provide a review to evaluate its suitability for publication:

Type of manuscript: Article
Title: Common fixed point theorems of generalized multivalued $(\psi,\phi)$contractions in complete metric spaces with application.

If you accept this invitation we would appreciate receiving your comments within 10 days. Mathematics has one of the most transparent, and reliable assessments of research available. Thank you very much for your consideration and we look forward to hearing from you.

Kind regards,

[name]
Assistant Editor

###################################################

From: [name]
Subject: [Mathematics] Manuscript ID: Review Request Reminder
Date: December 27, 2018 at 9:48:02 PM CST

Dear Professor Calegari,

On 26 December 2018 we invited you to review the following paper for Mathematics:

Type of manuscript: Article
Title: Common fixed point theorems of generalized multivalued $(\psi,\phi)$contractions in complete metric spaces with application.

You can find the abstract at the end of this message. As we have not yet heard from you, we would like to confirm that you received our e-mail.

###################################################

From: [name]
Subject: [Mathematics] Manuscript ID: Review Request Reminder
Date: January 3, 2019 at 7:48:15 PM CST

On 28 December 2018 we invited you to review the following paper for Mathematics:

Type of manuscript: Article
Title: Common fixed point theorems of generalized multivalued $(\psi,\phi)$contractions in complete metric spaces with application.

You can find the abstract at the end of this message. As we have not yet heard from you, we would like to confirm that you received our e-mail.

###################################################

From: Frank Calegari
Subject: Re: [Mathematics] Manuscript ID: mathematics Review Request Reminder
Date: January 3, 2019 at 11:08:03 PM CST

Dear [name],

Thank you for agreeing to enlist my professional reviewing services. My current rate is \$1000US an hour. Please send me the contract forms and payment details. I estimate somewhere between 2-5 hours will be required to review this paper.

###################################################

From: Frank Calegari
Subject: Re: [Mathematics] Manuscript ID: mathematics Review Request Reminder
Date: January 8, 2019 at 07:50:03 AM CST

Dear [name],

On January 3, I invited you to forward me the contract forms and payment details for my reviewing assignment.

However, if you are unable to provide payment because you are a predatory journal, please let me know quickly to avoid unnecessary reminders.

My previous message is included below:

Professor Francesco Calegari

###################################################

From: [name]
Subject: Re: [Mathematics] Manuscript ID: mathematics-412859 – Review Request Reminder
Date: January 8, 2019 at 9:25:25 PM CST

Dear Professor Calegari,

… Actually, the article process charge of this manuscript is only 350CHF. We need to invite at least two reviewers for each manuscript. We can’t bear the cost you proposed. So I will cancel the review invitation for you soon.

All the best to your work.

Kind regards,

###################################################

Posted in Mathematics, Rant, Uncategorized | | 7 Comments

## The Journal of Number Theory experiment

The Journal of Number Theory has been (for some time) the standard “specialist” journal for number theory papers. By that, I mean it was a home for reasonably good number theory papers which were not (necessarily) good enough for some of the top general journals. Of course, like every journal, there are better and worse papers. At least several papers in this journal have been referenced on this blog at some point, including those discussed here and here.

However, a number of recent changes have been taken place. JNT has introduced “JNT Prime” which seeks to publish

a small number of exceptional papers of high quality (at the level of Compositio or Duke).

(I’m not sure if free two-day delivery is also included in this package.) My question is: why bother?

I have several points of confusion.

