Hire my students!

Posted on November 18, 2020 by Persiflage

I have three students graduating this year: Shiva Chidambaram, Eric Stubley, and Noah Taylor. In light of the last post, I should give them a boost by reminding you of their (numerous) results which have been discussed on this blog. You can read about Shiva’s work here, here, and here, about Eric’s work here and here, and Noah’s work here and here. Alternatively, you can always click on the work of my students link.

But even this link is not complete! Here’s a result from Noah’s thesis which I haven’t discussed before:

Let \(N\) be prime, and let \(\mathbf{T}\) denote the \(\mathbf{Z}_2\)-Hecke algebra generated by \(T_l\) for \(l\) prime to \(2\), and let \(\widetilde{\mathbf{T}}\) denote the Hecke alegbra where \(T_2\) is also included. These Hecke algebras are famously not the same in general. For example, when \(N = 23\), the space of cusp forms is \(2\)-dimensional and has a pair of conjugate cusp forms as follows:

\(\displaystyle{q – \frac{\sqrt{5}+1}{2} q^2 + \sqrt{5} q^3 + \frac{\sqrt{5} – 1}{2} q^4 – (1 + \sqrt{5})q^5 + \ldots}\)

So \(\mathbf{T} = \mathbf{Z}[\sqrt{5}]\) whereas \(\widetilde{\mathbf{T}} = \displaystyle{\mathbf{Z} \left[ \frac{\sqrt{5}+1}{2} \right]}\). Noah gives a formula for the index:

Theorem: Let \(N\) be prime. Then the index \([\widetilde{\mathbf{T}}:\mathbf{T}]\) is given by the order of the space

\(S_1(\Gamma_0(N),\mathbf{F}_2)\)

of Katz modular forms of weight one and level \(\Gamma_0(N)\).

In particular, the index at level \(23\) is coming from the fact that there is a classical weight one form of this level. From this one sees that the index is non-trivial for all primes \(N \equiv 3 \bmod 4\) except for \(N = 3,7,11,19,43,67\) and \(163\). For primes \(N \equiv 1 \bmod 4\), on the other hand, I might guess that there would be a positive density of primes for which either the index was trivial or non-trivial. The question more or less hinges on the expected number of \(\mathbf{SL}_2(\mathbf{F}_{2^n})\) representations of \(\mathbf{Q}\) (with \(n \ge 2\)) which become unramified at all finite places over \(\mathbf{Q}(\sqrt{N})\).

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The upcoming jobs bloodbath

Posted on November 12, 2020 by Persiflage

Universities are losing lots of money this year. Even those schools with a sizeable endowment are very restricted in how those funds can be used, and the result is that many places will have hiring freezes. This is surely going to have an immediate impact in the jobs market in mathematics, at every level. In a usual year, Chicago hires as many as ten Dickson instructors (our named postdoctoral position). This year, I find it hard to imagine that we would hire half that number. In part, this is because we have moved to protect a number of our final year postdocs by extending their position for another year, although if enough other places do something similar then next year is going to be tough as well.

There are rumors that a number of places (including really top places) are not going to admit any graduate students in mathematics next year, and that others will have at the very least significantly smaller classes. I fully expect that we will have an incoming class but I don’t know how large it is going to be.

Does your department expect to reduce (significantly or moderately) your postdoc hiring this year? What about tenure track lines or graduate students? Let me know! During the last financial crisis, both the Simons foundation and the NSF (via the stimulus bill) had a real effect by adding more money to the postdoc pool — hopefully something like that will happen again.

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En Passant IX (I’m a Gnu)

Posted on October 28, 2020 by Persiflage

One feature of having an electric piano is the ability to record the accompaniment to songs which (for reasons of timing or otherwise) are quite hard to play and sing at the same time. A possible downside, however, is that this accompaniment is now available at any notice, and hence subject to the whims of any household member who perhaps does not appreciate what you wish to play and instead wants to listen to yet another rendition of the Gnu. And this is why the following song is the only live music performed at our house at the moment:

Everyone pretty much knows all the words at this point! (Hat Tip to Martin Rutherford for playing ill wind during music class in 1990)

In these times I recommend that everyone relax by taking a deep breath. And I mean the type of breath necessary for the following oboe part assuming that circular breathing (exhaling and breathing in at the same time) is not part of your daily repertoire:

Finally, a few tips on Australian fusion cuisine. If for some reason you find yourself going for long periods of time between trips to the grocery store, you might just consider opening that jar of Vegemite on the shelf, and then start eating it every day for breakfast. You may know the basic Vegemite tip (use buttered toast, don’t use too much), but you might be unsure what to do if your standard Italian bread is not available. This is advice for those times:

  1. Challah: This is not a good match. Toasted challah does not have the required firmness and it just doesn’t work. When eating toasted Challah, always stick to marmalade.
  2. Tortillas: A disaster: melted Vegemite on a tortilla running down the side in little puddles. Avoid.
  3. Injera: Jackpot! A pefect match. Probably a buckwheat crepe would also do as a pinch for a substitute.
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The eigencurve is (still) proper

Posted on October 23, 2020 by Persiflage

Although I don’t think about it so much anymore, the eigencurve of Coleman-Mazur was certainly one of my first loves. I can’t quite say I learnt about \(p\)-adic modular forms at my mother’s knee, but I did spend a formative summer before starting university thinking about (with Matthew Emerton) what in effect was the \(2\)-adic eigendecomposition of the (inverse) hauptmodul \(f = q \prod (1 +q^n)^{24}\) of \(X_0(2)\). I remember that we had a massive file called “tee-hee” which contained an absolutely huge number of Fourier coefficients which tested the memory limits of the University of Melbourne computer system (it was 10MB).

