Chenevier on the Eigencurve

Today I wanted to mention a theorem of Chenever about components of the Eigencurve. Let \mathcal{W} denote weight space (which is basically a union of discs), and let

\pi: \mathcal{E} \rightarrow \mathcal{W}

be the Coleman-Mazur eigencurve together with its natural map to \mathcal{W}. It will do well to also consider the versions of the eigencurve corresponding to quaternion algebras D/\mathbf{Q} as well.

Theorem: [Chenevier] Suppose that

  1. \mathcal{E} has “no holes” (that is, a family of finite slope forms over the punctured disc extends over the missing point),
  2. The “halo” of \mathcal{E} is given by a union of finite flat components whose slope tends to zero as x \in \mathcal{W} tends to the boundary of the disc.

Then every non-ordinary component of \mathcal{E} has infinite degree.

In particular, since both of these theorems are now known in many cases (properness by Hansheng Diao and Ruochuan Liu, and haloness by Ruochuan Liu, Daqing Wan, and Liang Xiao, at least in the definite quaternion algebra case), the conclusion is also known.

The proof is basically the following. Given a component C of finite degree, the first assumption implies that it actually is proper and finite. One may then consider the norm of U_p on C to the Iwasawa algebra to obtain a bounded (hence Iwasawa) function F = \mathrm{Norm}(U_p). This function cannot have any zeros (again by properness), and hence, by the Weierstrass preparation theorem, it is a power of p times a unit. But that implies that F has constant valuation near the boundary, which contradicts the fact that the slopes are tending to zero (except in the ordinary case).

Naturally one may ask whether \mathcal{E} has only finitely many components, although this seems somewhat harder to prove.

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What does it take to get a raise?

Gauss … [had] a salary that remained fixed from 1807 to 1824.

(see here.) What was Gauss’ salary? My limited google skills were not able to find this information, although I’m not sure how meaningful it would be to translate any such number into today’s dollars. More generally, although there is available data for academic salaries over the past 40 years or so, I’m curious for comparisons that go further back in time.

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The seven types of graduate student applicant

Yes, it’s that time of year again.

  1. Hide and Seek: Contacts you every day about the status of their application, then goes on radio silence the moment they receive an offer, never to be heard of again.
  2. The No Chancer: Has an offer from Harvard, Princeton, and MIT, but still plans to attend the prospective student day because they fancy a three day holiday in (wherever your university is located).
  3. The Copy & Paster: It has always been my dream to attend Michigan University, the best university in the world. Well, good luck with that.
  4. The Nervous Nellie: Has some questions about the graduate program — a lot of questions. Wants you (that is, me) to answer detailed questions about everything from the reasonable (exact duties of a TA, particulars on graduate student stipend and health insurance) to the less so (graduation statistics and data for the last 10 years of graduates, upcoming schedule of faculty sabbaticals for the next three years, tips on the best place in Evanston to purchase toothpaste, etc.).
  5. The Surprise: Never responds to any email query about whether they are interested in coming or whether they have offers from somewhere else, is completely discounted by the committee, but then ends up accepting on April 15.
  6. The Googler: Makes an effort to look at the department website to customize their application, but gets it all wrong: I would really like to work with X, Y, and Z where X is a postdoc, Y has retired, and Z moved to a different institution two years ago.
  7. The Unicorn: Actually accepts the offer well before April 15.

Tell me if I’ve missed anyone.

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Only Harvard Grads need apply

It’s hard to take articles in Slate too seriously, but I have to admit I was quite perplexed about the following article (with the concomitant research publication here).

The main thrust of the article seems to be as follows. A disproportionate number faculty at research universities in the US received PhDs from a small number of prestigious institutions, and hence (?) such hiring practices reflect profound social inequality. Is it just me, or does this appear to be utter bollocks? There is an obvious pair of hypotheses that would completely explain the data, namely:

  1. There is a hierarchical system of admission to graduate programs,
  2. Universities hire the strongest candidates they can, and admit the strongest graduate students they can.

