Clozel 70, Part II

Posted on September 30, 2023 by Persiflage

Many years ago, Khare asked me (as I think he asked many others at the time) whether I believed their existed an irreducible motive \(M\) over \(\mathbf{Z}\) (so good reduction everywhere) with Hodge-Tate weights \([0,1,2,\ldots,n-1]\) for any \(n > 1\). (Here the Motive is allowed to have coefficients.) When \(n=2\), the answer is no. Assuming all conjectures, such an \(M\) must be modular associated to a cusp form of weight \(2\) and level one, but no such cuspform exists. But the answer is also no unconditionally (for any notion of motive), and this fact is intertwined with the (inductive) proof by Khare and Wintenberger of Serre’s Conjecture. The hope might be that if no such motive existed for all \(n\), it could serve as the inductive basis for a more general form of Serre’s Conjecture.

My response at the time was that I guessed that no such motive existed for any \(n\). I generally feel that my intuition is quite good in these matters, so it was surprising to learn some time later a convincing meta-argument that such motives should really exist. This idea, which I can’t now remember whether I learned from Chenevier or Clozel, is related to trying to construct such forms which are in addition self-dual and so come from a classical group. In favourable situations, there exists a compact inner form on this group, so that computing these forms “reduces” to computing on certain finite sets. One such finite set turns out to be the set of positive definite lattices of discriminant one and dimension \(n\). As is well-known, they only occur in dimensions a multiple of 8. For \(n=8\) there is just \(E_8\), and for \(n=16\) there are two, and for \(n=24\) there turn out to be exactly \(24\), as classified by Niemeier, and which include the famous Leech lattice whose automorphism group is a central extension of the first sporadic group discovered Conway. Easier to compute is the weighted sum of such lattices by automorphisms; for \(n=24\), for example, this weighted sum is

\( \displaystyle{\frac{1027637932586061520960267}{129477933340026851560636148613120000000}}\)

which is very small, and of course is related to the fact that these lattices are quite symmetric. For \(n = 32\), however, the weighed sum is bigger than \(10^7\), and so there are lots of lattices. You might then think that the existence of these lattices (even just \(E_8\) when \(n=8\)) implies the existence of automorphic forms which then should give rise to the desired automorphic forms on \(\mathrm{GL}(n)\). But there are issues. One concerns the technical issue of trasfering forms between groups which is of course a subtle problem. But there is another. A form which is cuspidal on some group need no longer be cuspidal after transferring to \(\mathrm{GL}(n)\). So to see which forms are cuspidal you really need to do a computation. But these objects are of large complexity — already computing Hecke actions on supersingular points for \(X_0(11)\) is a non-trivial exercise; here the objects involve lattices of enormously high dimension. Chenevier and his co-authors, (including Lannes, Renard, and Taïbi) have done a remarkable job understanding what is going on here. The most basic example of the type of theorem they prove is as follows. When \(n = 16\) so there are two lattices; one can try to compute the action of a Hecke operator \(T_p\), and it turns out (see for example Theorem A here) that the answer involves Ramanujan’s function \(\tau(p)\). But this also tells you that the transfer to \(\mathrm{GL}_{16}\) will have some explicit isobaric decomposition corresponding to twists of the modular form \(\Delta\), and in particular the associated \(\pi\) will clearly not be cuspidal.

At the same time, there are some automorphic arguments which show that cusp forms of level one (and cohomologically trivial weight) cannot exist. Here the idea goes back to the (automorphic) proof of lower bounds for discriminants of number fields by Stark and Odlzyko. The idea in that case to use the explicit formula for \(\zeta_{K}(s)\) to construct an expression (and in particular the normalized version \(\Lambda_K(s)\) which satisfies the functional equation and involves \(N^s\) where \(N\) is the level which is directly related to the discriminant of \(K\)) and ultimately arrive at some expression which is provably non-negative unless the root discriminant of \(K\) is larger than some explicit constant (minus some explicit \(o(1)\) depending on the degree). Mestré generalized this argument to automorphic forms corresponding to other Motives, in particular proving that, assuming conjectures of Langlands type, that there did notexist any abelian varieties over \(\mathbf{Z}\) of dimension at least one (which was proved unconditionally by Fontaine) but also that the conductor of such an abelian variety had to be at least \(10^g\). This was then later generalized by Fermigier (a student of Mestré) and then by Stephen Miller (Rutgers!) to prove that there are no automorphic forms \(\pi\) for \(\mathrm{GL}_n/\mathbf{Q}\) of level one which are cohomological for the trivial representation when \(n < 27\). These are exactly the forms associated (conjecturally) to the motives of weight \([0,1,\ldots,n-1]\). Returning to the conference at Orsay: Chenevier gave a talk on understanding automorphic representations \(\pi\) of level one and low motivic weight, and once again raised the automorphic version of Khare’s question. Now I have known about this question for a long time, but somehow being reminded of a problem can sometimes be the spark to help one think about the question again.