Jumping forward in time, I learnt about Kevin Buzzard’s Arizona Winter School project on a special case of his slope conjectures. This turned out to be closely related to the explicit computations I had done when I was younger. I got in touch and we managed to solve the first special case of his conjectures. Kevin and I continued collaborating over the next few years on a number of papers related to the geometry of the eigencurve.

In the abstract theory of the eigencurve, it is not important how overconvergent a modular form is but merely that it is overconvergent. However, it has always seemed to me that the analytic theory of overconvergent modular forms deep into the supersingular annuli has many unrevealed mysteries. One problem Kevin and I thought about was whether the eigencurve was “proper” in the sense of whether any punctured disc of finite slope eigenforms could be filled in at the central point. (Coleman and Mazur raise this question in their original paper.) At one point we thought we had proved it — the idea was that (by Buzzard’s analytic continuation theorem) any finite slope eigenform would converge uniformly far into the supersingular annuli, and since this property would hold uniformly for all points on the punctured disc of finite slope eigenforms it would follow that the limiting form at the centre was also highly overconvergent. However, if that form had infinite slope, it would lie in the kernel of the \(U\) operator, and now there was an elementary argument to show that any such form had a natural radius which was not (as) highly overconvergent. Done! Except there was a problem: the results on overconvergence were only proved for forms of integral weight since they relied on geometric constructions, particularly on the fact that one could make sense of \(\omega^k\) (for an integer \(k\)) on the entire modular curve. Coleman’s definion of overconvergent forms of general weight used a trick involving Eisenstein series. The notion of radius of convergence arising from this construction ended up being related to ratios of certain Eisenstein series in weight zero, and these ratios are not very overconvergent for the most general weights. This meant that (for general weights) the radius only made sense in a small overconvergent region — in particular smaller than the radius necessary to rule out elements in the kernel of \(U\) — and the idea didn’t work. In some cases (for example the \((N,p)=(1,2)\) eigencurve) there were workarounds one could make to give ad hoc definitions of the radius in order to push things through (a proof of concept as it were), but the situation was otherwise not so great.

Some time later, Hansheng Diao and Ruochuan Liu proved that the eigencurve was indeed proper. There argument was completely different, and used local arguments and period rings. It was a very nice result, and possibly their argument should even be considered the “correct” one. However, to my delight, Lynnelle Ye has just posted to the arXiv a new proof of the properness of the eigencurve which does indeed proceed by exploiting the radius of overconvergence for finite slope forms and proving that it inconsistent with the radius of convergence of elements in the kernel of \(U\). As mentioned above, the immediate stumbling block for Kevin and I was that the definition of overconvergent forms of a general weight \(\kappa\) was not geometric. However, thanks to Pilloni and Andreatta-Iovita-Stevens there now are such definitions available. Ye takes these constructions and then pushes them further into the supersingular annuli. These efforts are then indeed enough to turn what was merely a heuristic into a completely rigorous proof!

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Chidambaram on Galois representations (not) associated to abelian varieties over Q

Posted on October 13, 2020 by Persiflage

Today’s post is about a new paper by my student Shiva. Suppose that \(A/\mathbf{Q}\) is a principally polarized abelian variety of dimension \(g\) and \(p\) is a prime. The Galois representation on the \(p\)-torsion points \(A[p]\) gives rise to a Galois representation:

\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_{2g}(\mathbf{F}_p)\)

with the property that the similitude character coincides with the mod-\(p\) cyclotomic character. A natural question to ask is whether the converse holds. Namely, given such a representation as above with the constraint on the similtude character, does it necessarily come from an abelian variety (principally polarized or not)?