Let’s examine these possibilities in the context of graduate school in mathematics. I have, on several occasions, been responsible for graduate admissions at my institution. I would say, on the whole, that prospective graduate students are among the most class conscious of anyone in academia. I would guess that, at least 75% of the time, a student will accept either the program that is the most highly ranked amongst those where they were admitted or a school within at most one or two places of their highest ranked option.

What about the second hypothesis? The worry here is that universities might view “undergraduate/graduate institution” as a proxy for “quality of candidate.” In my experience (being on hiring committees), this is utterly preposterous. I am not claiming that mathematical judgements are not a slippery thing — there are many variations which relate to matters of taste and inclination — but there are some reasonable objective criteria (GRE scores for graduate applications, publication record for job candidates) which would serve as a check against any implicit bias in this regard.

We here at Persiflage, however, are open to the idea that we may have missed something. So here are some other possibilities:

  1. You are talking about Mathematics, a field for which it is easier to make reliable judgements about the quality of research, and a field for which there is a more pronounced spike in talent at the top of the scale. Is this true? I honestly don’t know. Perhaps whatever field it is that produces papers like the one under consideration is not something for which talent of any kind is an asset, and so there is no real difference between graduates from Harvard or from Podunk U. Less sarcastically, suppose (say) I compare the English department faculty at the top ranked place (taking from this list) Berkeley and compare it to a place also ranked in the top 25 schools but closer to the bottom of that list, say UIUC. Then, if I knew something, could I confidently say that one department is much better than the other?
  2. You are talking about the experience of hiring/admitting students at a Group I university. Perhaps it is the case that, for lower ranked universities, there is insufficient expertise to hire on the basis of talent/output, and so PhD institution serves as a lazy way to evaluate the candidate. This seems to be a somewhat condescending argument, but it’s true that I don’t have any idea how hiring works at non-Group I universities. But surely the letters of recommendation would carry the most weight, and they would reflect the quality of research? At the very least, if you are going to claim this is what happens, you need to come up with a way to substantiate that claim.

Ultimately, I certainly don’t feel that I can rule out bias when it comes to hiring, but the fact that the paper under review uses “prestige” as a dirty word and doesn’t seem to acknowledge in any way that there is some correlation between prestige and quality of graduates is highly disturbing. Perhaps, as with this paper, the main goal is to substantiate the political beliefs of the authors rather than to undertake a serious academic inquiry. Still, even if the methodology is flawed, I would like people’s opinion on the conclusion.

Posted in Politics, Rant | Tagged , | 9 Comments

Review of Buzzard-Gee

This is a review of the paper “Slopes of Modular Forms” submitted for publication in a Simons symposium proceedings volume.

tl;dr: This paper is a nice survey article on questions concerning the slopes of modular forms. Buzzard has given a (very explicit) conjecture which predicts the slopes of classical modular (p-stabilized) eigenforms of level prime to p, at least under a certain regularity hypothesis. One consequence is that, under favourable circumstances, all the slopes are integers. The current paper describes the link between this and related problems to the p-adic Langlands program, as well as raising several further intriguing questions concerning the distributions of these slopes. The paper is well written, and is a welcome addition to the literature. I strongly recommend that this paper be accepted.

Review: Buzzard’s slope conjectures live somewhere in the world between 19th and 21st century mathematics. Suppose that one considers the space of over-convergent cusp forms of level N = 1 for p = 2. Then, using nothing more than classical identities between modular functions, one may prove that the smallest eigenvalue of the compact operator U_2 is at most \|2^3\|_2 = 1/8. On the other hand, it is now a “folklore” conjecture (Conjecture 4.1.1 of the paper under review) that, if p is odd and f is a classical modular form of level \Gamma_0(N) prime to p, then the residual representation:

\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(k)

is irreducible locally on the decomposition group at p whenever the valuation of a_p is not an integer. This problem seems to be a deep question in the p-adic Langlands program for \mathrm{GL}_2(\mathbf{Q}_p). The two cases where this is known, v(a_p) < 1 and v(a_p) sufficiently large, both require machinery from p-adic Hodge theory — in the former case, one needs the full local Langlands correspondence for \mathrm{GL}_2(\mathbf{Q}_p).

