# Persiflage > Galois Representations and more! ## Posts - [Dick Gross](https://galoisrepresentations.org/2026/01/15/dick-gross/): Here are some memories about Dick Gross, who sadly just passed away very recently. Nothing was quite as reassuring as having Dick Gross in your audience. Inevitably, when your talk was done, he would both compliment you on it and have something very interesting mathematical to say. The last time this happened to me was at the Tate 100 conference in March. Dick was a student of Tate, and although he wasn’t able to come in person, he gave some prerecorded reminiscences. (Many of the very nice things Dick says about Tate can also be said about Dick.) But after […] - [Arbeitsgemeinschaft 2026](https://galoisrepresentations.org/2025/06/04/arbeitsgemeinschaft-2026/): The April 2026 Oberwolfach Arbeitsgemeinschaft will be on Arithmetic Holonomy Bounds and Applications to Irrationality, and in particular will discuss some of the results of this paper. For those who don’t know, the Arbeitsgemeinschaft (“study group”) is different from usual Oberwolfach workshops (or workshops more generally) — the idea is that the participants learn the material and then teach it to each other. I have never actually been to one, or rather I almost did but it was first cancelled due to Covid and then went online due to Covid. That workshop asked a lot of participants in terms of […] - [En Passant: Mailbox and Tate 100](https://galoisrepresentations.org/2025/04/22/en-passant-mailbox-and-tate-100/): I lost the key to my office mailbox about a year ago (probably more), and just had it replaced. So today I got to enjoy the bounty of new mail, which consists of: A request from the AMS for money, dated Sep 14, 2024, A seasons greetings card from the dean of the college, year unknown, A brochure from Nasco education selling math toys, A poster for the 2025 Arizona Winter Schoo, dated Sep 25, 2024, The Fall 2024 University of Chicago library magazine, The Fall 2024 Berkeley Mathematics magazine. I might make another visit in another year! Leafing through […] - [It wasn't me!](https://galoisrepresentations.org/2025/03/13/it-wasnt-me/): I have a gentleman’s bet with another number theorist that I will be able to write a nonsense paper and get it published in respectable journal. To be honest, I barely have enough time to write actual papers let alone a nonsense paper, but it has crossed my mind from time to time. The respective mathematician recently inquired as to whether I was the author of a certain preprint, presumably written in an attempt to collect on our bet; I just want to say that it wasn’t me! First of all, I wouldn’t try to claim a major conjecture, the […] - [A new blog](https://galoisrepresentations.org/2025/03/03/a-new-blog/): For various hosting reasons, I have moved my old blog back to galoisrepresentations.org rather than galoisrepresentations.com. The forwarding link will die at some point so please update your links accordingly! Hopefully the previous content is still operational but let me know if you see anything broken. - [The Poincaré homology sphere](https://galoisrepresentations.org/2025/01/05/the-poincare-homology-sphere/): This illusion (from the Chicago museum of illusions, and duplicated, I believe, in other similar museums in other cities) “almost” appears to give a tiling of \(\mathbf{R}^3\) by regular dodecahedra, which for a number of reasons is not possible. (It also looks remarkably like knot not.) Moving one’s point of view slightly, one can observe that the interior faces don’t quite match up. But can this be remedied with curved mirrors? That is, is there a way to shape these mirrors so that one can is looking inside the Poincaré dodecahedral space? Your browser does not support the video tag. - [It's not a Lemma, it's a Proposition!](https://galoisrepresentations.org/2024/12/14/its-not-a-lemma-its-a-proposition/): Congratulations to Ken for winning the the Steele prize. I first met Ken on the Hearst mining circle. It was September of 1997, during the time I was applying for graduate schools. I was visiting Danny on the way to Bonn, and it was the one US university I visited. Coincidentally, to within a few months, Ken was the same age then as I am now. That’s