Since I don’t have maple, I can’t play with the following code:
https://sites.math.rutgers.edu/~zeilberg/tokhniot/MultiAlmkvistZeilberger.txt
But is ChatGPT now good enough to re-write this in either pari/gp or magma (or Mathematica). I’m not sure how realistic this might be (without some serious extra hands-on editing…)
Mark Kisin gave a talk at the number theory seminar last week where the following problem arose:
Let \(W\) be the Galois representation associated to the Tate module of an abelian variety \(A\) over a number field, and suppose that \(W = U \otimes V\). Now suppose that the Galois action on \(U\) is unramified at all primes above \(p\). Can you prove that the Galois action on \(U\) has finite image?
Of course this is a special case of the unramified Fontaine-Mazur conjecture. But here the representation \(U\) literally “comes from an abelian variety” although as a tensor factor rather than a direct factor. At first sight it seems like it should be much easier than the actual Fontaine-Mazur conjecture if you just find the right trick, but I don’t see how to do it! Here at least is a very special case.
Lemma: Suppose that \(A/K\) has ordinary reduction at a set of primes of density one, and
that \(U\) is a representation which is unramified at all primes dividing \(p\) of odd dimension which occurs as a tensor factor of \(W = H^1(A) = U \otimes V\). Then, after some finite extension of \(K\), \(U\) contains a copy of the trivial representation.
Proof: One may as well assume by induction that the action of the Galois group
on \(U\) is absolutely irreducible of odd dimension \(d\) and remains so for every finite extension (otherwise decompose it into such pieces and take one of odd dimension).
Now choose a prime \(v\) of \(K\). Let \(\alpha_i\) be the eigenvalues of Frobenius at \(v\) on \(U\),
and let \(\beta_j\) be the corresponding eigenvalues on \(V\). We know that \(\alpha_i \beta_j\) are algebraic numbers which are Weil numbers of norm \(N(v)\). The ratios of any two roots thus are also algebraic numbers with absolute value \(1\) at all real places, and so \(\alpha_i/\alpha_1\) has this property.
Let’s suppose that the ratios \(\alpha_i/\alpha_1\) are actually roots of unity for a set \(v\) of density one. Since \(W\) is be defined over a fixed finite extension \(E = \mathbf{Q}_p\), the degrees of these ratios has uniformly bounded order over \(E\), and the the orders of these roots of unity also have uniformly bounded order. But then (projectively) only finitely many characteristic polynomials will arise from Frobenii for a set of (edit: density one), which would imply that \(U\) has finite projective image, from which it easily follows that \(U\) becomes trivial over a finite extension (remember the determinant is unramified so of finite image). Hence it suffices to show that the \(\alpha_i/\alpha_1\) are all algebraic integers and then use Kronecker’s theorem.
For finite places not dividing \(N(v)\) this is clear because the valuations of the \(\alpha_i \beta_j\) are all trivial and so are their ratios. For finite places dividing \(N(v)\) now suppose in addition that \(A\) is ordinary. Fix a place above \(v\). If the \(\alpha_i/\alpha_1\) have valuation given by \(a_i\), and \(\beta_j/\beta_1\) have valuation \(b_i\), it follows that the quantities \(a_i + b_j\) take on precisely two values, zero and either \(1\) or \(-1\), and they take on each of these values exactly half the time. But then either \(a_i\) is constant and thus (considering \(i = 1\)) equal to \(0\), or the \(b_j\) are all zero, and then half the \(a_i\) are zero and half are \(1\) or \(-1\). But that’s clearly only possible if \(U\) has odd dimension. So done!
I suspect the case that \(\dim(U)=2\), even with an ordinary hypothesis, is probably quite hard. But I would be happy to be mistaken.
Plein Air is certainly the best cafe in Hyde Park. (Arguably Build Coffee is fine as well, but they are only open about 5 hours a week.) But it is also true to say that Plein Air is (at best) pretty inconsistent when it comes to espresso drinks; some baristas are definitely better than others, but frequently the result is honestly pretty disappointing. As a daily ritual, I really would hope for a lot more. So what better time to (finally) get serious about making coffee myself.