Comments: Here are some comments given in some order that bears little relation to the actual paper.

  • I don’t like the table on page 6, in particular, because certain ranges of numbers are bunched together, the output looks a little strange. Can one improve this in some way? Perhaps finish at 3^{10}? Perhaps include only selected powers bigger than 3^{10}? Perhaps normalize for the length of the range?
  • Corollary 5.1.2. Do you want to speculate on what happens in the reducible case? In some sense, in Buzzard’s conjecture, one doesn’t see the fact that the residual representations are globally reducible or not. On the other hand, weird stuff certainly happens for p = 2, as previously mentioned here. What happens in the reducible case for p = 3?
  • Conjecture 4.1.1 demands that k is even, but that is a consequence of the level being of the form \Gamma_0(N) — something which is noted immediately after the statement. So why include the condition in the statement of the conjecture? Also, perhaps it’s also worth remarking upon the case when a_p is a unit.
  • The authors (in Remark 4.1.3) point to the origins of this Conjecture 4.1.1 to around 2005. However, I feel like I remember some discussion of this conjecture in the Durham symposium of 2004. There were certainly hints of this conjecture on «la serviette de Kisin», upon which Mark gave a heuristic local argument for why the Eigencurve was proper — although the argument was slightly dodgy in that it collapsed if the napkin was rotated 90 degrees. (Of course, Mark was proved right when Hansheng Diao and Ruochuan Liu did indeed prove this result using local methods here.) Also, isn’t Conjecture 4.1.1 a consequence of Buzzard’s original conjecture as modified by Lisa Clay? Somehow it seems to me that what Remark 4.1.3 is referring to is the idea that Conjecture 4.1.1 is a consequence of a purely local conjecture, and refers to the period (2005?) when Breuil was formulating the first versions of the p-adic Langlands program.
  • For Conjecture 4.2.1, wouldn’t it make more sense to normalize the valuation in terms of the coefficient field \mathbf{Q}_p(\chi), so the statement once more becomes that a_p has integral valuation?
  • Why is the condition on Buzzard’s conjecture different when p = 2? (I understand it has to be modified in order to have a chance of being true, but I am asking if there is any explanation for why this is necessary.)
  • The authors remark (p.3) that it is not known whether there are infinitely many Buzzard irregular primes. Here is a short argument to prove that this is a consequence of standard conjectures of prime values of polynomials. We start with the observation that the first Buzzard irregular prime is p =59, and that the offending representation:

    \rho_{Q \Delta} : G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_{59})

    has exceptional image in the context of Serre and Swinnerton-Dyer (On l-adic representations and congruences for coefficients of modular forms, Antwerp III). Indeed, this particular example features prominently in that paper. I always thought this was not an entirely random coincidence, and since it seems relevant here, I thought I would finally bother to figure out what is going on. (For the next prime, p = 79, the corresponding representation has image containing \mathrm{SL}_2(\mathbf{F}_{79}), so it is somewhat of an accident.) The mod-59 representation above has projective image S_4. Now suppose that p \equiv 3 \mod 4 is a prime such that H/\mathbf{Q} is an S_4-extension which is unramified away from p. Such a representation will give rise (following an argument of Tate) to a mod-p representation of \mathbf{Q} which is unramified away from p. The congruence condition on p implies that it will be odd, and hence modular, by Langlands-Tunnell. Now let us suppose, in addition, that 4 divides the ramification index e_p. Under this assumption, the representation cannot be locally reducible, because the ramification index of any power of the cyclotomic character divides p-1 \equiv 2 \mod 4. Hence, if there are infinitely many such fields, there are infinitely many \mathrm{SL}_2(\mathbf{Z})-irregular primes. Consider the following fields studied by Darrin Doud (here):

    \displaystyle{K = \mathbf{Q}[u]/f(u), \quad f(u) = (u + x)^4 - p^* u},

    \displaystyle{(-1)^{(p-1)/2} p = p^* =  \frac{256 x^3 - y^2}{27}, \quad p^* \ne 1 + 4x}.

