How to reject a paper

Posted on March 31, 2020 by Persiflage

I just had the paper discussed in this post very quickly rejected. Since it was such a short paper, I thought it not too unreasonable to submit it to a strong journal, so I am not terribly disappointed. I can always say I chose such a journal for the benefit of my junior collaborator, of course. Rejections are almost always a bit of a kick in the gut, but I have to say that these were the nicest rejection letters I have ever received. It’s honestly more positive feedback than I receive on most of my papers which are accepted. Because of this, I thought I would share them (both) with you, in the hope that they will inspire (as a referee) how you phrase your future rejections. Remember that most authors get precious little positive feedback on their work. You of course have to evaluate the paper as you see it, but there’s almost always something positive and encouraging you can say. Maybe imagine that the author is your student… and you are giving them your opinion in person. The reports below have not been redacted in any way by me. (It may be possible that the editor stripped out less generous paragraphs, in which case kudos to the editor as well)

Report I: There is an old question about how often the Fourier coefficients of a modular form can vanish. I don’t know how important this question is, but it has intrigued people on and off for some time. The authors consider a variant of this question. they show that if you fix a prime p and a level N coprime to p then there are only finitely many non-CM Hecke eigenforms of level N and even weight for which the p’th Fourier coefficient vanishes. One might reasonably ask why one would care. The authors only answer seems to be the analogy to things like Lehmer’s conjecture. On the other hand the proof is a beautiful application of ideas from the deformation theory of Galois representations. I took me only a few minutes to understand the argument, which is very elegant and short. I guess is there is a case for publication it is that the proof is so nice. I’m not really sure what to recommend. The paper is very short, which I suppose is a plus, and it is a pleasure to read. As I say the downside is that it is not clear why one should care about the main theorem.

Report II: On the positive side, the result is simple to state and appealing, and the proof is indeed relatively short and elegant (although it is obtained more through an application of existing techniques in deformation theory of modular forms and Galois representations, than through the development of new techniques in this area.) The result is presented as being in the spirit of Lehmer’s conjecture, which asks how often a_p(f) can be zero for a fixed eigenform f of weight >2; in this paper one is fixing p and varying f instead over all forms of a fixed level but varying weights. This leads to a very different kind of problem, which can be tackled by different methods, involving the p-adic deformation theory of modular forms. To my knowledge, this “vertical Lehmer conjecture” had never been considered before, and the introduction, which is very brief, is somewhat lacking in motivation. And it seems very unlikely that the Vertical Lehmer conjecture which is proved will tell us anything about the original, horizontal version! I guess that, although the result is quite nice, its importance is perhaps not made clear enough, in the paper, to justify the appearance in a very top-flight journal like redacted.

I would see it well in a journal of a caliber right below redacted. Perhaps the authors could be asked make a somewhat better case in their introduction for the importance and eventual impact of this result . Assuming such a case can be made: they shouldn’t of course be asked to say more than they believe; if they just think the result is of interest in its own right without further applications or ramifications, that is perfectly fine as well, but then perhaps redacted is aiming a bit high.

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First of all, thank you to both reviewers for those kind words; it’s hard to find fault with anything you say. To answer the implicit question of the second reviewer, I personally am happy for the result to stand on its own, and didn’t particularly want to make a case for the importance of the result. I think of it more as an amuse-bouche. But good journals can publish amuse-bouches too! I guess one way to try to sell the paper would be to draw analogies where one replaces counting modular forms with \(a_p = 0\) by a condition at the infinite prime. One analogue (more fundamental of course) is counting Maass forms with \(\lambda = 1/4\). The narrative would be that these problems are fundamentally hard — for similar reasons — to study from an analytical point of view, and it is only when one can relate the problem to very arithmetic questions can one hope to make progress. Chacun à son goût as they say, at least they do in Austrian operetta, I have no sense if actual Francophone people say this or not (but please tell me in the comments).