Correcting what was a past failing of my own intuition, I was very happy that George Boxer, Toby Gee, and I were able to come up with a very simple argument to answer both questions; there does exist a compatible family of crystalline Galois representations with Hodge-Tate weights \([0,1,2,\ldots,n-1]\) for some \(n\); for example one can take \(n=105\). Moreover, this compatible system is even automorphic and associated to a cuspidal
\(\pi\) of level one and cohomological weight zero for \(\mathrm{GL}_n\). (With work, it is even “motivic” in the sense that the compatible system can be found inside an explicit algebraic variety over \(\mathbf{Q}\), so it is in particular also pure.) Now while the argument is very simple, it must also be said that is uses some extremely hard theorems; for a start, it uses both the full modularity lifting results of BLGGT (Barnet-Lamb, Gee, Geraghty, Taylor), following Clozel-Harris-Taylor and many others, *and* it uses the even more recent full symmetric power functoriality result for classical modular forms by Newton and Thorne. (Since the paper is only nine pages and the proof only half of that, I won’t explain it here.)

One would still like to prove, of course, that there are a huge number of self-dual forms for all sufficiently large \(n\). And one can naturally ask what is the smallest such \(n\), which we now know satisfies \(27 \le n \le 105\). The expectation is certainly that \(n\) is probably close to around \(32\). It would be nice to know!

Of course, there is an endless list of other tricky problems one can pose of this form. For example, does there exist a regular motive (with coefficients) over \(\mathbf{Z}\) with Hodge-Tate weights \([0,1,\ldots,n-1]\) for some \(n\) which is *not* essentially self-dual?

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Clozel 70, Part I

Posted on September 24, 2023 by Persiflage

I recently returned home from a trip to Paris for Clozel’s 70th birthday conference. Naturally I stayed in an airbnb downtown, and the RER B gods smiled on me with a hassle free commute for the entire week. Tekés was an interesting find, a fun (and surprisingly cheap) Israeli vegetarian restaurant right near where I was staying. But surely the food highlight of the week was the lunch spreads during the conference at Orsay — certainly the best conference food I’ve ever had! Great vegetarian food with amazing eggplant dishes, feta, figs, all the good stuff. (Rumor was it was chosen by Valentin Hernandez and paid for by Vincent Pilloni; I’m not sure if that’s true but a great job all round.)

There were quite a number of interesting talks, although as mentioned before I don’t like singling out because that sometimes seems like an implicit criticism of the other talks. But a few thoughts spurred by some of the talks (which you can more or less guess if you wanted to), some of which were already raised by others in conversation during the conference:

(Global) modules for Galois deformation rings. As a result of Taylor-Wiles patching, one usually constructs a CM-module \(M_{\infty}\) defined over some local deformation ring \(R\). Often quite a bit of mileage can be gained by exploiting the ring theoretic structure of \(R\) to conclude something about the module \(M_{\infty}\) and vice versa. Perhaps the ur-version of this argument is Diamond’s argument showing (in some circumstances) that the formal smoothness of \(R\) implies that \(M_{\infty}\) is free. A more recent example is in the work of Jeff Manning where he exploits the geometry of some particular \(R\) (by relating to a more geometric situation where one can perhaps understand the Picard group) to restrict the possible \(M_{\infty}\) to a very small number of possibilities from which one can then get some mileage. But one question raised is the extent to which this one can always do this. As one considers more and more complicated \(R\), is there some constraint on possible \(R\) which means that there are only going to ever exist a small number of faithful maximal rank one CM-modules \(M\), or are there going to be situations where \(R\) is very complicated and one can’t hope classify all such \(M\), but only (for some mysterious situation) a very small number of them turn up in global situations. Note that globally there is often a few possibilities of the type of cohomology one patches, and even for \(\mathrm{GL}(2)\) you can be in situations where you can force \(M\) to be free or self-dual by working in coherent cohomology or etale cohomology respectively and these modules are not always the same.

The work of Arthur: (Some) experts are at the point where they no longer expect Arthur to publish proofs of results he has claimed, leaving a huge gap in the literature. The summary seems to be that many very smart people are putting lots of effort into filling in some of these details, and that this seems to require new arguments. For example, it seems to be the case that one of Arthur’s proposed inductive arguments will not (at least naively) work. The mathematical community should be immensely grateful to people working on this!

Shimura Varieties: I once joked that today’s generation is more likely to learn about Galois representations associated to elliptic curves and modular curves before learning any class field theory. Well that generation has passed! We may be approaching a moment where people learn about Shimura varieties without ever thinking about modular curves, let alone getting close and personal with \(X_0(11)\). (Note: I don’t think that RLT knows the genus of \(X_0(11)\) and that never stopped him, of course.) Some people can look at the abstract definition of a Shimura variety and then start proving things; I am certainly not one of those people. Fortunately there are still many interesting open questions even about classical modular forms.