    Doud shows that K has discriminant (p^*)^3 and Galois group S_4, and it is easy to see that the splitting field H/\mathbf{Q} has all the required properties needed above as long as p \equiv 3 \mod 4. The formula relating tame ramification and the discriminant implies that e_p(K/\mathbf{Q}) = 4. Standard conjectures now predict that there are infinitely many such primes of this form — if y is odd, then p^* < 0. The first few such primes which are 3 \mod 4 are

    59, 107, 139, 283, \ldots

    which compares with the first few Buzzard irregular primes (taken from Buzzard’s paper):

    59, 79, 107, 131, 139, 151, 173 \ldots

    On the other hand, an unconditional proof by these means seems out of reach, because any modular S_4-extension unramified outside p \equiv 3 \mod 4 forces \mathbf{Q}(\sqrt{-p}) to have class number divisible by 3, and we don’t know if there exist infinitely many such primes.

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Random Photos

Lunt Hall, the Northwestern mathematics building,  recently underwent an upgrade of the fire alarm system. This includes introducing informative new signage, such as the following:

Sign

Another change is that the “internal” window to my office was boarded up. The window overlooked the internal stairwell and had previously been covered up with posters, but I always had the idea of removing them and using the window for office hours; students would look up at the window and ask questions, and I would offer them Delphic pronouncements in return. Alas, I will no longer be able to do this. Here’s a picture from the brief period between when the posters were removed and the window was boarded up; the photo is taken from the top of the stairs; the spot directly next to the window is a couple of feet lower:

office

Finally, I haven’t yet managed to get around to blogging about the Sarnak conference! So here at least is a picture, which is an action shot of me advocating (unsuccessfully, it turns out) for a certain paper to be accepted into Duke. (Pictured are Jonathan Wahl (Duke supremo), RLT, and me.)

Finally, here is an amusing box of free books I occasionally pass when I walk to work:

Tardis

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H_2(Gamma_N(p),Z)

In this post (which is a follow-up to the last post), I wanted to compute the group H_2(\Gamma_N(p),\mathbf{Z}), where \Gamma_N(p) is the congruence subgroup of \mathrm{SL}_N(\mathbf{Z}) for large enough N and p is prime. In fact, to make my life easier, I will also assume that p > 3, and in addition, ignore 2-torsion. The first problem is to compute the prime to p torsion. By Charney’s theorem, this will come from the cohomology of the homotopy fibre X of the map

\displaystyle{   SK(\mathbf{Z}) \rightarrow SK(\mathbf{F}_p).}

The relevant part of the Serre long exact sequence is, using classical computations of the first few K-groups of the integers together with Quillen’s computation of K_*(\mathbf{F}_p),

\displaystyle{   0 \rightarrow \pi_3(X) \rightarrow \mathbf{Z}/48 \mathbf{Z} \rightarrow   \mathbf{Z}/(p^2 - 1) \mathbf{Z} \rightarrow \pi_2(X) \rightarrow \mathbf{Z}/2 \mathbf{Z}  \rightarrow 0.}

Here is where it is convenient to invert primes dividing 6; from Hurewicz theorem and Charney’s theorem we may deduce that, where \sim denotes an equality up to a finite group of order dividing 48,

\displaystyle{   H_2(\Gamma_N(p),\mathbf{Z}[1/p])  \sim \mathbf{Z}/(p^2 - 1) \mathbf{Z}.}

In order to deal with 3-torsion, then we also have to show that the map K_3(\mathbf{Z}) \otimes \mathbf{Z}/3 \mathbf{Z} \rightarrow K_3(\mathbf{F}_p) is injective for p \ne 3. I have a sketch of this which I will omit from this discussion but it is not too hard (assuming Quillen-Lichtenbaum). It remains to compute the homology with coefficients in \mathbf{Z}_p. I previously computed that there was an isomorphism

\displaystyle{   H_2(\Gamma_N(p),\mathbf{F}_p) = \wedge^2 \mathfrak{g} \oplus \mathbf{F}_p  = H_2(G_N(p),\mathbf{F}_p) \oplus \mathbf{F}_p,}

where G_N = \mathrm{SL}_N(\mathbf{Z}_p) and \mathfrak{g} = H_1(\Gamma_N(p),\mathbf{F}_p) is the adjoint representation.