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You are welcome, Northwestern junior faculty

Posted on March 27, 2020 by Persiflage

Nobody has even accused Persiflage of being afraid to speak truth to power. After Chicago made their announcement that they were extending tenure by one year for all (qualified) faculty, I thought that, as a good local citizen of Evanston, I should put the pressure on Northwestern to do this same. Naturally, I first considered tweeting the official Northwestern account, or even better, the twitter feed of Morty Schapiro (Northwestern’s President). Alas, he didn’t seem to have a twitter feed, so I looked up the email address instead. The best I could find was nu-president@northwestern.edu, which I presumed would just be read by a staff member in the president’s office. Still, it was the best I could find, so I forwarded the email announcement from the University of Chicago’s provost to this address, adding a few choice words: Your turn, Northwestern! Surely you’re not going to let the *University of Chicago* (and Harvard, and etc.) outdo you here? There is no better way to persuade a university to do something than to point out the bold actions of “peer institutions.” Then, less than 10 minutes later, I got a response:

And now, the very same week, Northwestern has gone ahead an taken up my suggestion and extended the tenure clock. Naturally, I am too modest to take 100% of the credit; I will give Morty Schapiro at least 10% for listening and then another 10% for being gracious enough to reply immediately to an email which was written in a manner somewhat more casual than (I imagine) the usual letter to a university president. Of course, Morty is very well known amongst the students for being far more approachable than the average university president. I think that if I had written and sent a similar email to Bob Zimmer’s office, I would have phrased it slightly more politely.

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More on Lehmer’s Conjecture

Posted on March 21, 2020 by Persiflage

Lehmer said it was a “natural question” whether there existed an integer such that \(\tau(n)=0\) or not. I’ve wondered a little bit recently about how reasonable this is. (See this post.) The historical context is presumably related to the fact that, by the multiplicativity of coefficients, the vanishing of \(\tau(p)\) for one prime guarantees that a positive proportion of other coefficients vanish. From the perspective of Galois representations, however, I’m a little confused as to whether we expect any sort of “automorphic” Lehmer’s conjecture to hold. To recall, we have

\(\Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n.\)

Deligne’s bound says that \(|\tau(p)| \le p^{11/2}\), so a probabilistic argument suggests that there should only be finitely many primes for which \(\tau(p)=0\). Since there aren’t any such primes in the first few thousand primes, that’s a fairly convincing heuristic for why it might be true. But it’s basically impossible to prove anything this way, so one might hope to formulate a more general conjecture which is true for a more systematic reason.

A first attempt might be to ask that \(a_p(f) \ne 0\) for any eigenform \(f\) of weight \(k \ge 3\) and level prime to \(p\) which is not CM. (When \(k = 2\), of course, there are plenty of modular elliptic curves without CM, and (thanks to Noam) there are plenty of primes \(p\) with \(a_p(f) =0\)). At first thought this seems a little strong; after all, if we just work in weight \(12\) (say) then we know that \(|a_2(f)| < 2^{11/2} < 46\), so surely if you take enough such forms you should find one with \(a_2(f) = 0\). However, this secretly assumes that there are many weight \(12\) forms with coefficients in \(\mathbf{Z}\), and it seems that there are only finitely many such forms. So, for most forms, the coefficients would lie in (presumably) larger and larger number fields, and there would be more possibilities for \(a_2(f)\). For those who did the computation and might be worried, note that the probabilistic heuristic above only applies when the weight \(k \ge 4\). On the other hand, an easy exercise shows that when the weight is odd and the coefficients are integral then the form has CM. The conjecture that there only finitely many non-CM forms with rational coefficients in large even weight is certainly made in this paper, although Dave seems to be hesitant on numerical grounds to make the conjecture for \(k = 4\). There seem to be enough forms of weight \(k = 4\) and integer coefficients that perhaps there exists a form of odd level with \(a_2(f)=0\). In fact, it should be easy to search for such forms if you can search the LMFDB with a fixed Hecke eigenvalue, which I remember John Voight demonstrating at the Simons Institute general meeting, but I couldn’t work out just know when writing this post. Ah, but I guess one can just search for forms with coefficients in \(\mathbf{Q}\) and just look at them by hand. It appears that there is a form

\(f = q + 4 q^3 – 8 q^4 – 5 q^5 – 22 q^7 – 11 q^9 \ldots \in S_4(\Gamma_0(95),\mathbf{Q})\)

with \(a_2(f) = 0\). Are there any examples in higher \(>4\) weight?