actually somewhat reassuring, since it seems like a lifetime ago and Ken still seems pretty sprightly today. Continuing with this theme, I guess 1976 in today’s currency would be 2003. For another perspective, […] - ["Fields of definition"](https://galoisrepresentations.org/2024/12/07/fields-of-definition/): Can you help settle a disagreement? This is a debate about notation I am having with a colleague; I will try to present it without prejudice (and probably fail). Let \(G\) be a group, and let \(V\) be a finite dimensional complex irreducible representation of \(G\). Suppose that the traces of \(G\) actually lie inside a number field \(K\). The field \(K\) may well be much smaller than \(\mathbf{C}\). For example, if \(G\) is a finite group, then \(K\) will be a number field. But it is not always the case that the representation itself is conjugate to a representation […] - [Giving a good mathematics talk](https://galoisrepresentations.org/2024/11/02/giving-a-good-mathematics-talk/): Last week, Tadashi Tokieda came to Chicago to give a colloquium. If you have seen him speak, you will not be surprised to learn that it was absolutely delightful talk. I carried the talk around with me in my mind for many days afterwards, not only for its content, but also with the nagging question: how I can I make my own talks better? I think it’s very easy to feel that our subject (mathematics) is so technical that no talk can both convey depth and yet be accessible. But then why did this talk feel otherwise? There is certainly […] - [The Arthurian Legend](https://galoisrepresentations.org/2024/10/18/the-arthurian-legend/): Some time back, Kevin Buzzard (friend of the blog!) gave a series of talks in which he criticized certain aspects of the mathematical culture when it came to accepting proof. In addition to obvious targets like the classification of finite simple groups, he took aim at my paper with Boxer, Gee, and Pilloni, an in particular this passage: It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of \(\mathrm{GSp}_4\), as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT19], but this proof […] - [Walter Neumann](https://galoisrepresentations.org/2024/09/30/walter-neumann/): I recently learnt the sad news that Walter Neumann just passed away. Although I don’t think I have seen him in person for over 25 years, Walter was a pretty significant influence in my mathematical life. Here are some of my recollections. (See also this celebration of Walter on his retirement from people who knew him much better than me!) Walter was lured to Australia in 1993 by Melbourne University. Although Australia was unable to keep him (he moved back to the US in 1999), he tenure included the entire time I was an undergraduate. The first thing Walter did […] - [Am I taking students?](https://galoisrepresentations.org/2024/09/27/am-i-taking-students/): I receive many (many) unsolicited emails about the possibility of working with me at grad school. Some are clearly bulk emails sent no doubt to a large number of professors. Some are customized to include phrases like “I was really fascinated by you paper [most recent paper] and I want to learn more”. (I get a surprising number of emails also from predatory publishers eager for me to write a book about my paper “Correction to: Modularity lifting beyond the Taylor-Wiles method.”) Some are much more personalized, relevant, and interesting. But it seemed worthwhile to write a short blog post […] - [Not quite what I meant](https://galoisrepresentations.org/2024/08/15/not-quite-what-i-meant/): Weibo Fu wrote an interesting paper on upper bounds for spaces of Bianchi modular forms, pushing previous results of Simon Marshall and Yongquan Hu to get more or less optimal results in the weight aspect. More generally, for any number field \(F\) which is not totally real, and for the space of regular algebraic cuspidal automorhpic representations of fixed level and parallel weight \(k\), he obtains the bound (see Theorem 1.2): \[ \mathrm{dim} S_k = O(k^{d-1})\] where \(d = [F:\mathbf{Q}]\) (The “easy” bound is \(O(k^d)\)). This is a great result! I do however have one tiny quibble. Fu makes the […] - [Persiflage, 2012-2024](https://galoisrepresentations.org/2024/07/23/persiflage-2012-2024/): No, not