Some facts concerning the cohomology of G_N(p):

There are short exact sequences:

\displaystyle{   0 \rightarrow H_2(G_N(p),\mathbf{Z}_p)/p \rightarrow H_2(G_N(p),\mathbf{Z}/ p \mathbf{Z})  \rightarrow H_1(G_N(p),\mathbf{Z}_p)[p] \rightarrow 0,}

\displaystyle{   0 \rightarrow H_2(G_N(p),\mathbf{Z}_p)/p^2 \rightarrow H_2(G_N(p),\mathbf{Z}/ p^2 \mathbf{Z})  \rightarrow H_1(G_N(p),\mathbf{Z}_p)[p^2] \rightarrow 0.}

Since H_1(G_N(p),\mathbf{Z}_p) = \mathfrak{g} is annihilated by p, we may deduce that

\displaystyle{   H_2(G_N(p),\mathbf{Z}_p)/p = H_2(G_N(p),\mathbf{Z}_p)/p^2}

as long as

\displaystyle{   |H_2(G_N(p),\mathbf{Z}/p^2 \mathbf{Z}) | = | H_2(G_N(p),\mathbf{Z}/ p \mathbf{Z})|.}

Such an equality (for any group) is a claim about the Bockstein maps having a big an image as possible. Indeed, for any group \Phi, there is an exact sequence:

\displaystyle{  H_3(\Phi,{\mathbf{Z}}/p {\mathbf{Z}}) \rightarrow H_2(\Phi,{\mathbf{Z}}/p {\mathbf{Z}}) \rightarrow H_2(\Phi,{\mathbf{Z}}/p^2 {\mathbf{Z}})  \rightarrow H_2(\Phi,{\mathbf{Z}}/p {\mathbf{Z}}) \rightarrow H_1(\Phi,{\mathbf{Z}}/p {\mathbf{Z}})}

The first and last maps here are the Bockstein maps \beta_2 and \beta_1. Since p is odd, \beta_1 \circ \beta_2 = 0. On the other hand, we see that the orders of the cohomology groups with coefficients in \mathbf{Z}/p \mathbf{Z} and \mathbf{Z}/p^2 \mathbf{Z} will have the same order if and only if

\displaystyle{   \ker(\beta_1) = \mathrm{im}(\beta_2).}

Hence we have reduced to the following claim. Take the complex

\displaystyle{  H_*(G_N(p),\mathbf{F}_p) = \wedge^* \mathfrak{g} }

where the differentials are given by the Bockstein maps. Then we have to show that the cohomology of this complex vanishes in degree two. But what are the Bockstein map is in this case? Note that since H_1(G_N(p),\mathbf{Z}_p) = \mathfrak{g} is annihilated by p, the Bockstein map \beta_1 will be a surjective map:

\displaystyle{   \beta_1: \wedge^2 \mathfrak{g} \rightarrow \mathfrak{g}.}

To compute this explicitly, recall that the isomorphism H_1(G_N(p),{\mathbf{Z}}_p) = \mathfrak{g} comes from the identification of \mathfrak{g} with G_N(p)/G_N(p^2). Then, computation omitted due to laziness, we find that the Bockstein is precisely the Lie bracket. Moreover, since the (co-)homology is generated in degree one, the higher Bockstein maps can be computed from the first using the cup product formula. So the Bockstein complex above is, and I haven’t checked this because it must be true, the complex computing the mod-p Lie algebra cohomology of \mathfrak{g}. And this cohomology vanishes in degrees one and two, so we are done. One consequence of this computation is that

\displaystyle{  H_2(G_N(p),\mathbf{Z}_p) = H_2(G_N(p),\mathbf{Z}_p)/p}

is annihilated by p. Moreover, the last term can be identified with the kernel of the Lie bracket (Bockstein) on H_2(G_N(p),\mathbf{F}_p) = \wedge^2 \mathfrak{g}.