When \(g=1\), the answer is that all such representations come from elliptic curves when \(p \le 5\), but that for \(p \ge 7\) there exist representations for any \(p\) which do not. For \(p \le 5\), more is true: the twisted modular curves \(X(\rho)\) all are isomorphic to \(\mathbf{P}^1\). When \(p \ge 7\), the curves \(X(\rho)\) are of general type, so one might expect a “random” such example to have no rational points. Dieulefait was the first person to find explicit representations (for any such \(p\)) which do not come from elliptic curves (and there is a similar result in my paper here). Both of these arguments exploit the Hasse bound. Namely, if

\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)

is unramified at \(l \ne p \ge 5\) and \(\rho\) comes from \(E/\mathbf{Q}\), then \(E\) must have either good or multiplicative reduction at \(l\). But this puts a constraint on the possible trace of Frobenius at the prime \(l\). For \(l = 2\), for example, this leads to explicit examples of non-elliptic mod-\(p\) representations for \(p \ge 11\). The case \(p = 7\), however, requires a different argument. More generally, while the Hasse argument does generalize to larger \(g\), it only works when \(p\) is large compared to \(g\). On the other hand, the Siegel modular varieties \(\mathcal{A}_g(p)\) of principal level \(p\) are rational over \(\mathbf{C}\) for only very few values of \(g\) and \(p\). Indeed, they are rational only for

\((g,p) = (1,2), (1,3), (1,5), (2,2), (2,3), (3,2)\)

whereas \(\mathcal{A}_g(p)\) turns out to be of general type for all other such pairs. When \((g,p)\) is on this list, then, as discussed in these posts, the twists \(\mathcal{A}_g(\rho)\) can all be shown to be unirational over \(\mathbf{Q}\) and so any such representation \(\rho\) does indeed come from infinitely many (principally polarized) abelian varieties.

Thus one is left to consider all the remaining pairs. This is exactly the question resolved by Shiva:

Theorem [Chidambaram]: Suppose that \((g,p)\) is not one of the six pairs above such that \(\mathcal{A}_g(p)/\mathbf{C}\) is rational. Then there exists a representation:

\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_{2g}(\mathbf{F}_p)\)

with cyclotomic similitude character which does not come from an abelian variety over \(\mathbf{Q}\).

Shiva’s argument does not use the Weil bound. Instead, the starting point for his argument is based on the following idea. Start by assuming that \(\rho\) comes from an abelian variety \(A\). Suppose also that \(\rho\) is ramified at \(v \ne p\) and the image of the inertia group at \(v\) contains an element of order \(n\) for some \((n,p) = 1\). Using this, one deduces (using independence of \(p\) arguments) that

\(|\mathrm{Sp}_{2g}(\mathbf{F}_l)| = l^{g^2} \prod_{m=1}^{g} l^{2m} – 1\)

is divisible by \(n\) for all large enough primes \(l\), and hence divides the greatest common divisor \(K_g\) of all these orders. This is actually a very restrictive condition on \(n\). For example, using Dirichlet’s theorem, the number \(K_g\) is only divisible by primes at most \(2g+1\). But now if the order of the group \(\mathrm{Sp}_{2g}(\mathbf{F}_p)\) for any particular \(p\) is divisible by a prime power \(n\) with \(n\) not dividing \(K_g\), then one can hope to construct a mod-\(p\) Galois representation whose inertial image at some prime \(v\) has order divisible by this \(n\), and this representation cannot come from an abelian variety over \(\mathbf{Q}\).

The good news is that one can show that (most) symplectic groups have orders divisible by large primes using Zsigmondy’s theorem. Combined with a few extra tricks and calculations for some boundary cases, the groups \(\mathrm{Sp}_{2g}(\mathbf{F}_p)\) contain elements of “forbidden” orders exactly when one is not in the case of the six exceptional pairs \((g,p)\). Note that Zsigmondy’s theorem already arises in the literature in this context in order to understand prime factors of the (corresponding) simple groups.

So now one would be “done” if one could (for example) solve the inverse Galois problem for \(\mathrm{GSp}_{2g}(\mathbf{F}_p)\) with local conditions. The inverse Galois problem is solved for these groups, but only because there is an obvious source of such representations coming from abelian varieties. Of course, these are precisely the representations Shiva wants to avoid.

Instead Shiva looks for solvable groups inside \(\mathrm{GSp}_{2g}(\mathbf{F}_p)\) containing elements of order \(n\) for suitable large prime powers \(n\). Note that the obvious thing would simply be to take the cyclic group generated by the element of the corresponding order. The problem is that there is no way to turn the corresponding representation into a Galois representation whose similitude character is cyclotomic. The groups Shiva actually uses are constructed as follows. Start by finding prime powers \(n | p^{m} + 1\) for some \(m \le g\), then embed the non-split Cartan subgroup of \(\mathrm{SL}_2(\mathbf{F}_{p^m})\) into \(\mathrm{GSp}_{2g}(\mathbf{F}_p)\), and then consider the normalizer of this image. One finds a particularly nice metabelian subgroup whose similitude character surjects onto \(\mathbf{F}^{\times}_p\). Shiva then has to prove the existence of a number field whose Galois group is this metabelian extension with the desired ramification properties at some auxiliary prime \(v\) but also crucially satisfying the cylotomic similitude character condition. This translates into a (typically) non-split embedding problem — such problems can be quite subtle! Shiva solves it by a nice trick where he relates the obstruction to a similar one which can be shown to vanish using methods related to the proof of the Grunwald-Wang Theorem. Very nice! In retrospect, the case of \(g = 1\) and \(p = 7\) in my original paper is a special example of Shiva’s argument, except it falls into one of the “easy” cases where the relevant metabelian extension actually is a split extension over the cyclotomic field. In general, this only happens when the the maximum power of \(2\) dividing \(g\) is strictly smaller than the maximum power of \(2\) dividing \(p-1\) which is automatic when \(g\) is odd. (The case when \(p = 2\) is easier because the cyclotomic similitude character condition disappears!)

