SL_n versus GL_n

Posted on July 18, 2024 by Persiflage

I recently wrote a paper (with Toby Gee and George Boxer, see also here) on constructing regular algebraic automorphic representations \(\pi\) of (cohomological) weight zero and level one, and therefore also cuspidal cohomology classes in the cohomology of \(\mathrm{GL}_n(\mathbf{Z})\) for some values of \(n\).

There was one slightly subtle point which we had to address concerning the relation between the cohomology of \(\mathrm{SL}_n(\mathbf{Z})\) and \(\mathrm{GL}_n(\mathbf{Z})\), or at least the relationship between the parts of cohomology which come from cuspidal modular forms. I have observed this issue turn up in some different contexts, and that is what I wanted to talk about today. The main message is that from the perspective of the Langlands program, the cohomology of \(\mathrm{GL}_n(\mathcal{O}_F)\) is more fundamental than tbe cohomology of \(\mathrm{SL}_n(\mathcal{O}_F)\). When \(F = \mathbf{Q}\) these groups are “more or less” the same (more on that below), but the differences are more pronounced and significant when \(F \ne \mathbf{Q}\). But let’s start by talking about the case of classical modular forms, where there is already something a little bit interesting to say. A regular algebraic automorphic representation \(\pi\) for \(\mathrm{GL}(2)/\mathbf{Q}\) of level one corresponds to a cuspidal modular eigenform of weight \(k \ge 2\) and level one. We know that cuspidal modular forms of weight \(k \ge 2\) and level one contribute via Eichler-Shimura to the Betti cohomology groups of the modular curve. As an orbifold, the modular curve can be realized as \(\mathbf{H}/\Gamma\) where now \(\Gamma = \mathrm{SL}_2(\mathbf{Z})\) rather than \(\mathrm{GL}_2(\mathbf{Z})\). In this situation at least, we understand quite well what is happening. These eigenforms give rise to a two-dimensional space inside \(H^1\) of the modular curve, and thus inside \(H^1(\Gamma)\), and we understand what the “extra” action of the element
\[ \displaystyle{ \left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right) } \]
is; namely under the Eichler-Shimura isomorphism, it corresponds to the action of complex conjugation (so from the perspective of the Hodge filtration, it takes the holomorphic forms to the antiholomorphic forms and vice-versa). It acts on the relevant piece of cohomology with trace zero. Note that this no longer holds on non-cuspidal cohomology, for example \(H^0\) is one dimensional in both cases. Of course in cohomological weight zero (which corresponds to weight \(k = 2\)), there turn out to be no such forms, but the point is that the vanishing of the cuspidal cohomology for \(\mathrm{GL}_2(\mathbf{Z})\) is equivalent to the same statement for \(\mathrm{SL}_2(\mathbf{Z})\). (Something similar is also true in higher weight as well when there really do exist such forms.)

For larger \(n\) there is a similar equivalence; but now the behavior depends on the parity of \(n\). For \(n\) odd, the cohomology of \(\mathrm{GL}_n(\mathbf{Z})\) and \(\mathrm{SL}_n(\mathbf{Z})\) is (rationally) the same because \(\mathrm{GL}_n(\mathbf{Z}) \simeq \mathrm{SL}_n(\mathbf{Z}) \times \mathbf{Z}/2 \mathbf{Z}\) (then by the Künneth formula). But for \(n\) even, a level one weight zero \(\pi\) gives rise to two copies of the exterior algebra

\[ \bigwedge^* \mathbf{C}^{\ell_0} \]

in degrees \([q_0,\ldots,q_0 + \ell_0]\), with \(\ell_0 = (n-2)/2\), and the action of the “extra” element acts freely on these two copies. All this comes down to the differences in the real representation theory of \(\mathrm{GL}_n(\mathbf{R})\) and \(\mathrm{GL}_n(\mathbf{R})^{+}\) which is discussed briefly in the paper but which I won’t talk about here.