1. It’s easier to start from scratch. It is much easier (as far as developing a reputation goes) to start a new journal and set the standards from the beginning, than to steer a massive oil tanker like the Journal of Number Theory with its own firmly established brand. Consider the Journal of the Institute of Mathematics of Jussieu. This early paper (maybe in the very first issue) by Richard Taylor set the tone early on that this was a serious journal. Similarly, Algebra & Number Theory in a very few number of years became a reasonably prestigious journal and certainly more prestigious than the Journal of Number theory has ever been during my career.
2. The previous standards of JNT served the community well. Not every journal can be the Annals. Not every journal can be “better than all but the best one or two journals” either, although it is pretty much a running joke at this point that every referee request nowadays comes with such a description. There is plenty of interesting research in number theory that deserves to be published in a strong reputable journal but which is better suited to a specialist journal rather than Inventiones. Journal of Number Theory: it does what it says on the tin. Before the boutique A&NT came along, it was arguably the most prestigious specialist journal in the area. It is true that it was less prestigious than some specialist journals in other fields, but that reflects the reality that number theory papers make up a regular proportion of papers in almost all of the top journals, which is not true of all fields. So where do those papers go if JNT becomes all fancy?
3. Elsevier. Changing the Journal of Number Theory is going to take a lot of work, and that work is going to be done (more or less) by mathematicians. So why bother making all that effort on behalf of Elsevier? Yes, Elsevier continues to “make an effort” with respect to Journal of Number Theory, including, apparently, even sponsoring a conference. But (to put it mildly) Elsevier is not a charity, and nobody should expect them to start behaving like one.

So I guess my question is: who is better off if the Journal of Number Theory becomes (or heads in the direction of becoming) a “top-tier journal” besides (possibly) Elsevier?

Posted in Mathematics, Politics | | 8 Comments

## Dembélé on Abelian Surfaces with good reduction everywhere

New paper by Dembélé (friend of the blog) on abelian surfaces with good reduction everywhere (or rather, the lack of them for many real quadratic fields of small discriminant). I have nothing profound to say about the question of which fields admit non-trivial abelian varieties with everywhere good reduction, but looking at the paper somehow dislodged the following question from my brain, which I often like to ask and don’t mind repeating here:

Question: Fix a prime p. Does there exist a non-solvable extension of $\mathbf{Q}$ unramified everywhere except for p?

There is a (very) related question of Gross, who (and I can’t track down the precise reference) was generous and allowed ramification at infinity. That makes the question easy to answer for big enough p just by taking the mod-p Galois representations associated to either the weight 12 or weight 16 cusp form of level 1. But what if you impose the condition that the extension has to be unramified at the infinite prime as well (so totally real) then you are completely out of luck as far as Galois representations from algebraic automorphic forms go, because for those, complex conjugation will always be non-trivial. (Things don’t get any easier if you even allow regular algebraic automorphic representations, as Caraiani and Le Hung showed). Except, that is, for the case when p = 2. There is a different paper by Lassina on this topic, which solved Gross’ question for p=2 by finding a level one Hilbert modular form over the totally real field $\mathbf{Q}(\zeta + \zeta^{-1})$ for a 32nd root of unity with a non-solvable mod-2 representation. But (as he shows) this extension is ramified at infinity — in fact, the Odlyzko discriminant bounds show that to get a totally real extension (assuming GRH) one would have to take the totally real field to be at least as large as $\mathbf{Q}(\zeta + \zeta^{-1})$ for a 128th root of unity. Is it even possible to compute Hilbert Modular Forms for a field this big?

Leaving aside the computational question, there is also a theoretical one as well, even for classical modular forms. Given a Hilbert modular form, or even a classical modular form, is there any easy way to compute the image of complex conjugation modulo 2? One reason this is subtle is that the answer depends on the lattice so it really only makes sense for a residually absolutely irreducible representation. For example:

Question: For every n, does there exist a (modular) surjective Galois representation

$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow \mathrm{SL}_2(\mathbf{F})$

for a finite field of order divisible by $2^n$ which is also totally real? Compare this to Corollary 1.3 of this paper of Wiese.

I don’t even have a guess as to the answer for the first question, but the second one certainly should have a positive answer, at least assuming the inverse Galois problem. As usual, an Aperol Spritz is on offer to both the second question and to the first in the special case of p=2.

## Levi L. Conant Prize

Nominations for the 2020 Levi L. Conant Prize are now open!

To quote directly from the relevant blurb,

This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding five years.

Have you read any paper in either of these journals recently which you thought was really engaging, enlightening, or just neat? Perhaps something on a topic on which you were not so familiar which left you with the feeling that you gained some insight into what the key ideas or problems were? If so, please nominate them before June 30, 2019!