All of this becomes similarly confusing on the level of Galois representations. The modular forms with \(a_p(f) = 0\) have the special property that the local \(p\)-adic Galois representation \(\rho_f\) is induced from the unique unramified quadratic extension of \(\mathbf{Q}_p\). From this perspective, the Lehmer conjecture looks a little bit like Greenberg’s conjecture that an ordinary modular form is split if and only if it has complex multiplication. But whereas that conjecture (or at least a stronger version where one assumes such a splitting at all primes of the coefficient field) does follow from plausible conjectures about motives as explained by Matt. I wonder if Matt has any more opinions on what happens if one makes the assumption for only one prime of the coefficient field. Note that if you read Matt’s paper, you might be confused why you can’t also use Serre-Tate theory to prove that elliptic curves with \(a_p = 0\) have CM. I think Florian Herzig gives a nice explanation here of what is going on.

This is also related to the question raised in this post. While that conjecture is not unreasonable, it does skirt the border of conjectures which are actually false, for example, the conjecture that any exceptional splitting in a local Galois representation is caused by (more or less) some global splitting. After all, taken to it’s logical conclusion it would imply not only Lehmer’s conjecture but also (combined with Elkies’ theorem) that all elliptic curves are CM. Greenberg’s conjecture excludes the case of weight one forms, since certainly any form with finite image has many primes for which the local Galois representation is split but the global representation is not if the image is of exceptional type. One can still argue, however, that these forms are potentially CM. On the other hand, non-CM Hilbert modular forms of partial weight one, induced to \(\mathbf{Q}\), also admit some exceptional splitting on inertia. (Note that non-CM Hilbert modular forms actually exist, as follows from the computation of Moy and Specter described here). While these induced forms are not of regular weight (the HT weights are, up to twist \([0,2,2,4]\), the splitting of the local Galois representation is not explained by any correspondences over any finite extension.

I guess the summary is that all of this dicussion points to the fact that Lehmer’s conjecture is not true for any good reason beyond random probability grounds and so is kind of rubbish. Actually this reminds me of an experience one occasionally has after giving a seminar in which you feel like you proved a snazzy result and then the questions from the audience are somewhat deflating. Rest assured, this happens to the best of us — I remember watching a talk online where RLT was talking about his (joint) proof of the Sato-Tate conjecture for \(\Delta\), and the only question from the audience was does this have any implications for Lehmer’s conjecture?

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The Hausdorff Trimester has been indefinitely postponed

Posted on March 18, 2020 by Persiflage

I’m not sure if this is 100% official yet, but the Hausdorff Trimester scheduled for this summer has been (unsurprisingly) postponed. This is probably no great surprise to many of you. We have hopes to reschedule it again (perhaps for 2022) if possible. Together with this update, this is the final unfortunate installment of this series of posts.

On the other hand, mathematics departments have already started to hold online seminars. I was chatting with TG and GB last night and one of them pointed me towards the MIT NT seminar which held it’s first online talk yesterday. If you have zoom installed, which you probably do since you are now expected to teach on zoom next quarter, it is really easy to connect. Bjorn announced that the plan was to continue the seminar online next quarter.

If anyone else plans to hold number theory seminars online, please leave a comment and I will add it to the list.

1. MIT NT seminar (online)

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NSF Proposal, Graduate Fellowship Edition

Posted on March 17, 2020 by Persiflage

Note: I feel as a service to the number theory entertainment complex that I should blog more often in these times, even if it means being less coherent than usual. I might even try to get a few guest posts since I won’t be going to any conferences any time soon…

I recently linked to my first NSF proposal here, but just today I stumbled upon my graduate NSF fellowship application from 1998. There is really only one page which involves any proposal (rather than a list of courses I took or references), and I include the mathematical portion here in full (the only changes from the original are one or two spelling errors and changing fake LaTeX (\rho) to real LaTeX (\(\rho\)).

===============================================================

My research interests center mainly around the study of two dimensional Galois representations, the connection of such representations to Modular forms, and application of these connections to the arithmetic of Elliptic curves. Here are several possible questions which are of interest to me.