Thus the latest addition to my office: a Silvia Pro X, a Baratza Sette 270 grinder, shims added, and some other accoutrements, perhaps most gratuitously a Luna Acaia scale. A few days in and I’m already very pleased; not only is the taste better than anything else one can get in Hyde Park, but even the ritual of waiting for the machine to warm up is not at all an unpleasant experience. Also, as far as midlife crises go, neither very expensive nor excessively time consuming!
I’ve always been a little resistant to going down this path — my experiences with Jared Wunsch’s Silvia at Northwestern suggested a certain learning curve was required, and I would not call myself mechanically adept. But a few things have changed. First of all, modern espresso had moved a bit more towards “science” than “art” over the past few years: a meticuluous approach via weighting, ratios, tamping, the science of extraction, etc. have made consistency and reproducibility much more possible. Add to that a souped up version of the Silvia with more controls to make things easier, and an infinite number of online resources (James Hoffmann youtube videos) available online.
My intent it to experiment with milk drinks at some point, but for now I’m quite happy just concentrating on espressos. Arriving soon: extra cups for visitors!
Here is the setup. Those with a keen eye will note the novel application of potential modularity…
Postdoc hiring season will be upon us soon! I have two excellent graduate students who will be applying for academic jobs soon, Chengyang Bao and Andreea Iorga. I have mentioned Chengyang’s first project before here and an introduction to the results in Andreea’s thesis is here. Today I wanted to talk about Chengyang’s thesis.
Fix a local mod-p representation, say
\( \overline{\rho}: G_{\mathbf{Q}_p}
\rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)
given on inertia by \(\omega_2 \oplus \omega^p_2\). Associated to this residual representation is a Kisin deformation ring \(R\) corresponding to fixed determinant crystalline lifts of weights \([0,k-1]\), for some fixed positive integer \(k \equiv 2 \bmod (p-1)\). The special fibres \(R/p\) of these rings have dimension one, and so, if one denotes their maximal ideal by \(\mathfrak{m})\), then the Hilbert series
\(H_k(x) = \sum \dim(\mathfrak{m}^k/\mathfrak{m}^{k+1}) X^k\)
has the form
\(H_k(x) = \displaystyle{\frac{P_k(x)}{1 – x}}\)
where \(P_k(x)\) is a polynomial. The Hilbert-Samuel multiplicities of these rings are given by the Breuil-Mezard conjecture (also proved by Kisin). These numbers are explicitly given by the values \(P_k(1)\).
It seems quite surprising that understanding the seemingly simple number \(P_k(1)\) is so intimately linked to the proof of the Fontaine-Mazur conjecture. At the same time, we know very little about these rings \(R\) (or their special fibres) when \(k\) is large.
In the example above, the unrestricted (fixed determinant) local deformation ring \(R^{\mathrm{loc}}\) is is formally smooth of dimension three over \(\mathbf{Z}_p\). Although the rings \(R/p\) only have dimension one, one expects that for larger and larger \(k\) they start to “fill out” the unrestricted deformation ring. It is natural to wonder: how fast does this happen?
More explicitly, the Hilbert-Samuel series of the special fibre of the unrestricted deformation ring with fixed determinant is
\(\displaystyle{\frac{1}{(1-x)^3}} = 1 + 3 x + 6 x^2 + 10 x^3 + \ldots \)
So one can ask: what weight does one have to go to to see all three dimensions of the tangent space? How far does one have to go to see all of \(R^{\mathrm{loc}}/(p,\mathfrak{m}^n)\)?
This was the thesis problem of Chengyang Bao, which grew out of (in part) questions arising during her work here. In this particular case, it seems that one has to go to weight \(k = p^2 + 1\) to see the entire tangent space. Actually, an even more basic question (which also came up here), is whether there exists surjective map
\(R_{k+p-1}/p \rightarrow R_k/p,\)
this seems very tricky and is still open (but Chengyang’s work stongly suggests that it is true).