A result of Garland (at least) two talks reminded me of a vanishing result of Garland. Suppose that \(\Gamma\) is (for example) a lattice in \(\mathrm{SL}_n(\mathbf{Q}_p)\). Then the cohomology (with coefficients in \(\mathbf{Q}\)) vanish in positive degrees below \((n-1)\). But I think that much more should be true, namely, that the cohomology should all have a “trivial” Hecke action in the expected ways, i.e. the completed cohomology groups should all be finite in this range, as they are for \(\mathrm{SL}_n(\mathbf{Z})\) (more or less, let’s not be precise about ranges and what eactly is known). It feels like conjectures of this sort are not completely out of reach. Is this too optimistic? This is already an interesting problem in the case of \(H^2(\Gamma,\mathbf{F}_p)\).

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Kouign-Amann, Chicago Tasting

Posted on June 25, 2023 by Persiflage

A free morning on the north side this week meant a chance for a bike ride and a new cafe; nothing out of the ordinary. But this time I prepared an itinerary to hit up some of the most highly rated Kouign-Amann in Chicago. First stop, the Good Ambler, then on to Aya Pastry, and then to the Publican bakery; one order of Kouign-Amann at each stop! Then onto Metric coffee for a (good) cortado and a Kouign-Amann taste off:

Video Introduction

First of all; these were all good pastries! But none were in the neighbourhood of transcendent. Here’s a little more detail.

Good Ambler: Very crunchy and caramelized on the outside. I noted the inside was “bready”. I guess this is supposed to contrast not only with a cake but also with “buttery”. This was perhaps a complaint one could have made about each of them. I imagine that the relative ratio of both butter and sugar used in these versions is less than in the original. I also wonder about the relative quality of the butter. What’s the best butter one can get in Chicago?

Publican: From my notes: “If I had to, I could definitely eat all of this, but I won’t”. I think that same comment could honestly have been applied to all of them. Now that doesn’t sound very enthusiastic, but for context the amount of food I typically consume before lunch is half a piece of toast (and this was probably before 10:00AM). From the picture below, the interior was certainly in the ballpark of a croissant.

Aya: This was much denser than the other two, although again the closest point of comparison for the inside was a still sweet croissant, and while not a million miles away from the reality is not what I am going for. Both the Aya and the Publican had a deliberate layer of caramelization on the bottom, but in both cases this was very thin and didn’t make that much difference to the overall tast.

To get more of a hint of what I am looking for in my dreams, I think this video hopefully conjures up some idea (19:03 in to the video). It almost makes me want to try (again) to make it myself!

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Haebler and Gilberto

Posted on June 8, 2023 by Persiflage

Two obituaries in the NYT within one week for musicians in my music collection.

I can’t quite say that Mozart is my composer of choice, especially when it comes to the piano, although I could listen to Mitsuko Uchida play all day. But every now and again, Mozart knocks one out of the park. The G minor piano quartet, for example. While that quartet has a great arrangement for two pianos, here’s a Mozart fugue actually originally written for two pianos; here is my recording, performed by Ingrid Haebler and Ludwig Hoffmann

I once had some Brazilian students surprised during office hours that I was listening to Getz/Gilberto (I was surprised by their surprise in turn). But at any rate this is what I now assume all Dutch TV is like (featuring Astrud Gilberto):

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Google and Franck

Posted on May 15, 2023 by Persiflage

I have a google play (which plays streaming music) and it’s really terrible for classical music. If you choose virtually any piece of classical music and then flip forward three songs you invariably end up with Claire de Lune or the Moonlight sonata. For science, I just tested this again right now, and here are the unfiltered results:

  1. Initial Selection: Art of Fugue
  2. Next Piece: Toccata and Fugue in D minor (ha ha; semantically similar I guess)
  3. Next Piece: First movement, Moonlight sonata
  1. Initial Selection: Brahms, Op 119 no.1
  2. The initial selection continued with Op 119 no.2 on the same track; I don’t think that counts
  3. Prelude and Fugue in B flat minor #22 from Well-tempered Clavier Book 2. Pretty good! Mind you, when I asked google what it was playing to confirm, it said “Country road, by John Denver.” In fact, this appears to happen when I ask about any piece of music at all.
  4. Moonlight sonata (but third movement!)
  5. What sounds like cheap Debussy but google play still claims is John Denver.
  6. Another Brahms intermezzo! (Op 118)
  7. Flight of the Valkyries (blech)
  1. Initial Selection: Schubert Op 929, second movement.
  2. The Swan (arranged for violin and Piano)
  3. Schubert Op 929, second movement. Hmmm.
  4. The Nutcracker suite (perhaps complete? from the beginning recorded live.)
  5. Air on a G-string.

I say: hypothesis confirmed.

Some other strange results: I once asked for the Franck violin sonata and somehow got the piano accompaniment without the violin part. Actually that may have been the most interesting selection it ever played.