a eulogy! I’ve been a bit concerned for a while about how stable wordpress is as a website — various plugins are always updating on their own, and I have sometimes noticed that old blog posts do not always render latex correctly (at some point there was a change in how latex was handled). For a while I thought I should do something to make sure that all my past math posts did not suddenly disappear. This feeling was hastened when the subversion platform I was using, xp-dev, suddenly went down when the owner (and apparently only employee) […] - [SL_n versus GL_n](https://galoisrepresentations.org/2024/07/18/sl_n-versus-gl_n/): 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 […] - [A talk on my new work with Vesselin Dimitrov and Yunqing Tang on irrationality](https://galoisrepresentations.org/2024/06/16/a-talk-on-my-new-work-with-vesselin-dimitrov-and-yunqing-tang-on-irrationality/): 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 […] - [Zeilberger + ChatGPT](https://galoisrepresentations.org/2024/06/01/zeilberger-chatgpt/): 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…) - [Unramified Fontaine-Mazur for representations coming from abelian varieties](https://galoisrepresentations.org/2024/05/09/unramified-fontaine-mazur-for-representations-coming-from-abelian-varieties/): 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 […] - [Midlife crisis](https://galoisrepresentations.org/2024/02/12/midlife-crisis/): 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 […] - [The horizontal Breuil-Mezard conjecture](https://galoisrepresentations.org/2023/10/19/the-horizontal-breuil-mezard-conjecture/): 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 […] - [Magma Instability](https://galoisrepresentations.org/2023/10/13/magma-instability/): 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 […] - [Clozel 70, Part II](https://galoisrepresentations.org/2023/09/30/clozel-70-part-ii/): 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 […] - [Clozel 70, Part I](https://galoisrepresentations.org/2023/09/24/clozel-70-part-i/): 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 […] - [Kouign-Amann, Chicago Tasting](https://galoisrepresentations.org/2023/06/25/kouign-amann-chicago-tasting/): 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 […] - [Haebler and Gilberto](https://galoisrepresentations.org/2023/06/08/haebler-and-gilberto/): Two obituaries in the NYT within one week for musicians in my music collection. I can’t quite say that Mozart is my composer of choice, especially when it comes to the piano, although I could listen to Mitsuko Uchida play all day. But every now and again, Mozart knocks one out of the park. The G minor piano quartet, for example. While that quartet has a great arrangement for two pianos, here’s a Mozart fugue actually originally written for two pianos; here is my recording, performed by Ingrid Haebler and Ludwig Hoffmann I once had some Brazilian students surprised during […] - [Google and Franck](https://galoisrepresentations.org/2023/05/15/google-and-franck/): I have a google play (which plays streaming music) and it’s really terrible for classical music. If you choose virtually any piece of classical music and then flip forward three songs you invariably end up with Claire de Lune or the Moonlight sonata. For science, I just tested this again right now, and here are the unfiltered results: Initial Selection: Art of Fugue Next Piece: Toccata and Fugue in D minor (ha ha; semantically similar I guess) Next Piece: First movement, Moonlight sonata Initial Selection: Brahms, Op 119 no.1 The initial selection continued with Op 119 no.2 on the same […] - [Quadratic Reciprocity](https://galoisrepresentations.org/2023/05/02/quadratic-reciprocity/): I accidentally proved quadratic reciprocity in class today, or at least three quarters of a proof. Can you finish it off? Here’s the proof: start with a real quadratic field \(K\), and the sequence \(1 \rightarrow \mathcal{O}^{\times}_K \rightarrow K^{\times} \rightarrow K^{\times}/\mathcal{O}^{\times} \rightarrow 1 \) then take cohomology. If \(P_{K}\) is the group of principal ideals of \(K\), then from Hilbert Theorem 90 you deduce that \(P^{G}_K/P_{\mathbf{Q}} \simeq