Mathematische Zeitschrift (Part I: for reviewers)

Posted on August 3, 2019 by Persiflage

I am an associate editor for Math Zeitschrift. I thought that here could be a good place to record a few useful comments that I often pass on to reviewers. It is my intention for future referee requests to include a link to this post. I have tried to keep to issues specifically related to Math Zeitschrift rather than issues common to all reviews, although much of this advice could be applied more broadly. In particular, I have avoided discussing problems like “how much of the proof am I supposed to check” because that is not really an answerable question (beyond noting that is is the authors who are ultimately responsible for the validity of their arguments).

This post should also be useful for authors as well, although I will have a few more specific comments from them in the next post. The first thing a reviewer needs to understand is:

What is the level of the journal? This is probably the most important question to answer. The usual way that journals position themselves is to say that they are looking for submissions “similar to the level of Duke” which ranges from accurate to ambitious at best but is not necessarily useful for the reviewer unless they take it at face value (which they shouldn’t). Another way to proceed is to discuss the acceptance rate. At Math. Zeit, I believe the acceptance rates are in the range of 20% or so. Again, this is not so useful unless you have access to all the papers which are submitted — which you don’t. Slightly more informative is the following experimental observation: early on in my tenure as associate editor, approximately 60% or so of papers had favourable “quick opinion” requests. So, most likely, the standards are higher than you think. This reflects the unfortunate fact that more decent papers are submitted than can be accepted. Perhaps the most useful measure of quality, however, is to give a list of papers I have managed which have been accepted.

This is by no means a complete list, but I believe it is a complete list from some contiguous period of time (so no cherry picking going on, slipping in Jack Thorne papers just to look good):

https://arxiv.org/abs/1709.05759
https://arxiv.org/abs/1507.03238
https://arxiv.org/abs/1111.3654
https://arxiv.org/abs/1606.02341
https://arxiv.org/abs/1601.03131
https://arxiv.org/abs/1512.00586
https://arxiv.org/abs/1303.4630
https://arxiv.org/abs/1509.00320

You may form your own judgement from this. More importantly, as the quality keeps increasing, it might be useful to compare the paper I send to you with the papers above in areas most familiar to you and ask: “is this paper better? is it at least as good?“

What are you looking for from a quick opinion? At this point, I have decided not to convey any information from rejections in quick opinions to the authors. The reason is that enough authors respond with complaints that it is too much trouble, especially since (negative) quick opinions are mostly opinions concerning relative quality, not in depth reviews with useful suggestions. (Of course if the quick opinion comes with a useful suggestion I will pass it on.) So mostly I am looking for two things: is the paper at the level of Math Zeit, and (if so) do you have suggestions for possible referees for this paper? On the other hand, if it’s not up to snuff, could you give me a brief summary of why you came to that opinion (this is for my own benefit). Importantly, I am looking to reject most papers quickly at this stage. Mathematics has a problem with slow reviewing times, but it is especially painful to get your paper rejected (without any real useful feedback) after some especially long period of time. My dream is that all the papers that I reject will be rejected within a month. The journal process will always lead to good papers being overlooked and rejected; it’s hard for me to solve that problem but at least I can reject them quickly! I should say that I am pleased with how quickly I manage to reject most papers that are ultimately rejected (but by no means am I 100% compliant with my goal, unfortunately).

What if I don’t respond to your email asking for a quick opinion? Then you are added to the list of names destined for a sulfurous end, or, if you are lucky, merely a accidental prod from a poisoned umbrella next time we are at the same conference.

What if I write back but decline to offer an opinion, either with some excuse (busy, unqualified, uninterested, mortal enemies with one of the authors, or no excuse at all)? That’s perfectly acceptable.

What if I’m actually the advisor/mentor of one of the authors submitting the paper and don’t respond because I’m annoyed you didn’t check that before hand? Oops, sorry! An acceptable excuse but I would still prefer an email.