Speaking of Franck, I only recently learnt that he composed his violin sonata when he was 64. It’s not so usual for great composers to write great music late in their lives (Art of Fugue, unfinished when Bach was 65), but a little unusual, I think, for one’s most famous piece of music (perhaps by some distance) to be composed relatively late in life. I hope I prove my most famous theorem when I’m 64!

My go to recording is by Perlman and Ashkenazy; here they are (very young; from 1968, apparently) in the recording studio (apparently I am unable to embed this video, but I promise this is a real and interesting link!)

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Quadratic Reciprocity

Posted on May 2, 2023 by Persiflage

I accidentally proved quadratic reciprocity in class today, or at least three quarters of a proof. Can you finish it off? Here’s the proof: start with a real quadratic field \(K\), and the sequence

\(1 \rightarrow \mathcal{O}^{\times}_K \rightarrow K^{\times} \rightarrow K^{\times}/\mathcal{O}^{\times} \rightarrow 1 \)

then take cohomology. If \(P_{K}\) is the group of principal ideals of \(K\), then from Hilbert Theorem 90 you deduce that

\(P^{G}_K/P_{\mathbf{Q}} \simeq H^1(G,\mathcal{O}^{\times}_K)\).

If \(I_K\) is the group of all ideals of \(K\), the left hand side is a subgroup of \(I^G_K/P_{\mathbf{Q}}\) which is a product of groups of order two for each ramified prime. Now if \(K = \mathbf{Q}(\sqrt{pq})\) with \(p \equiv q \equiv 3 \bmod 4\), there is no unit of norm \(-1\) because \((-1/p) = (-1/q) = -1\), so you deduce that the cohomology of the unit group has order \(4\) and so from order considerations that the prime ideals of norm \(p\) and \(q\) are principal. Write \(\mathfrak{p} = (\alpha)\) and \(\mathfrak{q} = (\beta)\). One has \(x\) and \(y\) with

\(x^2 – p q y^2 = N(\alpha)\)

Here \(N(\alpha) = \pm p\). But the right hand side must be a square modulo \(q\), and so \(N(\alpha) = p\) if \((p/q)=+1\) and \(-p\) if \((p/q)=-1\). Equivalently, \(N(\alpha) = (p/q)p\), and similarly \(N(\beta) = (q/p)q\). But \(\mathfrak{p} \mathfrak{q} = (\sqrt{pq})\), so \(\alpha \beta = \varepsilon \sqrt{pq}\) for some unit \(\varepsilon\), and since all units in \(K\) have norm one, it follows that

\( pq(p/q)(q/p) = N(\alpha \beta) = N(\varepsilon) N(\sqrt{p q}) = – p q,\)

which is quadratic reciprocity! There is a similar argument for \(p \equiv 1 \bmod 4\) and \(q \equiv -1 \bmod 4\) (though now using facts about \((2/p)\) rather than \((-1/p)\)). However, it is not so clear if one can prove the case \(p \equiv q \equiv 1 \bmod 4\) using this argument. Is there one? Of course the challenge here is to “only” use Hilbert 90 and unique factorization of ideals, but never to make any arguments about how primes are splitting.

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En Passant VIII

Posted on April 28, 2023 by Persiflage

Coffee in Hyde Park: I had the misfortune of being stuck in down town Hyde Park needing a coffee without access to a car. I started at Sip and Savour where I ordered a cortado. They did not know what a cortado was. I paused, considered the situation, and slowly walked out. Attempt number two was at a cafe which was nominally a Peets but was actually a “Capital One” cafe. They did have a fine looking Marzocco machine which was a good sign, but they also had a very bored looking barista and my attempt to order a cortado lead to strike two: they also had no idea what I was taking about. I then peeked into Philz Coffee which I know is not really an espresso place; indeed they did not seem to have an espresso machine at all. I considered just getting a regular coffee, but I didn’t like the vibe and ended up at “Cafe 53.” Here I somehow ended up with two small (to go) espresso cups, one consisting of a very watered down and undrinkable espesso, and the other filled with frothed milk. I politely put them on the table, worked on my laptop for 30 minutes, then discretely through them in the trash. Thank god for Plein Air near campus!