H^1(G,\mathcal{O}^{\times}_K)\). If \(I_K\) is the group of all ideals of \(K\), the left hand side is a subgroup of \(I^G_K/P_{\mathbf{Q}}\) which is a product of groups of order two for each ramified prime. […] - [En Passant VIII](https://galoisrepresentations.org/2023/04/28/en-passant-viii-2/): Coffee in Hyde Park: I had the misfortune of being stuck in down town Hyde Park needing a coffee without access to a car. I started at Sip and Savour where I ordered a cortado. They did not know what a cortado was. I paused, considered the situation, and slowly walked out. Attempt number two was at a cafe which was nominally a Peets but was actually a “Capital One” cafe. They did have a fine looking Marzocco machine which was a good sign, but they also had a very bored looking barista and my attempt to order a cortado […] - [Deciphering Quanta](https://galoisrepresentations.org/2023/03/23/deciphering-quanta/): Sometimes it is claimed that Quanta articles are so watered down of mathematical content that they become meaningless. That presents a challenge: do I understand the quanta article on my own work? Here goes: New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’ I can confirm that I did not see this article in any form before it appeared. Overall I would say that it is faithful to the facts and I can interpret what everything means. I’m not quite sure why there is an Alex Kontorovich explainer about the Langlands Program in there but why not? I did, however, have […] - [Boxes for Boxer update](https://galoisrepresentations.org/2023/02/28/boxers-for-boxer-update/): As noted in this post, exactly 42 reprints of [BCGP] were recovered in January of 2022 from boxes left out in the snow outside Eckhart Hall addressed to George Boxer. As mentioned there, the packaging (5 boxes of 8 plus a further smaller box with 2) suggested that there was a “missing box” with 8 more copies of the paper, and that one should be on the lookout at Hyde Park used book stores for the remaining copies. These reprints are after all roughly of a similar scarcity to the extant Gutenberg Bibles. Well there has been an update! My […] - [What the slopes are](https://galoisrepresentations.org/2023/02/25/what-the-slopes-are/): Let \(f\) be a classical modular eigenform of weight \(k\), for example, \(f = \Delta\). The Ramanujan conjecture states that the Hecke eigenvalues \(a_p\) satisfy the bound \(|a_p| \le 2 p^{(k-1)/2}.\) A slightly fancier but cleaner way of saying this is as follows. Associated to \(f\) of weight \(k\), level \(N\) prime to \(p\) and finite order Nebentypus character \(\chi\) is a polynomial \(X^2 – a_p X + p^{k-1} \chi(p).\) (For \(\Delta\) one has \(\chi(p) = 1\) for all \(p\), but in general it can be some other root of unity.) This is the characteristic polynomial of Frobenius on the […] - [Check the arXiV regularly!](https://galoisrepresentations.org/2023/02/18/5346/): In a previous post, I discussed a new result of Smith which addressed the following question: given a measure \(\mu\) on \(\mathbf{R}\) supported on some finite union of intervals \(\Sigma\), under what conditions do there exist polynomials of arbitrarily large degree whose roots all lie in \(\Sigma\) whose distribution (in the limit) converge to \(\mu\)? A natural generalization is to replace \(\Sigma\) by a subset of \(\mathbf{C}\) subject to certain natural constrains, including that \(\mu\) is invariant under complex conjugation. I decided that this had a chance of being a good thesis problem and scheduled a meeting with one of […] - [Report from Australia, Part I, Coffee](https://galoisrepresentations.org/2023/02/15/report-from-australia-part-i-coffee/): My travel often involves making some effort to find good local coffee. From Palo Alto to Portland, a little effort finds quality cafes with reliable espresso drinks. How does the rest of the world then stack up with Australia, the acknowledged home of coffee? My first stop was Sydney, where my airbnb was conveniently located a stone’s throw from Skittle Lane Coffee. Many other cafes were on my list (Cabrito Coffee Traders, St Dreux Espresso Bar, and so on). What consistently stood out was not necessarily how far above in quality the coffee was from elsewhere in the world, but […] - [Potential Modularity of K3 surfaces](https://galoisrepresentations.org/2022/11/15/potential-modularity-of-k3-surfaces/): This post is to report on results of my student Chao Gu who is graduating this (academic) year. If \(A/F\) is an abelian surface, then one can associate to \(A\) a K3 surface \(X\) (the Kummer surface) by blowing up \(A/[-1]\) at the \(16\) singular points (corresponding to \(2\)-torsion points of \(A\). If \(F\) is a totally real field, then one knows that \(A\) is (potentially) automorphic, and it follows that \(X\) is also (potentially) automorphic, which in particular implies the Hasse-Weil conjecture for \(X\). It also proves that \( \rho(X/F) = – \mathrm{ord}_{s=1} L(H^2(X/\overline{F},\mathbf{Q}_p(1)),s),\) where \(H^2(X/\overline{F},\mathbf{Q}_p(1))\) is the etale […] - [Peak Hyde Park](https://galoisrepresentations.org/2022/11/12/peak-hyde-park/): Me dressed as a crocodile chasing the Groke while being chased by Drinfeld down Harper Ave on Halloween (all in front of a 16 foot inflatable pumpkin). Sadly, Drinfeld was not dressed as a Shtuka. - [The future is now; recap from Cetraro](https://galoisrepresentations.org/2022/07/31/the-future-is-now-recap-from-cetraro/): I’ve just returned from the second Journal of Number Theory biennial conference in Italy. It’s always nice to get a chance to see slices of number theory one wouldn’t otherwise see at the conferences I usually go to (although this was the first conference of any kind I attended in person since 2019). Here is a brief and incomplete recap. There were more talks that mentioned the Manin-Mumford conjecture and its various generalizations (particularly to uniform bounds in families) than I have ever previously attended in my life. There were probably equally many talks which mentioned Ax-Schanuel as well. It […] - [30 years of modularity: number theory since the proof of Fermat](https://galoisrepresentations.org/2022/07/08/30-years-of-modularity-number-theory-since-the-proof-of-fermat/): It’s probably fair to say that the target audience for this blog is close to orthogonal to the target audience for my talk, but just in case anyone wants to watch it in HD (and with the audio synced to the video) on I have uploaded it to youtube here: - [Locally induced representations](https://galoisrepresentations.org/2022/06/16/locally-induced-representations/): Today is a post about work of my student Chengyang Bao. Recall that Lehmer’s conjecture asks whether \(\tau(p) \ne 0\) for all primes \(p\), where \(\Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum \tau(n) q^n\) is Ramanujan’s modular form. You might recall that Naser Talebizadeh Sardari and I studied a “vertical” version of Lehmer’s conjecture where instead of fixing a modular form, we fixed a prime \(p\) and a tame level \(N\) and showed that there were only finitely many normalized eigenforms \(f\) of level \(N\) and even weight \(k\) with \(a_p(f) = 0\) which were not CM. We […] - [Joël Bellaïche](https://galoisrepresentations.org/2022/06/14/joel-bellaiche/): Very sad to hear that Joël Bellaïche has just died. He got his PhD at the same time as me, and I first got to know him during the Durham conference in 2004 and later at the eigenvarieties semester at Harvard (was that in 2005 or 2006?). Joël was an original mathematician, and his papers (many written with Gaëtan Chenevier) contain many really good ideas. As a postdoc, I was totally immersed in thinking about Galois deformations of reducible representations when the paper lisseté de la courbe de Hecke de \(\mathrm{GL}_2\) aux points Eisenstein critiques appeared on the arXiV. In […] - [Murphy's Law for Galois Deformation Rings](https://galoisrepresentations.org/2022/04/23/murphys-law-for-galois-deformation-rings/): Today’s post is about work of my student Andreea Iorga! A theorem of Ozaki from 2011, perhaps not as widely known as expected, says the following: Theorem: Let \(p\) be prime, and let \(G\) be a finite \(p\)-group. Then there exists a number field \(F\) and an extension \(H/F\) such that: \(H/F\) is the maximal pro-\(p\) extension of \(F\) which is everywhere unramified. \(\mathrm{Gal}(H/F) = G\). Since any non-trivial \(p\)-group \(G\) has a non-trivial center, it can be written