Conferences New and Old: Coronavirus Edition

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

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

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!

Posted in Chess | Tagged , , | 2 Comments

Vesselin Dimitrov on Schinzel–Zassenhaus

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.)

Posted in Mathematics | Tagged , , , , , , | 1 Comment

Job Dedication

Earlier last quarter, I had suffered a fairly poor night from some sort of stomach bug. Unfortunately, I made the ill-advised decision not to cancel my classes and went to work, which involves driving from Evanston to Hyde Park. Things did not go well; I was feeling so grim during honours group theory that for a 20-minute period I had to sit down with my head slumped on the table, occasionally able to utter a few sentences explaining the Orbit-Stabilizer theorem (I think I did a pretty good job in the circumstances). I somehow managed to survive through the full 50 minutes, by which time I had realized that I had to immediately cancel my graduate class and go straight home. Now one of the pleasant aspects of my drive is that there is a large uninterrupted stretch on LSD in which one does not have to stop. On the flip side, it also turns out that there is a large uninterrupted stretch on LSD in which one is not able to stop. That fact was more pertinent in the current situation. And so, ladies and gentlemen, for approximately 3-4 minutes while traveling approximately 40 miles an hour, I recreated almost in full one of cinema’s (and Terry Jones, RIP) most iconic scenes, with the seats, windows, and steering wheel of my newly purchased second-hand car playing the role of the bucket. With that anecdote out of the way, let me post that clip here now:

Posted in Travel, Waffle | Tagged , , , , | 1 Comment

Inter-universal Teichmüller theory explained

Normally a message such as the one below would go immediately to the rubbish bin. Fortunately for me, I happened to open it accidentally and thereupon discovered the most cogent explanation to date of IUT. I hereby share with you (in its entirety) the following email which was sent to me by a gentleman going by the name Samarium Beesix.

PRIVATE AND CONFIDENTIAL:

16.01.20

DEAR SIR,

PLEASE FORGIVE THIS INTRODUCTION. THIS MORNING I WAS LOOKING AT IMAGES FOR INTER UNIVERSAL TEICHMULLER THEORY.

I DISCOVERED ONE PARTICULAR IMAGE, A SKETCH FROM SOMEONE AT MSRI, WHICH CONTAINS A FAMILIAR LOOKING GEOMETRY.

IN DECEMBER OF 2014 I WAS SENT A PHOTOGRAPH OF A STRANGE OBJECT WHICH APPEARED IN THE SKY OVER BEECHEN CLIFF IN BATH.

THE PERSON WHO TOOK THE PHOTOGRAPH WAS STANDING AT THE ABBEY CHURCHYARD IN BATH – IN BETWEEN THE RECENTLY ERECTED WOODEN HUTS FOR THE BATH CHRISTMAS MARKET.

THERE WAS ALSO A MALE AND FEMALE WITNESS TO THE EVENT WHO WERE WALKING PAST, THEY WERE COMPLETELY NONCHALANT AND UNPHASED BY THE INCIDENT.

THE MALE ACTUALLY AFFIRMED WHAT THEY HAD SEEN AND SAID:

‘SO WHAT?’

THE EYE WITNESS SAID THAT THE LIGHTS IN THE PHOTOGRAPH WERE ATTACHED TO A VERY LARGE AND NON-SYMMETRICAL MUSTARD COLOURED FRAME. AND NO, THEY WERE NOT CHRISTMAS LIGHTS.

THE SURFACE OF THE GIGANTIC FRAME WAS PITTED WITH WHAT LOOKED LIKE DARK IRREGULAR POOLS OF WATER, RATHER LIKE THE WATER THAT RESTS ON A SURFACE AFTER RAINFALL.

THE EYE WITNESS ALSO SAID THAT THE LATTICE FRAME HAD SOME FORM OF HEAT HAZE AROUND IT.

THE CAMERA USED TO TAKE THE PHOTOGRAPH WAS NOT OF THE BEST QUALITY. THE SLIGHT DISTORTION OF THE IMAGE WAS CAUSED BY CAMERA SHAKE, AS WELL AS THE LENS BEING FULLY EXTENDED TO CAPACITY.