The quest for a good Kouign-Amann: In 2009 I went to a conference in Roscoff organized by Tilouine and Hida (mentioned previously here). I have a number of fond memories from that conference: The mathematical discussions; the tides; the speaker who wore loose hanging pants and then kept bending over very low to the ground to write on the extremities of the fairly low white boards to reveal a generous plumber’s crack to those of us fortunate to be sitting in the front row; but also the food. The culinary highlight for me (gizzards aside) was surely the Kouign-Amann, a super indulgent buttery cake which is a speciality of Brittany and of which I got to try several versions (all excellent). Little did I know how hard it would be to reproduce the same dish anywhere else, where very few people seemed to have heard of it. The less said about my attempts to make it myself the better. Now enter Dominique Ansel (spit). This third-rate fraud is probably best known for inventing the cronut, sold at his trendy eponymous bakery in NYC which took a certain section of the US culinary world by storm around 2013. But for me he will be eternally known as the chef who introduce the “Kouign-Amann” to the US. I use quotes here deliberately, since Ansel’s version is something resembling a slightly caramelized croissant and nothing than anything I had had in Brittany. I mean the guy has probably not been west of Rouen let alone actually gone to Brittany. It’s as if a Mexican chef in Baja boiled a tortilla with a hole cut out of it and called it a bagel. Of course, bakeries the entire world over started selling Kouign-Amanns based not on the original but on copies of Ansel’s version, which was a sad simulacrum of the original. I have now been on an almost 15-year quest around the world to find a cake sold by that name which in some way was as amazing as the first ones I tried but to no avail. From East coast to West coast, from Blé Sacré in Paris to, most recently, an hour waits in line at Lune croissanterie on Collins street. Mostly the results have been bad, but even when they have been quite decent they have not been Kouign-Amanns. The quest continues…

Chess: For those watching Ding-Nepo, what a match it has been so far! One thing I have appreciated about watching live on youtube is that the commentators (including Caruana and Giri) analyze the positions in real time instead of using stockfish. That helps you distinguish from an advantage which is obvious to a super GM from an advantage which is obvious only to a super computer! (By the time it is obvious to me the players have already resigned). I’m looking forward to tomorrow.

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Deciphering Quanta

Posted on March 23, 2023 by Persiflage

Sometimes it is claimed that Quanta articles are so watered down of mathematical content that they become meaningless. That presents a challenge: do I understand the quanta article on my own work?

Here goes:

New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’

I can confirm that I did not see this article in any form before it appeared. Overall I would say that it is faithful to the facts and I can interpret what everything means. I’m not quite sure why there is an Alex Kontorovich explainer about the Langlands Program in there but why not?

I did, however, have to stop and wonder when I saw the following picture:

This appeared with the caption *Congruence modular forms (left) have additional structure that noncongruence modular forms (right) lack.* OK, here is the challenge: can I work out what this picture actually represents?

In both pictures, there is clearly some color gradient between yellow and blue, and this has discontinuities along some regions we call \(L\) and \(R\) for left and right. These are also clearly pictures inside \(\mathbf{H}^2\) in the Poincaré disc model.

Here \(L\) looks at least superficially like a fundamental domain for some Fuchsian group. The rays of \(L\) going towards the point \(i\) on the boundary do look like geodesics in the hyperbolic metric (which are circular arcs meeting the boundary at right angles). There is a corresponding invariance for the figure by the parabolic element (with some cusp width) through \(i\). Consider a conformal map from the upper half plane to this model which sends \(\infty\) to \(i\):

Using the standard conformal map with the upper half plane:

\(\phi: \mathbf{H} \rightarrow D(0,1), \quad z \mapsto \displaystyle{\frac{1 + i z}{1 + z}} \)

From the picture, we see the images of the geodesics from the infinite cusp to \(n \alpha\) for \(n\) odd. All parabolic elements are conjugate, but if we are to guess what \(\Gamma\) is by looking at the picture then working out \(\alpha\) for this specific model is important. I tried to eyeball it for a while before printing it out and trying to compute the angle using continued fractions, which wasn’t so accurate but gave \(\tan \theta \sim 1/(1 + 1/3)\) with \(\theta\) in the mid \(30\)s. Then using some angle tools in keynote it came out to somewhere between \(36\) and \(37\) degrees, closer to the latter. So maybe it was exactly a \(10\)th root of unity, which would make \(\alpha\) live in a degree \(4\) field, or (much better!) maybe \(\alpha = 1/2\) in which case the angle is \(36.8698 \ldots\), and then \(L\) (or rather \(\phi(L)\) is invariant under exactly

\(A = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right).\)

Maybe I should have guessed this in retrospect! Let’s look at the geodesic from \(i \infty\) to \(1/2\), which in the picture goes from \(i\) to \(4/5 – 3i/5\). There appears to be a point with \(4\)-fold symmetry. That suggests invariance under some order four element corresponding to an elliptic point. But this is a little worrying, since \(\mathrm{PSL}_2(\mathbf{Z})\) does not have any elements of order \(4\)! Now there are some ways around this. For example, this could be the level set of a function which satisfies an extra invariance property under the normalizer of some congruence subgroup, and so does not itself come from an element of \(\mathrm{SL}_2(\mathbf{Z})\) but \(\mathrm{GL}_2(\mathbf{Q})^{+}\). For example, the Fricke involution on \(X_0(N)\) is \(\tau \mapsto -1/N \tau\) which corresponds to. a matrix in \(\mathrm{GL}_2({\mathbf{Q}})\) with determinant \(N\), although that has order \(2\). Any element of order \(4\) has to have eigenvalues of the form \(\alpha\) and \(\alpha i\), with \(\alpha + i \alpha \in {\mathbf{Z}}\), and thus eigenvalues \(1+i\) and \(1-i\) up to rational multiples. That means the determinant should be \(2\) times a square. The unique element such element up to conjugacy is