as a central extension of a smaller \(p\)-group \(G’\) by \(\mathbf{Z}/p \mathbf{Z}\), and thus the proof is (as one might imagine) […] - [What would Deuring do?](https://galoisrepresentations.org/2022/04/13/what-would-deuring-do/): This is an incredibly lazy post, but why not! Matt is running a seminar this quarter on the Weil conjectures. It came up that one possible way to prove the Weil conjectures for elliptic curves over finite fields is to lift them to CM elliptic curves using Deuring’s theorem. But after some discussions we couldn’t quite work out whether this was circular or not. Certainly if you can lift to a CM elliptic curve and lift Frobenius to an endomorphism \(\phi\) of the lift you get Weil immediately; the degree of \(\phi\) is \(p\) which implies the norm of \(\phi\) […] - [A random curve over Q](https://galoisrepresentations.org/2022/04/02/a-random-curve-over-q/): Let \(X/\mathbf{Q}\) be a smooth projective curve. I would like to be able to say that the motive \(M\) associated to \(X\) “generally” determines \(X\). That is, I would like to say it in a talk without feeling like I’m telling too much of a fib. But is this true? There are two issues. Recall that, by the Torelli Theorem, the Jacobian together with a principle polarization determines \(X/\mathbf{C}\). So there are two things to worry about: Knowing \(M\) only recovers the Jacobian up to isogeny, and you can certainly have two different curves with isogenous Jacobians, even isomorphic Jacobians […] - [ArXiv x 3](https://galoisrepresentations.org/2022/03/04/arxiv-x-3/): Three recent arXiv preprints this week caught my interest and seemed worth mentioning here. The first is a paper by Oscar Randal-Williams, which considers (among other things) the cohomology of congruence subgroups of \(\mathrm{SL}_N(\mathbf{Z})\) in the stable range. This is definitely something I have talked on the blog about a number of times, including here and here. To recall; Matthew Emerton and I proved that the completed cohomology groups \(\widetilde{H}^d(\mathbf{F}_p) = \lim H^d(\mathrm{SL}_N(\mathbf{Z},p^n),\mathbf{F}_p)\) are independent of \(N\) for \(N\) sufficiently large with respect to \(d\), and are moreover finite vector spaces with a trivial action of \(G = \mathrm{SL}_N(\mathbf{Z}_p)\). I […] - [What would a good ICM talk look like?](https://galoisrepresentations.org/2022/02/27/what-would-a-good-icm-talk-look-like/): Now that the ICM has (unsurprisingly) become a virtual event, it might be worthwhile thinking a little bit about what would constitute a good talk in this new setting. There’s a certain electricity to talks given in person, and I think that many speakers give better talks when they have an opportunity to read the audience. Certainly the zoom talks I have most enjoyed watching are those where I’ve been able to interact with the speaker, but that clearly becomes impossible once the audience is large enough. So an ICM of zoom talks (on a St Petersburg schedule in the […] - [Boxes for Boxer](https://galoisrepresentations.org/2022/01/04/boxes-for-boxer/): My brother texted me on Monday saying that there were seven (or so) boxes pilled up (outside!) in front of the mathematics department and all addressed to George Boxer. My first thought was that this was a transatlantic move gone horribly wrong, so I emailed the department looking for volunteer graduate students to haul the boxes inside. I managed to acquire the boxes the next day: Now the boxes turned out not to contain the entire sum of George’s possessions (nor a large pile of cash, unfortunately), but somewhat more hilariously it consisted of reprints from our recently published paper […] - [Simons Annual Meeting](https://galoisrepresentations.org/2021/12/23/simons-annual-meeting/): The last time I traveled for math was when I gave the Coble lectures at UIUC pre-pandemic (at least pre-pandemic as far as the US goes). A few months ago it seemed like one could begin to start traveling again, so I agreed to go to the Simons Conference scheduled for Jan 13-14 in NYC. While I’m prepared as the next person to acknowledge that we have to start living with the coronavirus and live our lives accordingly, traveling during might be the absolute peak of omicron in NYC seems a little unwise. Hence I sadly cancelled my trip today. […] - [Schur-Siegel-Smyth-Serre-Smith](https://galoisrepresentations.org/2021/11/25/schur-siegel-smyth-serre-smith/): If \(\alpha\) is an algebraic number, the normlized trace of \(\alpha\) is defined to be \( \displaystyle{T(\alpha):=\frac{\mathrm{Tr}(\alpha)}{[\mathbf{Q}(\alpha):\mathbf{Q}].