1. Serre’s conjectures predict that for any odd, absolutely irreducible Galois representation \(\rho\) into \(\mathrm{GL}_2(\mathbf{F}_q)\), there exists a modular form \(f\) which gives rise to \(\rho\), in the sense of Deligne/Shimura/Deligne-Serre. The characteristic zero representation \(\rho_f\), however, need not be defined into \(\mathrm{GL}_2(W(\mathbf{F}_q))\), (\(W(\mathbf{F}_q)\) = Witt–vectors of \(\mathbf{F}_q\)), but perhaps in \(\mathrm{GL}_2(\mathcal{O})\), for some ramified extension \(\mathcal{O}\) of \(W(\mathbf{F}_q)\). Is there any sense in which one can quantify the ramification of \(\mathcal{O}\) in advance? Is there perhaps a clear cohomological obstruction to a modular lift over \(W(\mathbf{F}_q)\)? Can one quantify this in terms of some \(H^2(G_{\mathbf{Q}},*)\), or perhaps in terms of \(R^{\mathrm{univ}}\), where \(R^{\mathrm{univ}}\) is the universal deformation ring of \(\rho\)? Perhaps if this is too difficult, some qualitative result in this direction?

2. Applications of the above ideas to rational cuspidal eigenforms of Level \(1\). Such forms
are only known to exist if \(\mathrm{dim} S_{2k}(1) = 1\). Can one use ideas from representations to show that no other cuspidal eigenform can be defined over \(\mathbf{Q}\)?

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My first thought is “I guess I haven’t changed that much as a mathematician over 22 years” followed by “not a bad problem but too optimistic.” The funny thing is that I do think of myself as a number theorist with a certain amount of breadth (despite protestations to the opposite from my most dyspeptic collaborator), so I guess I have to claim that I work on a large circle of ideas and sometimes return to very similar points on the circle. There are also echos in the first proposal of future work with Matt where we studied the ramification of \(\mathcal{O}\) for the reducible representation occurring in weight \(2\) and prime level \(N\), as studied by Mazur. The most definitive result in that paper was computing the exact ramification degree when \(p=2\), where in the case that the ring was a DVR one had \(e = 2^{h-1} – 1\), where \(2^h\) was the order of the \(2\)-part of the class group of \(\mathbf{Q}(\sqrt{-N})\). Other progress on this problem more in the spirit of the formulation above has been done by Lundell and Ramakrishna (MR2770582), although I still think there are many open questions around this problem which are of interest.

On the other hand, the second problem is too optimistic. One reason is related to Buzzard’s observation that, in high weight with \(p\) fixed, the representations all seemed to be defined over rings with very little ramification. (He goes on to make a conjecture along these lines for which nobody has made any progress.) So it seems unlikely to rule out forms of large weight with coefficients in \(\mathbf{Q}\) by showing that there are no such forms over \(\mathbf{Q}_p\) because the latter seems to be false in a strong way. The problem of showing there are no more eigenforms over \(\mathbf{Q}\) when the weight is at least 24, which is very close to Maeda’s Conjecture, is something on which virtually no progress has been made since my proposal, so I guess I don’t have to feel bad for not making any progress myself. On the other hand, I don’t actually think it is an impossible problem. Maybe I should work on it!

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The Arbeitsgemeinschaft has been indefinitely postponed

Posted on March 11, 2020 by Persiflage

An update on the last post: As you probably already know by now if you are a participant, the Arbeitsgemeinschaft has been indefinitely postponed. The recommendation to do so was made by the organizers as universities rapidly began to recommend the cancellation of all work related travel.

I hope that as many of you as possible can cancel your travel plans and get fully reimbursed. For those of you who have difficulties with this, especially those to whom I committed travel funding, please stay tuned. The last message I received from our administration on this matter was the following:

I have forwarded your message to our grants experts. The American Physical Society has asked funding agencies to allow reimbursement for travel expenses that are not reimbursable by airlines and are usually not allowed on grants as they cancelled the very large march meeting the day before it was to begin. I hope to hear if their request has been acted on yet.

Posted in Mathematics, Travel | Tagged | 4 Comments

Conferences New and Old: Coronavirus Edition

Posted on March 7, 2020 by Persiflage

A number of people have asked me whether the various conferences and workshops I am organizing this summer are still running. I thought I would have a blogpost containing all the current information, which I can update when and if necessary.

Question: Will the Arbeitsgemeinschaft and HIM trimester proceed as planned?