How long should a quick opinion take? Anywhere from a day to a month depending on exactly what you write back or where you are (I have one reviewer who always seems to get my emails while at the beach, for example.) That said, I have reviewers who always seem to write back several pages of detailed notes within 24 hours.

Should I summarize the paper? I mostly think that is not really necessary for any review. It is useful to explain what is novel or interesting about the paper, although the author should really have made an effort to do that as well.

Should I send my referee report as a pdf or a text file? I think I would prefer that, if it is a pdf file, the pdf file not contain the recommendation of the reviewer to accept the paper or not. Especially in cases in which the recommendation is not definitive, it puts me in an awkward position to reject the paper if the recommendation is generally (but not strongly) positive. So optimally send everything as a text file or send the summary as a text file and the detailed review (sans recommendation) in the pdf.

What happens if I always say no to quick opinion/referee requests? The immediate answer is that I will probably stop asking you to referee any more, which may sound like a positive to you. The down side is that the more people in your area do the same thing, the harder and harder it gets to find referees to do the job. At some point, the effect is going to be that I am just going to reject papers in that area unless they are obviously extremely strong, since it’s honestly a huge annoyance to email 10 people to get a single response, even when I am just asking for a quick opinion which could honestly take under 10 minutes. Some areas are definitely worse than others. Honestly, every time I get a paper related to the equivariant Tamagawa number conjecture I die a little, since it’s so hard to find anyone to review those papers.

How long should a review take? It would be nice if you could do it within about 3 months. At the point where I am finding people to review the paper, I am generally hopeful that the paper will be accepted. Under such circumstances, I am usually inclined to allow 6-9 months for the paper to be reviewed. I often ask referees to take a quick look early on to get a sense that they will be inclined to give a positive recommendation on the paper as well. Whether you think 6-9 months is way too long or relatively short probably depends on your own experiences. From my perspective, however, it’s such a difficult job getting people to agree to review that when I ask someone for a report in December and they say “I would be happy to, but I won’t get a chance to get to look at this paper in detail until the summer” I am almost always going to happily agree. Of course, some papers are harder to referee than others. Also, it may (and does) happen that, even after a positive quick opinion and a positive early indication from the referee the paper is rejected. The usual reason that this happens is that it is only when the reviewer looks at the paper in detail do they decide it is not as interesting as they first thought. The only way to avoid this is to only get referees who agree to write reports within 3 months. But if it takes me 6 months to find such a person, then it doesn’t really solve the problem.

What annoys you most about referee reports? Beggars can’t be choosers! But it does annoy me to wait 9 months for a review and get back a one sentence response “it is fine.” It’s honestly just embarrassing to me to have to send back such a report to the author after so long — can’t you at least find a few spelling mistakes/typos? I have received such reports on a number of occasions and then basically had to review the paper myself.

What pleases you most about referee reports? When referees time and time again selflessly put effort into helping the author make genuine improvements to their paper.

As editor, are you judge, jury, and executioner? The closer a paper is to my own expertise, the more likely it is that I have my own initial impressions of the paper. I don’t believe that my job is to be purely a functionary, and that I add value by having my own opinions. I have, on a few rare occasions, even solicited people to submit papers (and I would like to do so more frequently, although often by the time papers are on the arXiv they have already been submitted). In the end, I do believe that the process is impartial, although of course railway trains are impartial too, but if you lay down the lines for them that’s the way they go.

Posted in Mathematics | Tagged , , , , | 13 Comments

The Ramanujan Machine is all hype

Posted on July 17, 2019 by Persiflage

Edit (17/02/21) I changed the title of this post which was unnecessarily incendiary.

There’s a lot that I like about how mathematics operates as a social discipline. We have a great respect for the history of the subject, which in particular includes acknowledging the work that has come before us. In the end, we ultimately agree that it is the mathematics which does the talking. Each of us has our own tastes (of course) and some of us are more prone to be excited about our own work than others, but we are remarkably free from bullshit (about the actual mathematics, at least). This is what all of science should aspire to.