}}\) If \(\alpha\) is an algebraic integer that is totally positive, then the normalized trace is at least one. This follows from the AM-GM inequality, since the normalized trace is at least the \(n\)th root of the norm, and the norm of a non-zero integer is at least one. But it turns out that one can do better, as long as one excludes the special case \(\alpha = 1\). One reason you might suspect this to be true is as follows. The AM-GM inequality […] - [Polymath Proposal: 4-folds of Mumford's type](https://galoisrepresentations.org/2021/08/24/polymath-proposal-4-folds-of-mumfords-type/): Let \(A/K\) be an abelian variety of dimension \(g\) over a number field. If \(g \not\equiv 0 \bmod 4\) and \(\mathrm{End}(A/\mathbf{C}) = \mathbf{Z}\), then Serre proved that the Galois representations associated to \(A\) have open image in \(\mathrm{GSp}_{2g}(\mathbf{Z}_p)\). The result is not true, however, when \(g=4\), as first noted by Mumford (in this paper). The goal of this polymath project is to find an “explicit” example of such a Mumford \(4\)-fold over \(\mathbf{Q}\). There are a number of things I have in mind for what “explicit” might mean (this is, after all, supposed to be a polymath project so I’m […] - [59,281](https://galoisrepresentations.org/2021/08/19/59281/): The target audience of this blog (especially the mathematics) is usually professional mathematicians in the Langlands program. I do sometimes, however, have posts suitable for a broader mathematical audience. Very rarely though do I have anything (possibly) interesting to say to a popular audience. In my recent talk in the Number Theory Web Seminar, I gave a talk about some math that I’ve discussed with Soundararajan (and which will possibly be written up at some day) about the “average” digit of \(1/p\) in its decimal expansion, in particular, discussing the distribution of primes for which the average digit of \(1/p\) […] - [Divisors near sqrt(n)](https://galoisrepresentations.org/2021/06/08/divisors-near-sqrtn/): Analytic Number Theory Alert! An even more idle question than normal (that’s because it comes from twitter). Alex Kontorovich noted with pleasure the following pictorial representation of the integers from a Veritasium youtube video, where prime numbers are represented by \(1 \times n\) rectangles and all other numbers represented as \(a \times b\) rectangles (of area \(n\)) for some \(a > 1\). This leads to the natural followup questions. How much horizontal space does it take to graph the first \(X\) integers this way if one either: Plots the integers \(n\) as \(a \times b\) with \(a \le b\) as […] - [Potential Automorphy for GL(n)](https://galoisrepresentations.org/2021/04/29/potential-automorphy-for-gln/): Fresh on the arXiv, a nice new paper by Lie Qian proving potential automorphy results for ordinary Galois representations \(\rho: G_F \rightarrow \mathrm{GL}_n(\mathbf{Q}_p)\) of regular weight \([0,1,\ldots,n-1]\) for arbitrary CM fields \(F\). The key step in light of the 10-author paper is to construct suitable auxiliary compatible families of Galois representations for which: The mod-\(p\) representation coincides with the one coming from \(\rho\), The compatible family can itself be shown to be potentially automorphic. The main result then follows by an application of the p-q switch. Something similar was done by Harris–Shepherd-Barron–Taylor in the self-dual case. They ultimately found the […] - [Don't cite my paper!](https://galoisrepresentations.org/2021/04/26/dont-cite-my-paper/): The process of publishing a paper is an extremely long one, and it is not atypical to take several years from the first submission to the paper finally being accepted. The one part of the process that happens extremely quickly, however, is the moment when the journal sends you the galley proofs of the paper and then gives you 48 hours to make any final minor corrections. Despite