Returning to the main computation:

From the Hochschild–Serre spectral sequence and the computation of stable completed cohomology, one has an exact sequence:

\displaystyle{   0 \leftarrow H_2(G_N(p),\mathbf{Z}_p) \leftarrow H_2(\Gamma_N(p),\mathbf{Z}_p) \leftarrow \mathbf{Z}_p \leftarrow H_3(G_N(p),\mathbf{Z}_p).}

From known results in characteristic zero, we immediately deduce that there is some \alpha such that there is an exact sequence

\displaystyle{   0 \leftarrow H_2(G_N(p),\mathbf{Z}_p) \leftarrow H_2(\Gamma_N(p),\mathbf{Z}_p) \leftarrow \mathbf{Z}/p^{\alpha} \mathbf{Z} \leftarrow 0.}

we also deduce that there is an exact sequence:

\displaystyle{   0 \leftarrow H_2(G_N(p),\mathbf{Z}/p^n \mathbf{Z}) \leftarrow H_2(\Gamma_N(p),\mathbf{Z}/p^n \mathbf{Z}) \leftarrow \mathbf{Z}/p^{\mathrm{min}(\alpha,n)} \mathbf{Z} \leftarrow 0,}

There are spectral sequences:

\displaystyle{  H_i(\mathrm{SL}_N(\mathbf{F}_p),H_j(\Gamma_N(p),A)) \Rightarrow H_{i+j}(\Gamma_N,A)}

for A = \mathbf{Z}/p^n \mathbf{Z} and A = \mathbf{Z}_p.
For both of these rings, we have

\displaystyle{  H_1(\Gamma_N(p),A) = \mathfrak{g}, \qquad H_0(\Gamma_N(p),A) = A.}

Moreover, for sufficiently large N, we have

\displaystyle{  H_i(\mathrm{SL}_N(\mathbf{F}_p),A) = 0,}

this follows from and is equivalent to Quillen’s computation which implies that the K-groups of finite fields have order prime to p. Since H_2(\mathrm{SL}_N(\mathbf{Z}),\mathbf{Z}_p) is trivial for p > 2, we deduce that

\displaystyle{  H_0(\mathrm{SL}_N(\mathbf{F}_p),H_2(\Gamma_N(p),A)) = H_2(\mathrm{SL}_2(\mathbf{F}_p),\mathfrak{g}) = \mathbf{F}_p,}

where the last equality was already used in my paper. The compatibility of the spectral sequence above for different A implies that we also get an isomorphism

\displaystyle{  H_0(\mathrm{SL}_N(\mathbf{F}_p),H_2(\Gamma_N(p),\mathbf{Z}_p))   =  H_0(\mathrm{SL}_N(\mathbf{F}_p),H_2(\Gamma_N(p),\mathbf{Z}/p \mathbf{Z})).}

On the other hand, the invariant class must be an element of order p in

\displaystyle{  \mathbf{Z}/p^{\alpha} \mathbf{Z};}

and hence the reduction map

\displaystyle{  \mathbf{Z}/p^{\alpha} \mathbf{Z} \rightarrow \mathbf{Z}/p \mathbf{Z}}

sends an element of order p to an element of order p, and so \alpha = 1.

Putting things back together:

Assembling all the pieces, we see that we have proven the following:

Theorem: Let p > 3, and let N be sufficiently large. Let \mathfrak{g} be the Lie algebra \mathfrak{sl}_N over \mathbf{F}_p. Then, up to a finite group of order dividing 48, we have

\displaystyle{   \displaystyle{ H_2(\Gamma_N(p),\mathbf{Z}) \sim  \mathbf{Z}/(p^2 - 1) \mathbf{Z} \oplus \mathbf{Z}/p \mathbf{Z}    \oplus \ker \left( [\ , \ ] \wedge^2 \mathfrak{g} \rightarrow \mathfrak{g} \right)}.}

Moreover, still with p > 3, then up to a group of order dividing 16, we should have the same equality with p^2 - 1 replaced by (p^2 - 1)/3.

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