\(S = \left( \begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix} \right)\)

which has order \(4\) and fixes \(i\). If we want to fix a point on the geodesic \(1/2 + i t\), we can just make a hyperbolic scaling and then conjugate to get the matrix

\(S_t = \left( \displaystyle{ \begin{matrix}
1 – 1/2t & t + 1/4t \\
-1/t & 1 + 1/2t \end{matrix} } \right).\)

Here I started to run into a problem because it’s hard to approximate \(t\) to any precision from the picture, although \(t \sim 0.1\) numerically. There are a few natural integral matrices one can write down but it wasn’t clear what was going on.

Next I turned to the picture around the cusp \(-i\). This looks kind of weird because the biggest curves decidedly do not look like geodesics. But if we ignore this, we might decide that \(-i\) is another cusp. It’s also disturbing that the fundamental domain appears to contain enough of the boundary to suggest that \(\Gamma\) is a thin group, but let’s ignore this as well. So what is the cusp width (with out normalizations) at \(-i\)? Here numerically I simply get nonsense.because if there really were geodesics around \(i\) translated by a fixed parabolic element and the first one was of the size indicated in the picture then there would be many fewer visible ones. As a typical example, one should expect a picture like this:

which doesn’t look anything like the original picture.

So I’m ready to give up now. What about the picture on the right? Well here I don’t even know how to begin. There is not any obvious evidence of parabolic elements, which are not only present in congruence subgroups of \(\mathrm{SL}_2(\mathbf{Z})\) but of any non-congruence subgroup as well. Perhaps there is a cusp at \(i\) with some large cusp width. There also seem to be singularities inside the disc. But that just suggests this might be a level set of a meromorphic form. But why choose a meromorphic form rather than one that is holomorphic away from the cusps? I wonder if that gives hints about the first picture; perhaps the transition from yellow to blue occurs when \(\mathrm{Im}(f)\) goes from positive to negative, and where it might be \(0\) along some geodesic arc (For example: the Fourier coefficients are rational so the form is real along the purely imaginary axis) but the form can also have imaginary part \(0\) along some non-geodesic regions and that is what one is seeing around \(-i\). That might allow one to reinterpret the graphs as something related to \(\mathrm{Im}(f)\). This suggests that the point on the left that appears to have degree four instead is related to a “local” symmetry coming from a vanishing point of the derivative of the modular function, but I’m just guessing now.

I should probably spend no more time on this, so instead I open up speculations to the comments!

Full Disclosure The picture comes with an attribution to David Lowry-Duda but I did not try to follow that lead.

Update: Perhaps it was not as obvious as intended, but the request for speculation was intended to be “how do you interpret this from the picture alone” not “send me a link” which is something which I deliberately avoided trying to do. (I will no doubt look at the way they are generated at some point in more detail.) A brief look at the LMFDB link suggests that my second explanation is pretty close, although the graph on the left is of the argument of a modular function (or rather the modular form \(\Delta\)) rather than the imaginary part. But the guess as to the explanation for the “order four” symmetry is correct, it is explained by the vanishing of \(\Delta'(\tau)\) or equivalently the vanishing of \(E_2(\tau)\) at a point on the geodesic \(1/2 + i t\). The picture on the right is still a mystery, however, and original speculations are still welcome.

Update II: So here is an new example. Like the graphs above, it plots the argument of a two modular forms on (different) finite index subgroups of \(\mathbf{SL}_2(\mathbf{Z})\). I have normalized both pictures so that the cusp width at the cusp \(i\) is the same in both cases (under the normalizations above invariant under \(\tau \rightarrow \tau + 3\); making the cusp width too small makes the behavior at \(tau = i \infty\) dominate the picture as the covolume goes up. There are dome differences with this example however:

  1. I have used (holomorphic) modular functions rather than modular forms.
  2. For these particular choices of functions, they are non-zero everywhere.
  3. I made the picture myself so it is not as pretty as the ones above.
  4. The functions I chose have the property that they are real precisely on geodesics, and thus the singularities here do form a tesselation.


Now one of the forms is defined on a congruence subgroup and has a Fourier series in \(\mathbf{Z}[[q]]\), the other has a Fourier series in \(\mathbf{Q}[[q]]\) but not in \(\mathbf{Q} \otimes \mathbf{Z}[[q]]\) and is only defined on a non-congruence subgroup. But which is which? The pictures here are clearly much more similar than the pair of pictures above!