Posted in Mathematics, Students, Work of my students | Tagged , , , , , , , , | 5 Comments

Tips on becoming a computational number theorist

Posted on June 16, 2020 by Persiflage

How would you advise a student who has talents in both the computational and theoretical aspects of algebraic number theory? There is no hard border between computational and theoretical algebraic number theory, but there is a definite computational number theory community with its own norms and expectations. While I interact with this world, I am not a part of it, and so I don’t necessarily have the best practical advice to offer such a student. So instead of just guessing, I emailed a few people (including Drew Sutherland and John Voight) who graciously sent me a number of suggestions. Below is a lightly edited version of Drew’s email (incorporating some of John’s responses), posted with permission: I think there is some great advice here for both students and advisors! One comment I would like to amplify is the idea of spending time engaging with experts in the field. This is a useful suggestion for anyone in mathematics. It can certainly be terrifying for a fresh graduate student to put themselves out there and talk to the famous experts, but that of course is why it is important for senior people to make the effort to interact with junior people. (No doubt some areas of mathematics have more approachable famous people than others; computational number theory seems to be a pretty friendly place.) Finally, it is incumbent upon me as an advisor to note that my student Shiva Chidambaram (expected graduation 2021) will indeed be giving a talk at ANTS this year. If you are (virtually) attending ANTS this year, please consider both going to this talk and also saying hello to Shiva!

  1. Learn how to use Magma, Pari/GP, and Sage well. Each of these tools has particular strengths and there will almost certainly come a time when you want to be able to take direct advantage of them. In addition to your own research, if you are marketing yourself as a computational person you need to be prepared for random computational requests that may fall well outside your mathematical expertise; if you know the right button to push you can still come out looking like a genius.


  2. Should I be using a low level programming language like C/C++ and learning how to exploit libraries like GMP, Pari, FLINT, or NTL?
    This might eventually become important, but unless you have a project that would benefit from some serious computational horsepower, I wouldn’t burn a lot of time writing (or learning to write) low level code right now. But if you do have such a project (or have an advisor who can suggest one), then by all means, swing for the fences! Nothing better demonstrates your computational chops than computing something no one else has been able to compute. But I should note that this doesn’t necessarily require writing any low level code, a better algorithm that cleverly leverages functionality built in to Magma, Pari/GP, or Sage may achieve the same result.


  3. Make it your goal to deliver a talk at a major workshop or conference focused on computational number theory or explicit methods, such as the Algorithmic Number Theory Symposium (ANTS). Your future employer probably won’t be represented there, but you will get a chance to join a network of people who can really help you, both with your immediate job search and in the longer term. ANTS VIII was my first mathematics conference, and looking at the participant list twelve years later I can count at least twenty names of people who have played a direct role in my career as co-authors, co-editors, co-organizers, or mentors in some form. Ideally your talk will also lead to or be based on a publication, but making connections is really the crucial thing when you are getting started, I can trace eight of the papers I wrote in the years immediately following ANTS VIII directly to conversations or people I met there.

    It’s worth noting that as a young computational number theorist you are an ideal collaborator; you have the ability to make the mathematical ideas of others explicit in a way they may not be able to, and you likely have more time to devote to new projects than more senior collaborators (they will have other projects in progress, students to supervise, and myriad administrative obligations). But to make this work in your favor you have to network, and you need to be able to take a project and run with it.

  4. In order to achieve (3) you need an interesting research result to present. A good way to get started is to attend a project-based summer school or research workshop (not a conference), ideally one that has a mix of graduate students, post-docs, and more senior people. There are usually a few of these each year, often at places like MSRI or ICERM; one to keep an eye out for is the IAS/PCMI summer program on Number Theory Informed by Computation, which was meant to take place this summer but is now likely to happen in the summer of 2021 or 2022.


  5. Create a professional webpage that showcases your work and research interests, especially things you have computed. Don’t rely on the profile page you might have on your institution’s web site; you will soon be leaving and you need to establish your own brand. Every mathematician looking to attract invitations to give talks and eventually a job offer should do this, but if you are going to sell yourself as someone with computational expertise it is especially important that you have a credible web presence.


  6. Deposit your code and results on GitHub. Here I am simply going to repeat the excellent advice of my colleague John Voight: “this signals a level of seriousness, transparency, and a desire to share that is important in a collaborative working environment”.


  7. Prove theorems! If you are going to call yourself a mathematician this should go without saying, but it is worth emphasizing. As John Voight has also noted, proving theorems about your algorithms not only establishes results that may be of independent mathematical interest, it makes your algorithms better. For me, writing a paper is always (at least) a four step process: (a) come up with a new idea for computing something new, or computing something old in a new way; (b) implement the algorithm to see if it actually works and return to step (a) if it does not; (c) state and prove precise theorems showing that the algorithm does what it is supposed to do, and that it does so efficiently; (d) re-implement the algorithm to reflect all the ideas necessary to complete step (c), which will often make the code both clearer and faster.
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Families of Hilbert Modular Forms of Partial Weight One.

