## Referee Requests

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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

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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

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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 | | 5 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!

## Jean-Marc Fontaine, 1944-2019

The results which generate the most buzz in mathematics are usually those which can be expressed in an elementary (or at least pithy) way to a general mathematical audience. It is certainly true that such results may be profound (see Wiles, Andrew), but this is not always the case. An indirect consequence of this phenomenon is that there are mathematicians who are considered absolute titans of their own field, but who are less well-known by the broader mathematical community. Fontaine, who died this year, might be considered one of these people. Fontaine will forever be associated with p-adic Hodge theory, a subject which is absolutely central to algebraic number theory today. While the initial seed of this subject came from Tate’s paper on p-divisible groups, a huge part of its development was due to Fontaine over a period of 30 years (both in his solo papers and in his joint work). The usual audience for my posts is experts, but on the rare chance that someone who knows less p-adic Hodge theory than me reads this post, let me give the briefest hint of an introduction to the subject.

For a smooth manifold M, de Rham’s Theorem gives an isomorphism

$H^n_{\mathrm{dR}}(M) \rightarrow H^n(M,\mathbf{R}) = H_n(M,\mathbf{R})^{\vee}$

which can more naturally be phrased as that the natural pairing between (classes of) closed forms $[\omega]$ and (classes of) paths $[\gamma]$ given by

$\displaystyle{\langle [\omega],[\gamma] \rangle = \int_{\gamma} \omega}$

induces a perfect pairing on the corresponding (co-)homology groups. The class of paths in homology has a very natural integral basis coming from the paths themselves. For a general M, the de Rham cohomology has no such basis. On the other hand, if M is (say) the complex points of an algebraic variety over the rational numbers, then there are more algebraic ways to normalize the various flavours of differential forms. To take an example which doesn’t quite fit into the world of compact manifolds, take X to be the projective line minus two points, so M is the complex plane minus the origin. There is a particularly nice closed form $dz/z$ on this space which generates the holomorphic differentials. But now if one pairs the rational mutiples of this class with the rational multiples of the loop $\gamma$ around zero, the pairing does not land in the rational numbers, since

$\displaystyle{\int_{\gamma} \frac{dz}{z} = 2 \pi i}.$

In particular, to compare de Rham cohomology over the rationals with the usual Betti cohomology over the rationals, one first has to tensor with a bigger ring such as $\mathbf{C},$ or at least with a ring big enough to see all the integrals which arise in this form. Such integrals are usually called periods, so in order to have a comparison theorem between de Rham cohomology and Betti cohomology over $\mathbf{Q},$ one first has to tensor with a ring of periods.

It is too simplistic to say that p-adic Hodge theory (at least rationally) is a p-adic version of this story, but that is not the worst cartoon picture to keep in your mind. Returning to the example above, note that the period is a purely imaginary number. This is a reflection of the fact that some arithmetic information is still retained, namely, an action of complex conjugation on the complex points of a variety over the rationals is compatible (with a suitable twist) with the de Rham pairing. A fundamental point is that, in the local story, something similar occurs where now the group $\mathrm{Gal}(\mathbf{C}/\mathbf{R}) = \mathbf{Z}/2 \mathbf{Z}$ generated by complex conjugation is replaced by the much bigger and more interesting group $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p).$ Very (very) loosely, this is related to the fact that p-adic analysis behaves much better with respect to the Galois group, for example, the conjugate of an infinite (convergent) sum of p-adic numbers is the sum of the conjugates. In particular, there is a Galois action on the ring of all p-adic periods. So now there is a much richer group of symmetries acting on the entire picture. Moreover, the structure of the p-adic differentials can be related to how the variety X looks like when reduced modulo-p, because smoothness in algebraic geometry can naturally be interpreted in terms of differential forms.

So now if one wants to make a p-adic comparison conjecture between (algebraic) de Rham cohomology on the one side, and etale cohomology (the algebraic version of Betti cohomology) on the other side, one (optimally) wants the comparison theorem to respect (as much as possible) all the extra structures that exist in the p-adic world, in particular, the action of the local Galois group on etale cohomology, and the algebraic structures which exist on de Rham cohomology (the Hodge filtration and a Frobenius operator), and secondly, involve tensoring with a ring of periods B which is “as small as possible”.

Identifying the correct mechanisms to pass between de Rham cohomology and etale cohomology in a way that is compatible with all of this extra structure is very subtle, and one of the fundamental achievements of Fontaine was really to identify the correct framework in which to phrase the optimal comparison (both in this and also in many related contexts such as crystalline cohomology). (Of course, his work was also instrumental in proving many of these comparison theorems as well.) I think it is fair to say that often the most profound contributions to mathematics come from revealing the underlying structure of what is going on, even if only conjecturally. (To take another random example, take Thurston’s insight into the geometry of 3-manifolds.) Moreover, the reliance of modern arithmetic geometry on these tools can not be overestimated — studying global Galois representations without p-adic Hodge theory would be like studying abelian extensions of $\mathbf{Q}$ without using ramification groups.

Two further points I would be remiss in not mentioning: One sense in which the ring $B_{\mathrm{dR}}$ is “as small as possible” is the amazing conjecture of Fontaine-Mazur which captures which global Galois representations should come from motives. Secondly, Fontaine’s work on all local Galois representations in terms of $(\varphi,\Gamma)$ modules which is crucial even in understanding Motivic Galois representations though p-adic deformations, the fields of norms (with his student Wintenberger, who also sadly died recently), the proof of weak admissibility implies admissibility (with Colmez, another former student, who surprisingly to me only wrote this one joint paper with Fontaine), and the Fargues-Fontaine curve. (I guess this is more than two.)

Probably the first time I talked with Fontaine was at a conference in Brittany (Roscoff) in 2009. That was the first time I ever gave a talk on my work on even Galois representations and the Fontaine-Mazur conjecture, about which Fontaine had some very kind words to say. (One of the most rewarding parts of academia is getting the respect of people you admire.) I never got to know him too well, due (in equal parts) to my ignorance of the French language and p-adic Hodge theory. But he was always a regular presence at conferences at Luminy with his distinct sense of humour. Over a long career, his work continued to be original and deep. He will be greatly missed.

## The classics

I now have the complete collection from of light satirical music of the ’50s and ’60s from the two masters of the form from either side of the pond:

They are both similar and very different at the same time — Lehrer is definitely the more acerbic of the pair, as evidenced by the following pair of quotes concerning satire (themselves satirical):

When Kissinger won the Nobel peace prize, satire died.

The purpose of satire, it has been rightly said, is to strip off the veneer of comforting illusion and cosy half-truth. And our job, as I see it … is to put it back again!”