But what happens for general number fields \(F\)? There’s a confusion which I have seen in various places even for \(n=2\) about whether one should be considering the cohomology of \(\mathrm{SL}_n(\mathcal{O}_F)\) or \(\mathrm{GL}_n(\mathcal{O}_F)\). Of course it depends on what exactly one wants to do. But at least if one is interested in computing automorphic representations conjecturally associated to motives which have level one, one should really be considering the cohomology of \(\mathrm{GL}_2(\mathcal{O}_F)\). This confusion comes with good pedigree — It turns up in the Serre-Tate correspondence! Tate mentions (October 1969, page 382) a colloquium by Swan who “disappointed everybody” by computing that \(H_1(\mathrm{SL}_2(\mathbf{Z}[\sqrt{-14}]),\mathbf{Z})\) has rank three, compared to the lower bound (coming from the boundary tori) of two. (Side remark: Tate notes in a later letter [Nov 15] it should be \(\sqrt{-10}\), not \(\sqrt{-14}\).) Serre responds (October 15, page 384) that he doesn’t find this at all surprising, and in fact:

(via la théorie de Weil cela signifait qu’il existe de courbes elliptiques sur le corps en question qui n’ont pas de multiplication complexe — on n’en doute pas). En fait, vu Weil, il s’impose d’essayer de construire une courbe elliptique sur \(\mathbf{Q}(\sqrt{-56})\) ayant bonne réduction partout;

Now I confess that when I first read this quote I interpreted it as a misapprehension on Serre’s part, because (since this is \(\mathrm{SL}_2\) not \(\mathrm{GL}_2\)) there need not exist any such elliptic curve. But looking it up again now, I started to have my doubts, and Serre was perhaps more circumspect than I had assumed. Indeed chatgpt tells me:

The phrase “il s’impose d’essayer” in French does not have the same strict sense of necessity as “it is necessary” in English. A more nuanced translation could be “it is imperative to try” or “it is important to try.” It suggests a strong recommendation or importance, rather than an absolute necessity.

(Possibly Colmez can confirm this; AI has rendered his go playing superfluous but not yet his skills interpreting for anglophones the nuances of Serre’s words.) That’s also consistent with how Serre continues:

je connais trop mal la théorie de Weil pour être sûr que ça doit exister; mais il vaut la peine d’essayer

Later (note the remark on \(d=-56\) versus \(d=-40\) abpve), Serre says:

C’est bien \(\mathbf{Q}(\sqrt{-40})\) le corps où Mennicke a trouve que le rang de \(\mathrm{SL}_2\) rendu abélien est nombre de classes. Mais il a un corps encore plus beau: \(\mathbf{Q}(\sqrt{-109})\) où le \(\mathrm{GL}_2\) rendu abélien est infini (c’est une propriété plus forte S1 que la précédente). Ici aussi, on a envie de chercher des courbes elliptiques à bonne réduction.

Perhaps worth adding the modern footnote as well:

«via la théorie de Weil cela signifiait que…» je m’avançais beaucoup en disant ça (I was talking through my hat).

Of course, 45 years later things have been clarified, at least conjecturally. (We still have no general way to produce motives from cohomology, even for Hilbert modular forms of parallel weight \(2\).) One perspective which I think is helpful (at least to those who care more about Galois representations) is thinking about the differences between the Galois representations associated to automorphic forms on \(\mathrm{SL}_n\) versus \(\mathrm{GL}_n\). Given a \(\pi\) for the former (say cuspidal algebraic of weight zero and level one), you should think about this as giving a compatible family of projective representations:
\[\rho(\pi): G_F \rightarrow \mathrm{PGL}_n(\overline{\mathbf{Q}}_p)\]
which are absolutely irreducible and crystalline of the expected weights and unramified outside \(v|p\). Now in this situation,one knows (following for example Patrikis) that there exists for any such \(\rho\) a lift to a genuine representation of \(G_F\) which is crystalline at \(v|p\) of the right weight for all \(v|p\) — this generally requires some parity condition on the weight but we are assuming that here. What is not automatic, however, is that this lift has level \(N=1\) any more; that is, the image of inertia at other primes \(v\) may be non-trivial (though of course the image lies in the center). Here there is something special which happens only for \(F = \mathbf{Q}\); as observed by Tate, you can globalize these local characters and then twist to eliminate all the auxiliary ramification. (This argument is explained by Serre in his 1975 Durham paper which is always impossible to find online; it is used to show that a complex
projective representation can be lifted to an Artin representation ramified at the same set of primes.) For other fields, even if the class number is trivial, you get global obstructions coming (via class field theory) from the unit group. (Even for imaginary quadratic fields, where the unit group is not very big, this is still an issue, and the general problem can only be avoided for fields for which the unit group has order \(2\) and which have a real place, which is quite a restrictive condition when you think about it.) The direct automorphic argument is ultimately quite similar, but there are some traps waiting for the unwary (related to Grunwald-Wang); see the discussion in this paper.