PLEASE EXERCISE PROFESSIONAL DISCRETION. THANK YOU.

Posted in Rant | Tagged , , , , | 1 Comment

The last seven words of Kedlaya-Medvedovsky

New paper by my student Noah Taylor! It addresses some conjectures raised by Kedlaya and Medvedovsky in this paper. Let \(\mathbf{T}\) denote the Hecke algebra acting on modular forms of weight two and prime level \(N\) generated by Hecke operators \(T_p\) for \(p\) prime to \(N\) and \(2\) (the so-called “anemic” Hecke algebra). If \(\mathfrak{m}\) is a maximal ideal of \(\mathbf{T}\) of residue characteristic two, and \(\mathbf{T}/\mathfrak{m} = k\), there exists a corresponding Galois representation:

\( \overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{T}/\mathfrak{m}) = \mathrm{GL}_2(k).\)

If \(S\) denotes the space of modular forms modulo \(2\), then certainly \(\mathrm{dim}_{k}(S[\mathfrak{m}]) \ge 1\). Since there can exist congruences between modular forms, it is certainly possible that the generalized \(\mathfrak{m}\)-eigenspace of \(S\) has dimension greater than one. Kedlaya and Medvedovsky observe that if one assumes that \(\overline{\rho}\) has (projectively) dihedral image, then one can systematically predict lower bounds for this generalized eigenspace contingent on various properties of \(\overline{\rho}\). They prove a number of such results, but they finish the paper with what amounts to six more conjectures. Actually, one of the conjectures splits into two completely different cases, and so I like to think of it as seven conjectures.

Before stating the conjectures, first note that \(\overline{\rho}\) (when projectively dihedral) is necessarily induced from the field \(\mathbf{Q}(\sqrt{\pm N})\). The corresponding representation may or may not be ordinary at the prime 2. Also, let \(h(N)\) denote the even part of the class number of \(\mathbf{Q}(\sqrt{N})\). Now we can state the conjectures, which are now all proved by Noah:

  1. Suppose that \(N \equiv 1 \pmod 8\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 4.
  2. Suppose that \(N \equiv 1 \pmod 8\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{-N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least \(h(-N)\).
  3. Suppose that \(\mathfrak{m}\) is Eisenstein. Then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least \((h(-N) – 2)/2\).
  4. Suppose that \(N \equiv 5 \pmod 8\). If \(\mathfrak{m}\) is ordinary-\(\mathbf{Q}(\sqrt{N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 4.
  5. Suppose that \(N \equiv 5 \pmod 8\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{-N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 2.
  6. Suppose that \(N \equiv 3 \pmod 4\). If \(\mathfrak{m}\) is \(\mathbf{Q}(\sqrt{N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 2.
  7. Suppose that \(N \equiv 3 \pmod 4\). If \(\mathfrak{m}\) is ordinary-\(\mathbf{Q}(\sqrt{-N})\)-dihedral, then the generalized \(\mathfrak{m}\)-eigenspace has dimension at least 2.

Noah uses quite a number of different arguments to prove this theorem. One basic idea is that the extra dimensions are related to deformations of \(\overline{\rho}\), but only in some of the proofs is this connection transparent. More directly, Noah exploits the following:

  1. The existence of weight one dihedral representations. When \(\overline{\rho}\) is unramified at 2 it is natural to look to such forms. However, even when \(\overline{\rho}\) is ramified at two, the weight one forms, after giving rise via congruences to weight two forms, can often be level-lowered to level \(N\) using an argument similar to that employed by me and Matt in our paper on the modular degree of elliptic curves.
  2. Known properties of the real points of the Jacobian \(J_0(N)\), in particular the connectedness of \(J_0(N)(\mathbf{R})\) for prime \(N\) as proved by Merel. This can be used to give a lower bound of \(2\) when \(\overline{\rho}\) is totally real. In order to get a better bound in the even case (if necessary) one has to combine this with other arguments.
  3. The difference between the Hecke algebra \(\mathbf{T}\) and the Hecke algebra where the operator \(T_2\) is also included. If this Hecke algebra is strictly larger than \(\mathbf{T}\) after localization at \(\mathfrak{m}\), then one can show that the \(\mathfrak{m}\)-torsion of \(S\) has to be at least two, and moreover one can make this argument work nicely with some of the other methods for producing non-trivial lower bounds.