Answer: The answer is that there are no current plans to cancel or postpone either of these events. The MFO has issued a statement here. That said, the situation may change. There was an upcoming conference in Darmstadt which was cancelled today (March 6) with participants receiving the following message (see here):

Although we were hoping and expecting so far that the conference would
take place as planned, and although we still consider the risk of an
infection extremely low, the recommendations from the university TU
Darmstadt and from the German Research Foundation DFG have changed by now, and advise to cancel/postpone conferences and similar events at
this moment.

Note that same site lists a message from March 2 which says We definitively expect that the conference will take place as planned. Given that such plans can change rapidly, and yet buying a plane ticket is something that gets done significantly in advance, this leads naturally to the following question:

Question: Suppose the conference is cancelled. What does that mean for my non-refundable airfare/transport?

Answer: This is a good question, and one I have been trying my best to answer since I have some 25K in participant funds on my NSF grant I have committed to travel for US participants. The answer is pretty much that nobody knows, which is maybe not entirely surprising in the circumstances. I spoke with both my own finance office (who administer my grant) and my program officer at the NSF. The answer from the University of Chicago at this point is there are no definitive answers yet, beyond noting that anyone in this position should start by trying to work directly with the airline. The answer from NSF is also still a work in progress, although this question is directly addressed initem #11 on this list which I reproduce here:

A conference has been canceled, but I have nonrefundable travel and/or hotel costs. Can these be charged to a NSF Conference or Travel grant?

NSF is currently working internally as well as with our federal partners on a number of proposal and award-related issues pertaining to COVID-19. NSF will communicate with the community about these issues and will provide guidance as further information becomes available. In the meantime, please continue to follow all relevant policies and procedures, including those of your organization, and apply those practices consistently.

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I apologize that this email doesn’t really contain any definitive information, although it has definitive confirmation from the NSF that definitive answers are not yet available. I will certainly try to update this blogpost whenever I get further information, but please also feel free to post your own updates and questions.

Update, March 8: At least one US university have asked their faculty to avoid “non-essential travel to Germany”.

Update, March 10: Several more universities have banned “non-essential travel”. Chances of cancellation/postponement are now seriously high, hope to have more information in the next day or so.

Posted in Travel | Tagged , , , , , | 4 Comments

Counting solutions to a_p = λ, Part II

Posted on March 3, 2020 by Persiflage

This is a sequel to this post where the problem of counting eigenforms with \(a_p = \lambda\) and \(\lambda \ne 0\) was considered. Here we report on recent progress in the case \(\lambda = 0\).

It is a somewhat notorious conjecture attributed to Lehmer (who merely asked the question, naturally) that the coefficients of

\( \Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum \tau(n) q^n = q-24q^2+252q^3+\ldots \)

never vanish. One problem with this conjecture is that there really isn’t any compelling reason it should be true except (basically) on probability grounds given the growth of the coefficients. As with a number of problems concerning “horizontal” questions about modular eigenforms (fixing the form and varying the prime \(p\)), it is often easier to consider the analogous “vertical” question where one fixes \(p\) and varies the weight. Namely: fix a tame level, say \(\Gamma = \Gamma_1(N)\), fix a \(p\) prime to the level, and then consider the eigenforms of level \(\Gamma\) and varying weight with \(a_p = 0\). Unlike in the case of Lehmer’s conjecture, is certainly can happen that \(a_p = 0\), for example:

  1. If \(f\) is associated to a modular elliptic curve \(E\) with supersingular reduction at \(p \ge 5\).
  2. If \(f\) has CM by an imaginary quadratic field \(K\) in which \(p\) is inert.

Let \(S(X)\) denote the number of cuspforms of level \(\Gamma\) and weight \( \le X\) such that \(a_p = 0\). Consider bounds on this function. The trivial bound, given by counting all cuspforms, is \(S(X) \ll X^2\). If you try to improve this bound using analytic techniques, namely via the trace formula, you can only do very slightly better, say \(S(X) \ll X^2/\log(X)\). The problem is that the condition \(a_p = 0\) within the space of all spherical representations \(\pi_p\) which could possibly occur has measure zero, so any trace formula approach will have to use a test function where \(a_p\) has support in some non-trivial interval \([-\varepsilon,\varepsilon]\) depending on the weight. This is the same problem (more or less) which prevents the analytic approach from giving optimal bounds to the number of weight one modular forms (where now the measure zero condition is being imposed at the infinite prime instead of at the prime \(p\)). One approach to improving these bounds is to use non-commutative Iwasawa theoretic methods as employed in my paper with Matt and then used by Simon Marshall to give the first non-trivial bounds for spaces of modular forms for \(\mathrm{GL}(2)\) over imaginary quadratic fields of fixed level and varying weight. This approach should lead, in principle, to a power saving over the trivial bound.