Posted on June 3, 2020 by Persiflage

Today I would like to talk about a beautiful new theorem of my student Eric Stubley (see also here). The first version of Eric’s result assumed (unknown) cases of the general Ramanujan conjecture for Hilbert modular forms, and relied on a beautiful idea due to Hida. The final argument, however, is unconditional, and goes beyond Hida’s ideas in a way (I hope) that he would be delighted to see.

Suppose that \(F\) is a real quadratic field in which \(p = vw\) splits. If \(f\) is a Hilbert modular form of (paritious) weight \((1,2k+1)\) and level prime to \(p\), then the corresponding Galois representation (really only defined up to twist):

\(\rho_f: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p)\)

has the property that, for exactly one of the places \(v|p\), the restriction \(\rho_f |_{G_v}\) is unramified. Forms of partial weight one are slippery objects — one can construct such forms which are CM, but the existence of any such form which is not CM was open until an example was found by my students Richard Moy and Joel Specter (see here, here, and here). They behave in many ways like tempered cohomological automorphic forms for groups without discrete series, more specifically Bianchi modular forms or cohomological forms for \(\mathrm{GL}(3)/\mathbf{Q}\). In each of these cases, the invariant \(l_0\) as considered in Calegari-Geraghty (see for example section 2.8 of this paper) is equal to \(1\). Following work of Ash-Stevens and Calegari-Mazur, one might consider whether or not \(f\) deforms into a family of classical forms. For example, the form \(f\) will be ordinary at \(v\), and so it lives in a Hida family \(\mathcal{H}\) over \(\Lambda = \mathbf{Z}_p[[\mathcal{O}^{\times}_v(p)]] \simeq \mathbf{Z}_p[[T]]\) where we keep the weight and level at \(w\) fixed and consider (nearly) ordinary forms at \(v\). The specialization of this family to regular paritious weights will give a space of classical Hilbert modular forms. What can one say about the other specializations in partial weight one?

Theorem [Stubley]: only finitely many partial weight one specializations of the one variable \(v\)-adic Hida family \(\mathcal{H}\) associated to \(f\) are both classical and not CM.

This gives a completely general rigidity result for all partial weight one Hilbert modular forms in the split case. Over the past decade or so, the prevailing philosophy is that the only algebraic automorphic forms which are not exceedingly rare are either those coming from automorphic forms which are discrete series at infinity, or come from such forms on lower rank groups by functoriality. In this setting, this predicts that non-CM forms of partial weight one should be rare. It might even be plausible to conjecture that, up to twisting, there are only finitely many such forms of fixed tame level. However, such conjectures are completely open, and Stubley’s result is one of the first general theorems which points in that direction. (Stronger results for very specific \(F\) and \(p\) and tame level were obtained by Richard Moy and are discussed in some of the links above.)

One way to think about this theorem is in terms of the Galois representation associated to \(\mathcal{H}\). Assume for convenience of exposition that the family is free of rank one over \(\Lambda\). The Galois representation \(\rho_f\) extends to a family:

\(\rho: G_F \rightarrow \mathrm{GL}_2(\mathbf{Z}_p[[T]])\)

where \(\Lambda = \mathbf{Z}_p[[T]]\) represents weight space, so \(T = 0\) corresponds to the original specialization, and \(T = \zeta – 1\) for a \(p\)-power root of unity \(\zeta\) corresponds to a specialization to partial weight one with non-trivial level structure at \(v\). These representations are all nearly ordinary at \(v\). Is it possible that they could be split locally at \(v\) for infinitely many specializations to partial weight one? Since a non-zero Iwasawa function has only finitely many zeros, this would actually force the local representation to split for all \(T.\) Moreover, it should imply (and does in many cases) that the specializations \(T = \zeta – 1\) are all classical by modularity lifting theorems. Thus, by Stubley’s theorem, this can only happen when the family \(\rho\) is CM. In particular, Stubley’s result implies a theorem (assuming some Taylor-Wiles hypothesis) that a family of Galois representations which is (say) nearly ordinary at \(w\) of fixed weight and level and nearly ordinary at \(v\) is locally split at \(v\) if and only if it is CM.

Experts should recognize the similarity between the Galois theoretic version of Stubley’s theorem and the work of Ghate-Vatal, who prove that an ordinary family over \(\mathbf{Q}\) cannot be locally split unless it is CM. The main ingredient in their proof is the fact that there are only finitely many weight one forms of fixed tame level (up to twist) which are not CM, since these correspond either to \(A_4, S_4, A_5\) extensions of \(\mathbf{Q}\) unramified outside a fixed set of primes, which are clearly finite, or real multiplication forms, whose finiteness comes down to the finiteness of the ray class group of conductor \(N \mathfrak{p}^{\infty}\) for a split prime \(\mathfrak{p}\) in a real quadratic field. However, the analogous statement for partial weight one forms is completely open as mentioned above, so Stubley’s theorem requires a quite different argument.