One of the difficulties in this project is that close to nothing was known about the rings \(R\) for \(k\) anything larger than \(k = 2p\) or so (although there has certainly been quite a bit of work understanding the link between \(a_p\) and the residual representation, including work of Buzzard-Gee, Sandra Rozensztajn, and many others using p-adic Langlands for larger \(k\)).
Chengyang’s approach was, perhaps surprisingly, to use global methods. The basic summary of the Taylor-Wiles method as formulated by Kisin is that via patching one finds that a patched Hecke ring may be identified with a power series ring over \(R\). By reverse engineering this, if one finds a residual representation with sufficiently nice global properties, one can use explicit Taylor-Wiles primes to get arbitarily close approximations to the Kisin deformation ring \(R\). One of the tricks here is to be able to do this in a way where one can work efficiently after fixing a residual representation and then increasing the weight.
By doing these computations, Chengyang generated lots of explicit data about these rings \(R\) from which one can start making conjectures. I said before how the Breuil-Mezard conjecture amounts to predicting the value of \(P_k(x)\) at \(x = 1\). Chengyang has, at least in this particular case, been able to formulate an exact conjectural answer for the *entire* polynomial \(P_k(x)\). As a consequence, one can read off from this the answer of how large a weight one has to go to see all the directions in \(R^{\mathrm{loc}}/\mathfrak{m^n}\). I mentioned before that for \(n=2\) the answer is \(p^2+1\). and my guess was that the answer in general might be of order \(p^n\). But somehow the conjectural answer (up to constants which depend on \(p\)) turns out to be of order \(O(n^2)\), which is honestly completely different from anything that I would have guessed. I think of this conjecture as a new “horizonal Breuil-Mezard conjecture.” But really, it’s only half a conjecture; the hope is that one can understand and interpret Chengyang’s conjecture on the \(\mathrm{GL}_2(\mathbf{Q}_p)\)-side, and working this out is an exciting problem.
At the same time, there are lots of other things one can start to guess from looking at these explicit rings. Chengyang has a precise conjecture which says when the rings \(R\) in this setting are complete intersections or Gorenstein, and it also seems that they are always Cohen-Macaulay.
Even though we “know” \(p\)-adic Langlands for \(\mathrm{GL}_2(\mathbf{Q}_p)\) much better than in any other situation, there seems to be a real opportunity here to tease out many more precise and explicit conjectures from Chengyang’s work, and really to discover new phenomena which have hitherto never been noticed because computations of these rings has been so limited. (Another basic question: how many components does the generic fibre of \(R\) have in terms of \(k\)?).
I recommend that anyone interested in more details read Chenyang’s research statement!
I had occasion to return to some magma scripts I wrote in 2012. I the script used a number of pre-computed auxiliary files with computations, and was a little complicated, but didn’t use anything particularly complicated. So I was really surprised to run them in 2023 and find that they no longer worked. That is, they compiled, but the results they gave were different (and also incompatible with the truth). It was quite confusing to understand what has gone wrong, but eventually I traced it to the following. Early on in the file one has (having defined \(t\) as a variable using code that’s easy to write but which is somehow causing issues with wordpress):
F := NumberField(t^2 – 5);
ZF:=Integers(F);
So far, so good. But later on, the script called upon elements of \(F\) of the form \(\texttt{ZF![a,b]}\). But it turns out that if \(\texttt{x:=ZF[0,1]}\) that
\(x^2 + x = 1\)
in 2012, but
\(x^2 – x = 1\)
in 2023; that is, \(x\) was replaced by \(-x\), which is not an automorphism. I have no idea how or why that changed, but it certainly broke everything and took several days to fix. Annoying!
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 \(H^2(\Gamma,\mathbf{F}_p)\).
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:
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!
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):
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:
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!)