On the other hand. the best possible bound on \(S(X)\) will have the shape \(S(X) \ll X\), because the number of CM forms of each weight will be bounded independently of the weight, and there is no reason to imagine that the other exceptions will contribute anything of this order. Indeed, in my previous post, I conjectured that there should only be finitely many such forms of fixed level which are not CM as the weight varies.

When I last visited Madison in 2018, Naser Sardari was working on this problem, and in a preprint from late 2018, he proved exactly a bound of the optimal shape \(S(X) \ll X\), with the slight caveat that one should restrict to even weights. Quomodocumque blogged about it here.

Just a few weeks ago, Naser was in town in Chicago, and we got to talking about this problem again. Happily, we were able to come up with one more extra ingredient to push the original result to a (close to) optimal conclusion, and prove the aforementioned conjecture:

Theorem: (C, Sardari). Fix a prime \(p > 2\) and a tame level \(\Gamma_1(N)\). Then there are only finitely many eigenforms of level \(\Gamma\) and even weight with \(a_p = 0\) which are not CM.

This establishes a vertical version of Lehmer’s conjecture, up to a congruence on the weight, which arises for a technical reason discussed more below.

The first main idea of the proof is as follows. The \(p\)-adic Galois representation associated to \(\rho_f\) for a modular form can be very complicated viewed as a representation of the Galois group of \(\mathbf{Q}_p\). However, if \(a_p = 0\), then the local representation has a very special form: it is induced from an unramified extension \(K/\mathbf{Q}_p\). Breuil gave a precise formula for the representation, but a fairly soft argument shows that it is induced — 2-dimensional irreducible crystalline representations over \(\mathbf{Q}_p\) are determined by \(a_p\), and twisting by an unramified character fixes both the determinant and the condition \(a_p = 0\), hence \(V = V \otimes \eta_K\) is induced. That means that one can capture the locus of such representations by a local deformation condition. It is not the case that locally induced implies globally induced, as can be seen from the example of supersingular elliptic curves. This is related to the fact that the map

\(\pi: R^{\mathrm{loc}} \rightarrow R^{\mathrm{glob}}\)

of (unrestricted at \(p\)) local to global deformation rings is not a surjection. On the other hand, we know in some generality that this is a finite map. This was explored in this post, and then more properly taken up in a paper I wrote with Patrick Allen. The argument to this point is now enough to prove the original result of Sardari. Let \(R^{\mathrm{loc,ind}}\) denote the local deformation ring of induced representations. If \(R = R^{\mathrm{glob}} \otimes_{R^{\mathrm{loc}}} R^{\mathrm{loc,ind}}\) denotes the global deformation ring of locally induced representations, we know that the forms with \(a_p = 0\) and a fixed weight are the points of this deformation ring which lie in the fibre over some fixed point in local deformation space. Hence the finiteness of \(\pi\) gives a uniform bound on the number of points in this fibre, and hence a uniform bound over the number of such modular forms in any fixed weight. BTW, for those wondering why there is a restriction on the parity of the weight, it is only really there to prevent the residual representation from being globally reducible, a setting in which one doesn’t quite yet know the finiteness of \(\pi\). (When the optimal \(R = \mathbf{T}\) theorems become available in the reducible case, our methods should apply without any restrictions.)

Now comes the second ingredient. In order to explain it, let me describe the ring \(R^{\mathrm{loc,ind}}\) in more detail, or at least the part coming from inertia. This local deformation ring is basically equal to the deformation ring of the trivial character of \(G_K\), and in particular the ring has the form

\( \mathbf{Z}_p[[ \mathcal{O}^{\times}_K(p)]] \)

where \(A(p)\) denotes the maximal pro-\(p\) subgroup of \(A\). This ring is isomorphic (at least for odd p) to the Iwasawa algebra \(\mathbf{Z}_p[[X,Y]]\) after (via the p-adic logarithm) fixing a choice of multiplicative basis for \(\mathcal{O}^{\times}_K(p)\). Imagine that some component of the global deformation ring (with a locally induced condition) has infinitely many points which correspond to classical non-CM modular forms of level prime to \(p\). The points in weight space correspond to the algebraic characters of the following form:

\( \mathcal{O}_K \rightarrow K^{\times}, \qquad z \mapsto z^n \)

We now exploit the following fact which might (at first) be surprising: any infinite collection of these weights are Zariski dense! To make things a little more concrete, suppose we choose a basis of \(\mathcal{O}^{\times}_K(p)\) of the form \(1 + p\) and \((1+p)^{\eta}\), for a suitable \(\eta \in \mathcal{O}_K\) which will not be in \(\mathbf{Z}_p\), for example, \(\sqrt{u}\) for some non-quadratic-residue. The corresponding points with respect to the usual Iwasawa parameters have the shape:

\( X \mapsto (1 + p)^{n}-1, \qquad Y \mapsto (1 + p)^{\eta n}-1.\)

Instead of proving here why these are Zariski dense, it might be more useful to explain a very close analogy that Naser brought up with Lang’s Conjecture: if you take an infinite set of pairs of points of the form \((\exp(x),\exp(\eta x)) \subset (\mathbf{C}^{\times})^2\), then they will be Zariski dense whenever \(\eta \notin \mathbf{Q}\). In other words, the group subvarieties of the formal torus going through \((X,Y)=(0,0)\) basically all have to be of the form \((1+X)^{\eta} = (1+Y)\) for \(\eta \in \mathbf{Z}_p\). (Coincidentally, the arithmetic applications of Lang’s conjecture was the subject of the recent Ahlfors lecture by Peter Sarnak which you can watch here. Our result is yet another application!)

Once your non-CM points are Zariski dense, you are home and hosed: using an idea due to Ghate-Vatsal, you now specialize at lots of points which are inductions of finite order characters. The corresponding Galois representations have finite image on inertia and so are classical by known results. But then (apart from finitely many exceptions) they have to all be CM, because they are classical weight one forms, and the image of inertia is sufficiently large to rule out them having exceptional image.

One might ask whether the results are effective. I’m not so sure because of the following issue. Suppose you take \(p = 79\) and level one (I’m not sure this case will exhibit the required behavior but it might.) Then you might be able to prove that the global locally-induced deformation ring is (now over all weights) \(\mathbf{Z}_p = \Lambda/\mathfrak{P}\). But it might be very hard to tell if that weight \(\mathfrak{P}\) corresponds to a classical weight or a random weight, simply because \(\mathbf{Z}\) is dense in \(\mathbf{Z}_p\). This is not unlike the problem of showing that the zeros of the Kubota–Leopoldt zeta function are not in arithmetic weights.

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

Vale instantchess.com

Posted on February 27, 2020 by Persiflage

One of the few time wasting activities I still indulge in is speed chess. (1 minute per player for the entire game is the slowest time control I play online.) There are a number of excellent free online sites available, but one that wasn’t quite in that category was “instantchess.com.” One terrible aspect of this website was that your opponents were random, and in particular it completely disregarded ratings when assigning matches. Actually, it was slightly worse than this; it seemed to have a preference for setting up games between people who had played before, but the algorithm included games which one immediately abandoned because of the rating differential. So you would end up in cycles where you would abandon a game because of the mismatch, and the site would immediately assign you the same opponent. Even though lichess.org (and other places) are infinitely better and free, this website was much older and maintained mostly voluntarily, so one really shouldn’t complain. Moreover, even though it wasn’t a perfect site, there was a certain simplicity which meant that I often found myself playing there. Perhaps some of the appeal was the generosity of the rating system: the last time I played I was ranked 13th out of all 7000 or so lightning players, and had a “2400+” rating which I promise is greatly flattering to me:

I don’t think I ever played the top ranked player, but I did play most people on this list a few times. I would assume that none of them are GMs. I do wonder how many GMs I have inadvertently played on lichess — I’ve certainly been absolutely crushed often enough for it to be possible. (Definitely the best improvement on lichess was preventing anonymous users from using the chat feature.)

The flip side of this website going down is that it might push me to more useful ways of wasting my time!