Before discussing the proof, we first need to discuss a result of Hida (see this paper) about fields of definition of ordinary forms in families. Consider an ordinary family over \(\mathbf{Q}\), and consider specializations in some fixed weight, amounting (with some normalization) to specializing \(T\) to \(\zeta – 1\) for a \(p\)th power root of unity. The coefficient field will automatically contain \(\mathbf{Q}(\zeta)\). Suppose that for any prime \(q\), the degrees \([\mathbf{Q}(a_q,\zeta):\mathbf{Q}(\zeta)]\) are bounded for infinitely many specializations. Then Hida proves the family has to be a CM family. Let \(\alpha_q\) be one of the corresponding Frobenius eigenvalues. Hida’s key insight is to note that \(\alpha_q\) is a Weil number, and that Weil numbers over extensions of \(\mathbf{Q}(\zeta)\) of uniformly bounded degree are extremely restricted, and in particular given an infinite collection of such numbers then infinitely many of them have to be of the form \(\alpha \zeta\) for a fixed \(\alpha\). Using a rigidity lemma fashioned for this very purpose, he then deduces that \(\alpha_q\) in the Iwasawa algebra more or less has to equal \(\alpha (1+T)^s\) for some \(s \in \mathbf{Z}_p\), and this puts enough restrictions on \(a_q\) for him to be able to deduce the family is CM.

Stubley’s first idea is to use Hida’s result in the context of partial weight one forms. The key fact that is different in partial weight one is that when \(a_{v} \ne 0\), the form \(f\) is automatically ordinary at \(v\), and hence the \(\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\zeta))\) conjugates of \(f\) will still be ordinary at \(v\)! This is completely false in regular weights. However, in partial weight one, the only possible (finite) slope of any form at a split prime is \(0\). As a consequence, the boundedness assumption of Hida’s theorem is always going to be satisfied, because all of the conjugates have to lie on one of the finitely many Hida families which all have bounded rank over \(\Lambda\).

There is, however, a problem. Hida’s argument crucially uses the fact that \(\alpha_q\) is a Weil number, which uses the Ramanujan conjecture for forms of regular weight. The Ramanujan conjecture is completely open for partial weight one forms, since we have no idea how to prove they occur motivically (nor prove modularity of their symmetric powers). This is where Stubley’s second idea comes in. Instead of the Ramanujan conjecture, one does have standard bounds on the coefficients \(a_q\). This is not enough to deduce that \(\alpha_q\) has the form \(\alpha \zeta\) for some fixed \(\alpha\). Instead, Stubley shows that it does allow one to show that the trace of \(a_q\) (together with the trace of any if its powers) to \(\mathbf{Q}(\zeta)\) (which has uniformly bounded degree) can be written as a finite sum of roots of unity where the number of terms does not depend on \(\zeta\). Again for convenience of exposition and to avoid circumlocutions with traces, let us suppose that the rank of the Hida algebra is one and so \(\mathbf{Q}(\zeta,f) = \mathbf{Q}(\zeta)\). Then Eric shows that infinitely many of the \(a_q\) satisfy:

\(a_q = \alpha_1 \zeta_1 + \alpha_2 \zeta_2 + \ldots + \alpha_N \zeta_N\)

for varying \(p\)-power roots of unity \(\zeta_i\), but where \(\alpha_i\) and \(N\) are fixed. Then Stubley proves a new rigidity theorem in this context (not unrelated to results of Serban) showing that one must have an equality

\(a_q = \alpha_1 (1+T)^{s_1} + \alpha_2 (1+T)^{s_2} + \ldots + \alpha_N (1 + T)^{s_n}\)

over the Iwasawa algebra. This is probably enough to show the family has to be CM using ideas similar to Hida, but even that is not necessary — by using this formula for specializations in regular weight one deduces that the \(\alpha_i\) are in \(\overline{\mathbf{Q}}\), and then applying Hida’s theorem in this fixed regular weight one deduces that the family is CM.

Stubley’s theorem is the first result that gives general theoretical evidence towards the conjecture (if one is so bold to make such a conjecture) that there are only finitely many non-CM partial weight one forms of fixed tame level up to twist. It also shows that certain \(v\)-ordinary deformations of a non-CM partial weight one form \(f\) will not be classical. But there is also a second possible way to deform \(f\), namely, to vary the weight at \(w|p\) instead (or as well). For example, if the form \(f\) was also ordinary at \(w|p\), one could look at the ordinary at \(w\) Hida family. One might also conjecture that this family only contains finitely many non-CM points, but this is still open. (Boxer has raised this question.) I think this is an interesting but very hard question!