So for example, it is true that as \(F\) ranges over all imaginary quadratic fields, one has
\[H^1_{\mathrm{cusp}}(\mathrm{SL}_2(\mathcal{O}_F),\mathbf{C}) \ne 0\]
for all but finitely many \(F\). But the analogue for \(\mathrm{GL}_2(\mathcal{O}_F)\) is not only unknown, but
we certainly have:

Conjecture: There are infinitely many imaginary quadratic fields \(F\) with
\[H^1_{\mathrm{cusp}}(\mathrm{GL}_2(\mathcal{O}_F),\mathbf{C}) = 0.\]

By the way, from the perspective of Galois representations, one can see why the group above should be non-zero in the case of \(\mathrm{SL}_2(\mathcal{O}_F)\). Let \(F = \mathbf{Q}(\sqrt{-D})\). All we need to find are modular forms \(\pi\) of weight two with the property that, locally at primes \(p|D\), the corresponding Weil-(Deligne) representation on restriction to inertia becomes trivial after restriction to \(\mathbf{Q}_p(\sqrt{-D})\) up to twist. One easy way to achieve this is to take ramified principal series \(\mathrm{PS}(1,\chi)\) for some (local) ramified quadratic character \(\chi\). The problem is this leads (globally) to a sign difficulty; if \(F\) has prime discriminant, then globally you would want the weight of \(\pi\) to be two and the Nebentypus character to be the quadratic character of conductor \(\Delta_F\) which is odd, which is a problem. (Sometimes it is not; if \(F = \mathbf{Q}(\sqrt{-p})\) and \(p \equiv 1 \bmod 4\) then you can take the real character of conductor \(p\), but if \(p \equiv -1 \bmod 4\) this doesn’t work.) But instead of principal series, one can take certain supercuspidal representations: Assume that \(F_p/\mathbf{Q}_p\) is a ramified quadratic extension. Then if \(\chi\) is a totally ramified character of \(F^{\times}_p\) of order \(2^m\) where \(2^{m} \| p-1\), then the base change of this supercuspidal representation will be unramified up to twist, but the original representation will not be unramified up to twist. It’s now easy to construct such forms (and even compute how many of them there are), and see there are plenty of them when the discriminant of \(\Delta_K\) gets large (one has to avoid CM forms over \(K\) which can become non-cuspidal but these are easy to bound.) It’s also easy to see that while these base changes are unramified at every place up to a local twist they are not in general unramified everywhere up to a global twist.

The forms one finds in this way by base change are invariant under complex conjugation (now acting on the group), and there is another “geometric” way to show they exist which was originally done by Rohlfs (see this paper), who I believe was the first person to prove the non-vanishing claim above. (In fact, this is one way to start proving base change in this situation.)

When it comes to general number fields, one certainly expects (by functoriality!) that \(H^*_{\mathrm{cusp}}(\mathrm{GL}_n(\mathcal{O}_F),\mathbf{C})\) should be non-zero for \(n=79\) say and every number field \(F\), but this is hopeless for almost all fields. Using our arguments (and Newton-Thorne for totally real fields!) One certainly can prove it for many totally real and CM fields (some ramification conditions are required for the arguments to work) using the exact same argument. Of course, when for such fields there exists a cuspidal Hilbert modular form of weight two and level one then you can just used Newton-Thorne directly! For general fields, as usual, the problem of understanding automorphic forms eludes us.

Curiously enough, while writing this post, there appeared a very recent preprint by Darshan and Raghuram here which constructs, for example, cuspidal cohomology classes for \(\mathrm{GL}_n/F\) of (for example) cohomological weight zero for any number field \(F\) which is Galois over a totally real field \(F^{+}\) of some deep enough level, but with no further assumptions on \(F\). Clozel (in this paper) did something similar when \(n\) is even by automorphic induction, but already for \(n=3\) this no longer works. Assuming all conjectures, the simplest way to construct such forms for \(F = \mathbf{Q}\) or any totally real field is to take symmetric squares of Hilbert modular forms (these more or less constitute all the self-dual forms). It seems to me that the forms found by Darshan and Raghuram must be some shadow of these forms over the largest totally real subfield \(F^{+}\) of \(F\) and so one is seeing a hint of non-cyclic base change here which is intriguing! I hope to return to this later when I understand it better.