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Vesselin Dimitrov on Schinzel–Zassenhaus

Posted on February 10, 2020 by Persiflage

Suppose that \(P(x) \in \mathbf{Z}[x]\) is a monic polynomial. A well-known argument of Kronecker proves that if every complex root of \(P(x)\) has absolute value at most 1, then \(P(x)\) is cyclotomic. It trivially follows that, for a non-cyclotomic polynomial, the largest root \(\alpha\) in absolute value satisfies \(|\alpha| > 1\). Elementary considerations imply that this can be improved to

\(|\alpha| > 1 + c_n\)

for some real constant \(c_n > 0\) that only depends on the degree. What is the true rate of decay of this parameter as the degree increases? By considering the example \(x^n – 2\), the best one can hope for is that \(c_n\) can be taken to have the form \(c/n\) for some constant \(c\). This is exactly what is predicted by the Schinzel-Zassenhaus conjecture:

Conjecture [Schinzel-Zassenhaus] there is an absolute constant \(c\) and a bound

\(\displaystyle{|\alpha| > 1 + \frac{c}{n}} \)

for the largest root of all non-cyclotomic polynomials.

In fact, Schinzel-Zassenhaus don’t actually make this conjecture. Rather, they first prove a bound where \(c_n\) has the form \(2^{-n}\) up to a constant, and then go on to say that they “cannot disprove” the claim. And of course, this then gets turned into a conjecture named after them! The best bounds were rapidly improved from exponential to something much better, but the original conjecture remained open. That is, until Vesselin Dimitrov in this paper proved the following:

Theorem [Vesselin Dimitrov] The Schinzel-Zassenhaus conjecture is true.

Vesselin’s result is completely explicit, and gives the effective bound \(|\alpha| \ge 2^{1/4n}\), or

\(\displaystyle{c_n = 2^{1/4n} – 1 \sim \frac{\log(2)}{4n}.}\)

The actual proof is very short. Step 0 is to assume the polynomial is reciprocal, which is a quite reasonable assumption because the conjecture (and much more, including Lehmer’s conjecture) was already known by work of Smyth the non-reciprocal case (MR0289451). I’m not sure this step is even needed, since the conjecture is certainly true for polynomials whose constant term is not plus or minus one, and so one can simply replace the polynomial by the reciprocal polynomial in what comes below. Step 1 is to show the inclusion

\( \displaystyle{\sqrt{\prod (1 – \alpha^2_i/X)(1 – \alpha^4_i/X)} \in \mathbf{Z}[[1/X]].}\)

The argument here is elementary (the only prime to worry about is \(p = 2\)). If the original polynomial is cyclotomic, then this squareroot is actually a polynomial, but otherwise it is a power series which is not rational. But now one has a power series which has an analytical continuation outside a very specific region in the plane, namely the “hedgehog” (I would have called it a spider) consisting of rays from \(0\) to \(\alpha^4\) and \(\alpha^2\) in \(\mathbf{C}\). These rays may overlap, but that only improves the final bound. The complement of the Hedgehog is a simply connected region, and know one wants to say that any power series with integer coefficients that has an analytic continuation to such a region with sufficiently large transfinite diameter has to be rational. Step 2 is then to note that such theorems exist! The transfinite diameter of the region in question can be computed from results already computed in the literature, and the consequent bounds are enough to prove the main theorem, all in no more than a couple of pages! It is very nice argument indeed. As a comparison, to orient the reader not familiar with Bertrandias’ theorem (which is used to deduce rationality of the power series in question), it might be useful to give the following elementary variation. Suppose that instead of the hedgehog, one instead took the complement of the entire disc or radius \(r\) for some \(r < 1\). (Importantly, this does not contain the hedgehog above which has spikes outside the unit circle.) Replacing \(X\) by \(1/X\), one ends up with a power series on the complement which is a disc of radius greater than one. Now one can apply the following Trivial Theorem: A power series in \(\mathbf{Z}[X]\) with radius of convergence bigger than one is a polynomial.

Plugging this into Dimitrov’s setup, one deduces a new proof of Kronecker’s theorem! So the main technical point is that the “trivial theorem” above can be replaced by a more sophisticated version (to due Bertrandias and many others) where the region of analytic continuation can be taken to be something other than a disc. (For an exposition of some of these rationalization/algebraization theorems, a good point to start is this post of Matt Baker.)

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