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Picard Groups of Moduli Stacks update

Posted on May 27, 2020 by Persiflage

A tiny update on this post. I was chatting with Benson and realized that I may as well ask him directly for a presentation of the mapping class group of a genus two surface. Perhaps unsurprisingly, it can be found in his book with Dan Margalit (see page 122 of their book which might be downloadable from a Russian website) and is given as follows:

\(G \simeq \langle a_1,a_2,a_3,a_4,a_5 | \ [a_i,a_j] \ \text{for $|i-j|>1$}, a_i a_{i+1} a_i = a_{i+1} a_i a_{i+1},\)
\( (a_1 a_2 a_3)^4 = a^2_5, [(a_5 a_4 a_3 a_2 a_1 a_1 a_2 a_3 a_4 a_5),a_1], (a_5 a_4 a_3 a_2 a_1 a_1 a_2 a_3 a_4 a_5)^2, \rangle.\)

The next task is to find the representation

\(G \rightarrow \mathrm{Sp}_4(\mathbf{Z}) \rightarrow \mathrm{Sp}_4(\mathbf{F}_2) \simeq S_6\)

and then take the index \(6\) preimage \(\Gamma \subset G\) of the \(S_5 \subset S_6\) corresponding to fixing a Weierstrass point. Note there are two conjugacy classes of \(S_5\), the correct one is the one whose restriction to \(A_5\) still acts absolutely irreducibly on \((\mathbf{F}_2)^4\). Then one can use Reidemeister-Schreier to compute a presentation of \(\Gamma\) and then compute \(H_1(\Gamma,\mathbf{Z})\). This is all good in theory, and Farb-Margalit does have a chapter on the symplectic representation, but actually having to read the book in detail to extract the precise symplectic representation sounded like too much work, especially since all of this is ultimately just for a two sentence comment in a paper that might be removed for space reasons anyway. So instead I just fired up magma with the representation \(G\) and asked it to find *all* index six subgroups. It turns out that there are only two of them (up to conjugation), which must come exactly from the two subgroups of \(S_5 \subset S_6\). The abelianization of one is \(\mathbf{Z}/10 \mathbf{Z} \simeq G^{\mathrm{ab}}\), but the other group is

\(\Gamma = \langle a_1,a_2,a_3,a_4 \rangle\), and one finds that \(H_1(\Gamma,\mathbf{Z}) \simeq \mathbf{Z}/40 \mathbf{Z}.\)

Hence this is (in light of the previous dicussion) the correct subgroup, and this (unsurprisingly although not entirely independently) confirms the analysis of naf in the comments. Now I suspect that if you think a little harder than I am prepared to do (or if you just know a little bit more than me), you might be able to see directly from the definition of the \(a_i\) that \(a_1,a_2,a_3,a_4\) fix a Weierstrass point; if you are such a person please make a comment!

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Panel Discussion: Mathematics Research Online

Posted on May 13, 2020 by Persiflage

Odds are high that virtual conferences in mathematics are here to stay. It seems crucial, therefore, to think long and hard about ways to make them work for all participants. We need to have conversations as a community to better understand how to make this happen. Andrew Sutherland and Bianca Viray are organizing a panel (virtual of course!) on this very topic one week from today (May 20th), with a number of panelists who have already had experience running virtual workshops. I encourage you to join in and learn from there experience, but also to add your own voice to the conversation.

For the link, click here!

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Picard Groups of Moduli Stacks

Posted on April 30, 2020 by Persiflage

Here are some algebraic geometry musings related to the last post, most of which is hopefully correct. Everything below is secretly over \(\mathbf{Z}[1/6]\) but I think one may as well think about what is happening over \(\mathbf{C}\). Warning: I don’t know any algebraic geometry, please correct me if you see any nonsense.

As mentioned in the last post, if you fix a \(3\)-torsion representation with cyclotomic determinant and look at the corresponding moduli space of elliptic curves with this \(3\)-torsion, you get a \(\mathbf{P}^1\) (at least accounting for cusps). A natural followup question is: what geometric object do you get over the stack \(\mathcal{A}_{1} = \mathcal{M}_{1,1}\)?

Thinking about stacks in the most naive way, we just consider

\(y^2 = x^3 + a x + b\)

for \((a,b)\) in \(\mathbf{P}(4,6)\) minus \(\Delta = 0\) in the stacky sense. But just thinking about this as an elliptic curve over \(\mathbf{Q}(a,b)\), you can write down:

\(y^2 = x^3 + A x + B\)

where

\(\begin{eqnarray*}
3A(a,b,s,t) & = & 3 a s^4 +18 b s^3 t -6 a^2 s^2 t^2 -6 a b s t^3 -(a^3+9
b^2) t^4, \\
9B(a,b,s,t) & = & 9 b s^6-12 a^2 s^5 t-45 a b s^4 t^2-90 b^2 s^3 t^3 + 15 a^2 b s^2 t^4 \\
&& \qquad -2 a
(2 a^3+9 b^2 ) s t^5 -3 b (a^3+6 b^2 ) t^6,
\end{eqnarray*}
\)

Now one thing you notice straight away about these equations is that they change when one replaces \(a,b\) by \(a \lambda^4, b \lambda^6\), namely:

\(A(\lambda^4 a, \lambda^6 b,s,t) = A(a,b, \lambda s, \lambda^3 t)\)

and the same equation holds for \(B\). That is, the parametrization of \(\mathbf{P}^1\) changes, and so the family is not literally projective space over this stack. Of course, if

\(\Delta(a,b) = 16(-4 a^3 – 27 b^2),\)

then

\(\Delta(a \lambda^4,b \lambda^6) = \lambda^{12} \Delta(a,b),\)

where \(\Delta\) trivializes \(\omega^{12}\). In order to remove the ambiguity, one can then define

\(\displaystyle{A^*(a,b,s,t) = A \left(a,b, \frac{s}{\Delta^{1/12}},\frac{t}{\Delta^{3/12}}\right)}\)

and similarly with \(B^*\), then the equation is well defined, at least after addressing the issue of taking 12th roots correctly. This suggests that after pulling back to the space where you adjoin \(\Delta^{1/12}\) you get projective space, but that the original space is not projective space at all but maybe something like the projective bundle

\(\mathrm{Proj}(\mathcal{O}_X \oplus \omega^2) = \mathrm{Proj}(\omega \oplus \omega^3)\)

where \(\omega\) is the usual line bundle which has order \(12\) in the Picard group of \(\mathcal{A}_{1}\).

Something very similar happens for the equations for families of fixed three torsion over \(\mathcal{M}^{w}_2\), the moduli stack of genus two curves with a fixed Weierstrass point. In this case, the base looks like

\(y^2 = x^5 + a x^3 + b x^2 + c x + d\)

or \(\mathbf{P}(4,6,8,10)\) minus \(\Delta = 0\). (You need to be a little bit more careful at the prime \(5\).) Here the corresponding identity for \(A,B,C,D\) is

\(A(\lambda^4 a,\lambda^6 b,\lambda^8 c,\lambda^{10} d,s,t,u,v)
= A(a,b,c,d,\lambda s, \lambda^7 t,\lambda^{13} u,\lambda^{19} v)\)

and

\(\Delta(\lambda^4 a,\lambda^6 b,\lambda^8 c,\lambda^{10} d) = \lambda^{40} \Delta(a,b,c,d)\).

So now one wants to trivialize the family by taking the cover with various roots of \(\Delta\), including \(\Delta^{1/20}\). Except now I don’t really know what the Picard group of \(\mathcal{M}^{w}_2\) is. Somehow I first assumed that the Picard group would be the same as that of the corresponding moduli space of abelian surfaces \(\mathcal{A}^{w}_2\), and since \(\Delta\) seems to give a trivialization of some power of the determinant bundle it should be related to torsion in \(H_1(\Gamma,\mathbf{Z})\) for the corresponding congruence subgroup \(\Gamma\) of \(\mathrm{Sp}_4(\mathbf{Z})\). But because of the congruence subgroup property, presumably \(H_1(\mathrm{Sp}_4(\mathbf{Z}),\mathbf{Z})\) is equal to \(\mathbf{Z}/2 \mathbf{Z}\), and that’s not going to change by taking the map to \(S_6 = \mathrm{PSp_4(\mathbf{F}_2)}\) and taking the pre-image of \(S_5\). But it is pure folly to imagine the Picard group of \(\mathcal{M}^{w}_2\) and \(\mathcal{A}^{w}_2\) coincide. The latter contains an extra divisor, the Humbert divisor, consisting of direct sums of elliptic curves. Moreover, (I guess) the Siegel modular form corresponding to \(\Delta\) is probably very close to the Igusa form, which vanishes not only at the cusp but also along the Humbert divisor. So the line bundle \(\omega\) on \(\mathcal{A}_2\) has infinite order even though its pullback to \(\mathcal{M}_2\) does not because \(\Delta\) itself is giving a trivialization of some power of \(\omega\). So it is indeed plausible that abelianization of the corresponding (index five subgroup of) the \(g = 2\) Torelli group has \(20\)-torsion. One way to try to compute this is to explicitly compute the abelianization of the corresponding cover of the mapping class group (I guess there are explicit presentations?). So the first question is can someone confirm that \(\mathrm{Pic}(\mathcal{M}^{w}_2)\) does indeed have \(20\)-torsion? If only there was someone in my department who could prime me on the properties of mapping class groups… Actually, Andrew Putman is probably the obvious person to ask. The second problem is confirm that the family explicitly computed in the last post does indeed coincide with \(\mathrm{Proj}(\mathcal{O}_X \oplus \omega^6 \oplus \omega^{12} \oplus \omega^{18})\).

I confess my efforts to do a literature search in this case have not been very thorough. In my mind I somehow thought that the Picard group of the stack \(\mathcal{M}_g\) (for \(g \ge 2\)) was \(\mathbf{Z}\), but that is transparently false, at least for \(g = 2\). I got as far as doing a google search for Picard groups of moduli stacks and found a few pages of notes written by Daniel Litt. So I naturally zoomed in to Daniel Litt’s office hours once after he advertised them on twitter… but I soon realized that it would take too long to explain and he had better things to do like explaining modular forms to his students… so here it is now in blog form!

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