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A talk on my new work with Vesselin Dimitrov and Yunqing Tang on irrationality

Posted on June 16, 2024 by Persiflage

Here is a video of my talk from the recent 70th birthday conference of Peter Sarnak. During a talk one always forgets to say certain things, so I realized that my blog could be a good place to give some extra context on points I missed. There are three things off the top that I can add before rewatching the talk. The first is that I made a typo in one of my collaborator’s name (oops!). The second is that I didn’t mention the work of Bost-Charles, whose influence on our work is clear. Indeed the \(m = 0\) version of the holonomy theorem (version III) in this talk is a theorem in their monograph. The third is that my presentation of known irrationality results for *explicit* zeta values makes sense in the context of framing of my talk, but it’s good to note that the irrationality results of Rivoal, Ball-Rivoal, and Zudilin (for example, at least (edit: one) of \(\zeta(5), \zeta(7), \zeta(9), \zeta(11)\) is irrational) in a closely related direction are amazing theorems. There’s probably more to say, and I might add some extra comments if I watch the video again).

Some incidental remarks concerning history I thought about when preparing my talk: I know from popular accounts that Apéry’s result came as a complete surprise. Similarly, the result of Gelfand-Schneider was a complete shock as well. (Hilbert was reputed to say that he didn’t think this problem would be solved within his lifetime.) Now these two theorems are “recent enough” so that the memory of their resolution is still within the collective consciousness of mathematicians. In the first case, I still know a bunch of people (Henri Cohen and Frits Beukers) who were actually at Apéry’s infamous lecture. But what about (edit Lindemann’s) proof that \(\pi\) is transcendental? I have no sense as to what was the reaction at the time, in part due to my lack of historical knowledge but also to the lack (as far as I can see) of easily available informal discussions about contemporary mathematics from the 19th century (I assume that personal letters would be the best source). The best (?) I could find was the following (quoted from here):

In fact his [Lindemann’s] proof is based on the proof that \(e\) is transcendental together with the fact that \(e^{i \pi} = -1\). Many historians of science regret that Hermite, despite doing most of the hard work, failed to make the final step to prove the result concerning which would have brought him fame outside the world of mathematics. This fame was instead heaped on Lindemann but many feel that he was a mathematician clearly inferior to Hermite who, by good luck, stumbled on a famous result.

First, this seems pretty brutal towards Lindemann (to be fair, the continuation of the text does give some more grudging praise of Lindemann). Second, which historians are being referred to here? This seems far too judgemental for the historians I have ever spoken to in real life. If this text is at all accurate, it seems to suggest that Lindemann’s result was lauded but perhaps not considered surprising to his contemporaries? I feel that this is recent enough that one should be able to get a fuller idea of what was going on at the time.

Going back in time further, I also wonder what Lambert’s contemporaries thought of his proof (in the 1760s) that \(\pi\) was irrational. When I was giving a public talk on \(\pi\) in Sydney I looked up Lambert’s paper. The introduction is quite amusing, with the following remark that suggests a modern way of thinking not much different to how I think about things today:

Démontrer que le diametre du cercle n’est point à sa circonférence comme un nombre enteir à nombre entrier, c’est là une chose, dont les géometres ne seront gueres sorpris. On connoit les nombres de Ludolph, les rapports trouvés par Archimede, par Metius, etc. de même qu’un grand nombre de suites infinies, qui toures se rapportent à la quadrature du cercle. Et si la somme de ces suites est unq quantité rationelle, on doit assez naturellement conclure, qu’elle sera ou un nombre entier, ou one fraction très simple. Car, s’il y falloit une fraction fort composée, quoi raison y auroit-il, pourquoi plutôt relle que telle autre quelconque?

(Or in translation, errors some combination of mine and google translate):

We prove that the ratio of the diameter of the circle to its circumference is not rational; something that geometers will hardly be surprised by. We know the number \(pi\) of Ludolph, and expressions for this number found by Archimedes, by Metius, etc. in terms of a large number of infinite series of rational numbers, which all relate to the squaring of the circle. If the sum of these sequences was a rational quantity, we must quite naturally conclude that it will be either a whole number, or a very simple fraction. For, if a very complicated fraction were necessary, what reason would there be to be equal to such a number rather than any other real (irrational) number?

I guess Occam was from the 14th century!

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Zeilberger + ChatGPT

Posted on June 1, 2024 by Persiflage

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…)

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Unramified Fontaine-Mazur for representations coming from abelian varieties

Posted on May 9, 2024 by Persiflage

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.

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Midlife crisis

Posted on February 12, 2024 by Persiflage

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…

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The horizontal Breuil-Mezard conjecture

Posted on October 19, 2023 by Persiflage

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!

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Magma Instability

Posted on October 13, 2023 by Persiflage

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!

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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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