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Boxes for Boxer update

Posted on February 28, 2023 by Persiflage

As noted in this post, exactly 42 reprints of [BCGP] were recovered in January of 2022 from boxes left out in the snow outside Eckhart Hall addressed to George Boxer. As mentioned there, the packaging (5 boxes of 8 plus a further smaller box with 2) suggested that there was a “missing box” with 8 more copies of the paper, and that one should be on the lookout at Hyde Park used book stores for the remaining copies. These reprints are after all roughly of a similar scarcity to the extant Gutenberg Bibles. Well there has been an update! My student Abhijit noted that 7 copies had been left out in the hall as an offering for graduate students (and that he had secured a copy). I initially assumed that several of my copies had escaped my hands but they all seem to be accounted for (there were 39 in my office, at least one at home, and at least one I gave to TG). So the missing box did exist after all! I still haven’t determined what happened to the box. There could have been a reprint thief who, out of guilt or madness (depending on how much of it they tried to read), was driven them to give them to the world as an attempt of repentance. Then again, the box may have ended up in someone’s hands much more unwittingly and they had only just discovered the contents. Or finally it could have been a janitor who dumped them out from a dusty closet where they had inadvertently been placed. The plot, as they say, thickens! Anonymous tips from uchicago graduate students welcome in the comments.

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What the slopes are

Posted on February 25, 2023 by Persiflage

Let \(f\) be a classical modular eigenform of weight \(k\), for example, \(f = \Delta\). The Ramanujan conjecture states that the Hecke eigenvalues \(a_p\) satisfy the bound

\(|a_p| \le 2 p^{(k-1)/2}.\)

A slightly fancier but cleaner way of saying this is as follows. Associated to \(f\) of weight \(k\), level \(N\) prime to \(p\) and finite order Nebentypus character \(\chi\) is a polynomial

\(X^2 – a_p X + p^{k-1} \chi(p).\)

(For \(\Delta\) one has \(\chi(p) = 1\) for all \(p\), but in general it can be some other root of unity.) This is the characteristic polynomial of Frobenius on the \(\ell\)-adic representation associated to \(f\) for \(\ell \ne p\), or the characteristic polynomial of crystalline Frobenius for \(\ell = p\). The Ramanujan conjecture is now that the Frobenius eigenvalues \(\alpha_p\) and \(\beta_p\) satisfy

\(|\alpha_p| = |\beta_p| = p^{(k-1)/2}.\)

This conjecture was famously proved by Deligne. Having determined the complex valuation of these eigenvalues, one might ask about their \(\ell\)-adic valuations as well. If \(\ell \ne p\), then since the polynomial above has constant term prime to \(\ell\), then \(\alpha_p\) and \(\beta_p\) are \(\ell\)-adic units. So the remaining case is \(p = \ell\).

When \(p = \ell\), the \(p\)-adic valuation of the two roots \(\alpha_p\) and \(\beta_p\) certainly satisfy \(v(\alpha_p), v(\beta_p) \ge 0\) and \(v_p(\alpha) + v_p(\beta) = k-1\). Given \(f\), we call these valuations the *slopes* of \(f\). The first observation is that the slopes depend on more than just the weight \(k\). For example, suppose that \(f\) is the weight \(2\) eigenform associated to a (modular) elliptic curve \(E\) with good reduction at \(p\), then either \(E\) has ordinary reduction, in which case the slopes are \(0\) and \(1\), or \(E\) has supersingular reduction, and the slopes are both \(1/2\).

Instead of fixing \(f\) and varying \(p\), one can fix both \(p\) and the tame level \(N\) and ask how the slopes vary as the weight changes. For example if \(N=1\) and \(p=2\), then the first interesting case is when \(k=12\) and \(f = \Delta\). In this case \(\tau(2) = 24\), and the two slopes are \(3\) and \(11-3 = 8\).

I’ve already tried to suggest by analogy to the Ramanujan conjecture why determining the p-adic valuations (the slopes) might naturally be an interesting question. But let me mention two other natural reasons. The first is that, from \(p\)-adic Hodge Theory, the crystalline eigenvalues \(\alpha_p\) and \(\beta_p\) determine the restriction of the \(p\)-adic Galois representation associated to \(f\) to the local Galois group at \(p\). The valuation of these slopes while containing less information than the eigenvalues themselves still tell you a lot about the \(p\)-adic representation, although determining exactly what is still a question of active and open interest. Secondly, as Coleman observed (following Hida in the case when one of the slopes is \(0\)), one can deform the Galois representations associated to these finite slope forms into continuous \(p\)-adic families of Galois representations associated to cuspidal eigenforms, which leads to the story of the eigencurve of Coleman-Mazur and beyond.

Gouvêa was one of the first people to undertake a numerical study of the roots.
In this paper where the slopes are, Gouvêa observed a number of interesting behavior of the slopes which were all somewhat mysterious. First, the slopes were almost all integers. This is not surprising when \(a_p(f) \in \mathbf{Z}\), but in general \(a_p(f)\) will be a random algebraic integer. The slopes also seemed to be distributed in a number of surprisings way. For example, although the slopes in weight \(k\) associated to \(f\) add up to \(k-1\), they tended to be concentrated in the intervals \([0,(k-1)/(p+1)]\) and \([p(k-1)/(p+1),k-1]\).