Perhaps this is why I find the self-promotion surrounding the Ramanujan Machine so distasteful. (I wasn’t going to bother wasting any more time on this but here I am, last time I promise.) The idea of trying to automate methods for finding identities is an interesting one. But if want to claim that you have found something new, then some justification is required. For a start, you should at least be expected to do a cursory search of the literature. Perhaps you should even consult an expert? If the authors had been content to be more modest with their claims, merely explaining that automation was their main goal, and that they were merely hopeful to use these ideas to make new discoveries, I would have had no issue at all with their paper. Of course, nobody would have heard about the paper either. I already complained last time about the overblown rhetoric, but since then, my interactions with one of the authors indicated that the rot lies deeper still.

The author in question seemed happy (while listening to my previous complaints) to indicate that the novelty is not in producing new mathematics but in automatically generating formulas “without knowing Gauss’ work”. As I indicated above, that’s a reasonable and modest claim. But then, the same author will tweet out to the world the false claim that his program has discovered new and amazing mathematical conjectures. (A rather curious set of tweets to @elonmusk @yurimilner @stephen_wolfram @RHDijkgraaf; is the common thread people who have access to money that can be spent on math?) Don’t imagine for a minute that this is not a deliberate and conscious decision: the press stories generated about “automating the insight of Ramanujan” don’t happen on their own — they need the hook of “exciting conjectures” to be “newsworthy” and this is exactly what has been peddled via a concerted publicity campaign. The best way to describe this entire story is as follows: this is what happens when you import startup culture into mathematics. Maybe such Janus-faced interactions are commonplace in Silicon Valley where qualifiers are merely a distraction from a good sales pitch. But it’s an utterly abhorrent mindset that I think mathematicians must strive to banish at all costs. In particular, the choice to deliberately obscure the fact that the program has generated (as yet) nothing considered remotely new by an expert while simultaneously boasting of a triumph in automating intuition is absurd; Ramanujan would roll over in his grave.

Caveats: The paper has a number of young authors who I consider completely blameless. The possibility of redemption still awaits in the next version of the paper.

Random Continued Fraction: I give you one of the author’s lastest tweets, in which one of their amazing conjectures has been generalized! (By a 15 year old apparently, well done to him, I hope he doesn’t waste any more time on this). I’ll spare you the amazing gif animation that builds up to this final climax:

Well … OK I guess? But, pretty much exactly as pointed out last time, not only is the proof only one line, but the nature of the proof makes clear exactly how unoriginal this is to mathematics:

(There is actually something vaguely interesting about how certain specializations of complicated identities are harder to prove than the original identities, but that is only tangentially relevant to this post.)

Posted in Mathematics, Rant | Tagged , , , , , , , , , , , | 15 Comments

En Passant V

Posted on July 7, 2019 by Persiflage

(warning: today’s persiflage comes with possible extra snark due to sleep deprivation)

The Ramanujan Machine: I learnt from John Baez on twitter about The Ramanujan Machine, a project designed to “help reveal [the] underlying structure” of the “fundamental constants” of mathematics. It seems that more effort has been spent on hype rather than on learning anything about continued fractions, and there is nothing there that would be surprising to Gauss let alone Ramanujan. Despite the overblown rhetoric (sample nonsense from the website: Suggest a proof to any of the conjectures that were discovered by the Ramanujan Machine. Have a formula named after you!), the chances of finding anything novel by these methods seem slim. While I’m all for the use of computers in mathematics, at this point (for these type of identities) we are very much in a world where the guiding hand of human mathematical intuition is very much required. As for the “new conjectures” that are animated on their flashy website (screenshot below):

well, the less said the better. It’s harder to say which one is sillier to pretend is original. The latter is a specialization of a specialization of a specialization of a specialization of Gauss’ 2F1 continued fraction identity, sending the four parameters \(a,b,c,z\) to \(a = 0, b = 1/2, c = 5/2, z = -1.\) (Hint: when your result is a specialization of a formula on Wikipedia it does not count as original.) The former is even more elementary and proves itself by induction:

The rest of the paper is littered with mathematical trivialities and grandiose bombast. (All the other “new” identities are similar trivializations of Gauss’ hypergeometric continued fraction or just known infinite (or finite) sums in disguise via Euler’s continued fraction.) What would be a much more interesting project is to find a way of taking a continued fraction and recognizing it as a specialization of one of the (very many) known results. Lest I be considered a luddite, I should note that when it comes to infinite sums and integrals, this is something that Mathematica does amazingly well, so respect to the people who worked on that.

3Blue1Brown: I don’t really get the appeal, to be honest (yes, I know, I’m not the target audience). The presenter has a geometric perspective which gets shoehorned into everything whether it is appropriate or not. I watched two videos, both of which seemed to miss (or at least elide) at least one key underlying mathematical point. The first was a video on quaternions. The fundamental property of \(\mathbf{H}\) is that it is (the unique!) non-commutative division algebra over the real numbers. But the video really only talked about the multiplicative structure, in which case you may as well talk about \(\mathrm{SU}(2).\) Are you really “visualizing quaternions” when you only think about the multiplicative structure? Then there was this video on the Riemann hypothesis. The video does a reasonably good job of explaining analytic continuation in terms of conformal maps (not that I think of it that way, but this is a perfectly reasonable way to think about it). However, the entire video ignores (once again) the key point that what is amazing is that the zeta function has an analytic continuation at all rather than it is unique (time stamp 16m 45s):

https://www.youtube.com/watch?v=sD0NjbwqlYw?#t=16m45s

The closest the video comes to acknowledging this is the quote “which through more abstract derivation we know much exist” which is somewhere between wrong (suggesting that the extension exists for formal reasons) and misleading (using “abstract” to mean something like “beyond the scope of this video” or something). How can you start to think about the Riemann Hypothesis without appreciating why the analytic continuation exists? I guess the video helps bridge the gap between mathematics and physics; popular accounts of physics have long since enabled people to think that they understand something about physics while actually not really having any real idea what is going on, now people can think that way about the Riemann zeta function as well! (The number of views of these videos is in the millions.)

Hatcher on Class Groups: I learnt that Allen Hatcher, author of a wonderful and free textbook on algebraic topology, is writing a textbook on the geometry of binary quadratic forms. I’m sure it will be a great read, and I don’t quite mean to lump it together with the two examples above, although I do notice that it does not appear to mention (in any way) class field theory. That seems to be a strange omission: couldn’t the book at least have a sentence or two indicating that two centuries of algebraic number theory has been built on generalizing Gauss’ work on the class group?

Posted in Mathematics, Rant | Tagged , , , , , , , , , , , , , | 2 Comments

The stable cohomology of SL(F_p)

Posted on June 19, 2019 by Persiflage

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 a (added non-trivial irreducible) 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.

Added: Some of the discussions in the comments below contain some minor inaccuracies, but in back and forth conversations with Will via email he has formulated a pretty convincing conjectural answer to my questions (and also my secret unasked questions). Hopefully I will come back to this post later when these are all proved!

Posted in Mathematics, Uncategorized | Tagged , , , , , | 4 Comments

Harris versus Buzzard

Posted on June 12, 2019 by Persiflage

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 | Tagged , , , , , , , | 24 Comments

Choices

Posted on May 29, 2019 by Persiflage

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

Posted on April 11, 2019 by Persiflage

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

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.

Do not hesitate to contact me if you have any questions about this request.

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 | Tagged , , , | 7 Comments

The Journal of Number Theory experiment

Posted on April 4, 2019 by Persiflage

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 | Tagged , , , , , , , , , | 8 Comments

Dembélé on Abelian Surfaces with good reduction everywhere

Posted on March 27, 2019 by Persiflage

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.

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Levi L. Conant Prize

Posted on March 11, 2019 by Persiflage

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!

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