Concerning the third point: the difference between the Hecke algebra \(\mathbf{T}\) and the full Hecke algebra is the addition of the operators \(T_2\) and \(T_N\). Noah’s arguments crucially use this in the case of \(T_2\) but not of \(T_N\). But this is also explained in the paper: once you add the Hecke operator \(T_2\), it turns out that you have the full Hecke algebra! The fact that the Hecke algebra is integrally generated by \(T_p\) for \(p\) prime to the level is not true for general levels \(N\) but just happens to be true for \(N\) prime. It suffices to prove the result after localizing at any maximal ideal \(\mathfrak{m}\). Mazur proved it in the Eisenstein case by a somewhat subtle argument (it’s false in general for Eisenstein primes at non-prime level). Second, in the non-Eisenstein case, the argument uses the result that all irreducible representations modulo \(2\) are ramified at \(N\). If there were such a representation, it would be an absolutely irreducible and unramified away from \(2\), and Tate prove that no such representations exist!

Of course, apropos of the title, this post must finish with the following:

Posted in Mathematics, Music, Students | Tagged , , , , , , , , , , , , , , | 3 Comments

New Results in modularity, Christmas Update II

Just like last year, once again saint Nick has brought us a bounty of treasures related to Galois representations and automorphic forms in the final week of the year.

First there was this paper by Newton and Thorne, proving, among other things, the modularity of symmetric powers for a large range of holomorphic modular forms, including \(\Delta\) and any newform associated to a semistable elliptic curve. There is a lot to enjoy about this paper, not least of which is the nice application of an old computation of Buzzard and Kilford. But there are also some very nice new results on Selmer groups and reducible modularity lifting proved in the substantial related papers by Newton-Thorne and Allen-Newton-Thorne respectively. (Added: It was remiss of me not to also mention this paper by Thorne and Christos Anastassiades as well!) It’s often hard for the non-specialist to appreciate “technical” improvements on previous theorems, but in this case, they are all wrapped up neatly with a bow by such a clean application: \(\mathrm{Sym}^n(\Delta)\) is modular!

Moving on, we have this paper (monograph?) by Liu, Tian, Xiao, Zhang, and Zhu on the Bloch-Kato conjecture for a very general class of motives associated to Rankin-Selberg convolutions of forms on \(\mathrm{GL}_n\) and \(\mathrm{GL}_{n-1}\). I remember a few years ago talking to Yifeng during his interview at Northwestern (reader, we hired him) about this beautiful paper, giving a totally new argument to study questions of Selmer groups using cycles and level raising congruences. The current paper seems to be not only a version of that on steroids but also with a nice hot cup of tea with 3 lumps of potassium. It’s an amazing achievement which pulls together a lot of wonderful ideas, including Xiao-Zhu’s work on the Tate conjecture, not to mention all the previous work on the Gan-Gross-Prasad conjecture.

Well done to both groups of authors!

(In different times I would have given more details as to what these papers actually do, but as my free time nowadays consists of brief moments like this at 5:00AM in the morning you will have to forgive me, and anyway, these papers all seem to be very well written with nice introductions. That said, there will be some more technical mathematics posts coming up, not least of which relates to work of my own students. Stay tuned, Persiflage intends to keep posting!)