Kevin Buzzard went one step further and looked more closely at the slopes when \(N=1\) and \(p=2\). Ostensibly, according to Kevin, this was to test William Stein’s latest magma code for bugs! Kevin found that the slopes in this case satisfied a much more regular pattern. As mentioned above, when \(k=12\), there is a single eigenform whose smallest slope is \(3\). When \(k = 12 + 64\), there are six eigenforms whose smallest slopes are \(3, 7, 13, 15, 17, 25\), and when \(k = 12 + 2^{10}\), there are \(86\) forms whose slopes are

\(3, 7, 13, 15, 17, 25, 29, 31, 33, 37, 47, 49, 51, 57 \ldots \)

where all of these sequences are given explicitly by the \(2\)-adic valuation of \(2 ((3n)!/n!)^2\). Note that the Gouvêa-Mazur conjecture says that these sequences should have some initial segment in common, but in fact the Gouvêa-Mazur conjecture only implies that the slopes in weights \(16 + 2^6\) and \(16 + 2^{10}\) should be the same up to slope \(6\), and in practice they agree ridiculously further than this. This was all very mysterious. Kevin found a general algorithm which conjecturally computed by an inductive procedure all the slopes in all weights for a fixed tame level \(N\). (In what I always regarded as a missed opportunity, he did not call the paper “What the slopes are”).

In fact, Kevin’s conjectural answer required an assumption on \(N\) and \(p\) which for \(p>2\) was equivalent to asking that the local residual representations associated to all low-weight forms are locally reducible. For \(N=1\), the first case for which this does not happen is \(p = 59\), and this story is related to our counterexample to the original form of the Gouvêa-Mazur conjecture.

Kevin Buzzard was a speaker at the Arizona Winter School in 2001, and for his project he outlined a special case of his conjecture corresponding to overconvergent modular forms of weight \(k = 0\). Of course there are no classical modular cuspidal forms of weight \(k = 0\), but by Coleman’s theory there is a direct link between the questions about classical forms in all weights and overconvergent forms of finite slope in all weights. Kevin’s problem in its original formulation is described here.

I was a graduate student at the time, and although I wasn’t actually assigned to Kevin’s group, I did see his problem and had an idea which more or less amounted to the idea of using a slightly different explicit basis for this space than Kevin had considered and for which the \(U_2\) operator has a much nicer explicit form. In fact this basis was related to computations I had done in the summer of 93-94 and shortly afterwards during my first year at Melbourne uni, using what Matthew Emerton and I had come to definitely know as the “f” function

\(f = q \prod_{n=1} (1 + q^{n})^{24}.\)

(When It came to writing my paper with Kevin, I still felt strongly enough to insist we call it by this letter.)

Kevin and I put quite a bit of effort into proving his conjecture for more general \(k\), still with \(N=1\) and \(p=2\), but only succeeded in a few cases, including \(k = -12\), but also \(k = -84\) which was somewhat randomly chosen as a case where our approach failed but only by a little bit which could be overcome. There are a few other cases where \(X_0(p)\) has genus zero and one can do something similar, but otherwise very little progress was made on these conjectures in this situation.

There are a plethora of related conjectures which also came out (at least indirectly) from similar calculations. For example, Kevin’s algorithm certainly always produces integers, so, in light of the \(p\)-adic theory, there is the natural question of whether a crystalline representation with Hodge-Tate weights \([0,k-1]\) for \(k\) even whose residual representation is reducible always has finite slope, Then there are questions of the exact relationship between the slope and the Galois representation, and so on.

Concerning the exact slopes themselves, perhaps the biggest advance over time were refinements of Kevin’s conjecture. The Ghost Conjecture, formulated by Bergdall and Pollack, is some “master conjecture” of a combinatorial nature generalizing a number of previous conjectures concerning the slopes of classical overconvergent modular forms over the centre of weight space. (Although, perhaps amusingly, it’s not clear that these conjectures do actually imply Buzzard’s original conjecture.)

Now cut to the present day. In a recent preprint, Ruochuan Liu, Nha Xuan Truong, Liang Xiao, Bin Zhao have now proved all of these conjectures, at least up to some genericity hypotheses (excluding the case \(N=1\) and \(p=2\)!). The authors certainly employ technologies that didn’t exist 20 years ago (\(p\)-adic Langlands, for example), but that was not the only obstruction to previous progress: the paper contains a number of very original and clever ideas. Very amazingly and satisfyingly, it resolves a large number of the open problems discussed above, including Gouvêa’s conjectures about the distribution of slopes, the integrality properties of slopes for locally reducible representations, and even a version of the Gouvêa-Mazur conjecture. It is also satisfying that the arguments use \(p\)-adic local Langlands, given that some of the initial computations of slopes served at least in part as inspiration for some aspects of this program. I myself have not really worked on this circle of problems for almost 20 years but I am still very happy to see these questions answered!

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