Posted in Mathematics | Tagged , , , , , , , , , , , , , , , , , , , | 6 Comments

Background for the Hausdorff Summer School

For those attending the Haussdorf Summer School previously mentioned here, I followed up with the speakers to ask them a little about what background was optimal for getting the most out of their lectures. In particular, I asked them to send me a sentence along the following lines: it would be useful for participants in this course to know X and to have some familiarity with Y, but no knowledge of Z is assumed for various (set) values of X,Y, and Z. Here are the responses, which I hope will be useful for some of you. (Some light editing has taken place which may have introduced typos, as I’m pretty prone to those.)

  • Arthur-Cesar le Bras and Gabriel Dospinescu on p-adic geometry.

    It would be useful for participants in this course to know the basic formalism of adic spaces (most importantly the notion of continuous valuation, adic spectrum, rational domain and being aware of some of the perversities related to the structure presheaf not being a sheaf in general) and to have some familiarity with Fontaine rings (and what they are useful for; we recommend Berger’s paper An introduction to the theory of p-adic Galois representations, sections I-II for a brief overview) and p-divisible groups (the latter being most likely a prerequisite for other courses as well)

  • George Boxer and Vincent Pilloni on Higher Hida theory.

    A basic familiarity with modular forms, modular curves, Hecke operators, etc. together with basic familiarity with rigid analytic geometry would be enough to follow a significant part of the course.

  • Patrick Allen and James Newton on Automorphy lifting.

    It would be useful for participants in this course to know the basics of Galois cohomology and modular forms and to have some familiarity with automorphic representations and deformation theory of Galois representations (although we will give a quick summary of the deformation theory needed).

  • Eva Viehmann and Cong Xue on Shtukas.
  • We will assume algebraic geometry and some familiarity with linear algebraic groups.

  • Sophie Morel and Timo Richarz on Geometric Satake.

    It would be useful to know about schemes, etale cohomology, the six functors, algebraic groups, root systems, highest weight theory, classical Satake isomorphism — for the last four topics, knowing the theory in the case of \(\mathrm{GL}_n\) or \(\mathrm{SL}_n\) is already pretty good. It would be helpful to have some familiarity with the Tannakian formalism, perverse sheaves, and G-bundles. No knowledge is assumed of loop groups or affine Grassmannians.

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Mathjobs Application Tips Update

Previously I wrote about what I considered a “bug” in mathjobs: when letter writers submit their letter, the default time those letters are available is 18 months. But this leads to the following chain of events:

  1. An applicant applies for a job. Perhaps because of the vagaries of the market (or because they only apply to a limited number of places) they do not get an offer.
  2. The same applicant applies (perhaps more broadly) the next year.
  3. Because the letters have 18 month expiry dates, the applications all list THE OLD LETTERS as well as the new letters.
  4. Because letter writers are often busy and/or lazy, they typically do not update a letter very much from one year to the next. Hence the letter they submit is almost identical.

The result is that it becomes completely clear to the letter reader that the candidate is applying for the second straight year. This has the chance of conveying the message that not only did they fail to get a job last year, but they haven’t done very much in the subsequent year either since the letters are pretty much the same. This is why I encouraged reference letter writers to be particularly careful when either choosing the default expiry date or when writing for someone for a second consecutive year.

Two job seasons later, this still seems to happen pretty frequently (I noticed it quite a few times on the [11] applications that I looked at so far). I was thus motivated to write to the AMS about this issue. Their main response was to gently point out to me that mathjobs is not exclusively a system for applicants to apply to R1 research institutions and that the needs of applicants might vary “possibly more than you realize.” In particular, they pointed out the opposite problem of hearing from “frantic job applicants whose letters have expired at a time when they need/want them right away.” After a little more discussion, however, they did point out to me the useful tip that the applicants themselves have a way of avoiding this from happening:

You can exclude existing letters from being used in any applications by clicking on the green checkmarks after the letters on your coversheet form to turn them into red x’s.

I was assured that you could do this explicitly in the context in which there were multiple letters from the same person over two years. (That is, as an applicant, one can “red x” Professor X’s letter from 2018 and “green check” Professor X’s letter.) So my recommendation is to do this! Even better, if you are an applicant applying for a second year and you do this, please let me know in the comments (anonymous names are OK!) that it worked.

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