the journal having taken up to several years to referee the paper, these messages often come with breathless warnings that failure to respond within the time window puts your paper in danger […] - [The Arbeitsgemeinschaft has returned!](https://galoisrepresentations.org/2021/03/26/the-arbeitsgemeinschaft-has-returned/): An update on this post; the Arbeitsgemeinschaft on derived Galois deformation rings and the cohomology of arithmetic groups will now be taking place the week of April 5th. Here is some practical information if you are curious. Is there somewhere I can watch the lectures even though I am not a participant? No, the workshop is invitation only. Is there somewhere I can watch the lectures as a virtual participant? I assume so, but I don’t know the exact details. I predict you will find out at the same time I do. Is anyone attending in person? I believe so, […] - [Test Your Intuition: p-adic local Langlands edition](https://galoisrepresentations.org/2021/03/09/test-your-intuition-p-adic-local-langlands-edition/): Taking a page from Gil Kalai, here is a question to test your intuition about 2-dimensional crystalline deformation rings. Fix a representation: \(\rho: G_{\mathbf{Q}_p} \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)\) after twisting, let me assume that this representation has a crystalline lift of weight \([0,k]\) for some \(1 \le k \le p\). Let \(R\) denote the universal framed local deformation ring with fixed determinant. Now consider positive integers \(n \equiv k \bmod p-1\), and let \(R_n\) denote the Kisin crystalline deformation ring also with fixed determinant. Global considerations suggest that for \(n \equiv m \equiv k \bmod p-1\) and \(n \ge m\), there should […] - [Fermat Challenge](https://galoisrepresentations.org/2021/02/21/fermat-challenge/): A challenge inspired from a question of Doron Zeilberger. Do there exist arbitrarily large integers \(n\) with the following property: There exists an ordered field \(F\) such that \(x^n+ y^n = z^n\) has solutions in \(F\) with \(xyz \ne 0\). The only solutions in \(F\) to \(x^m + y^m = z^m\) for \(3 \le m < n\) satisfy \(xyz = 0\), To give a somewhat looser phrasing, you might try to prove Fermat over \(\mathbf{Q}\) by an inductive argument that only relies on positivity of squares together with the fact that Fermat was classical known for some small values of […] - [Ramanujan Machine Redux](https://galoisrepresentations.org/2021/02/11/ramanujan-machine-redux/): I had no intention to discuss the Ramanujan Machine again, but over the past few days there has been a flurry of (attempted) trollish comments on that post, so after taking a brief look at the latest version, I thought I would offer you my updates. (I promise for the last time.) Probably the nicest thing I have to say about the updated paper is that it is better than the original. My complaints about the tone of the paper remain the same, but I don’t think it is necessary for me to revisit them here. Concerning the intellectual merit, […] - [Hire my students!](https://galoisrepresentations.org/2020/11/18/hire-my-students/): 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 upcoming jobs bloodbath](https://galoisrepresentations.org/2020/11/12/the-upcoming-jobs-bloodbath/): 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 […] - [En Passant IX (I'm a Gnu)](https://galoisrepresentations.org/2020/10/28/en-passant-ix-im-a-gnu/): 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 […] - [The eigencurve is (still) proper](https://galoisrepresentations.org/2020/10/23/the-eigencurve-is-still-proper/): 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 […] - [Chidambaram on Galois representations (not) associated to abelian varieties over Q](https://galoisrepresentations.org/2020/10/13/chidambaram-on-galois-representations-not-associated-to-abelian-varieties-over-q/): 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 […] - [Tips on becoming a computational number theorist](https://galoisrepresentations.org/2020/06/16/tips-on-becoming-a-computational-number-theorist/): 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 […] - [Families of Hilbert Modular Forms of Partial Weight One.](https://galoisrepresentations.org/2020/06/03/families-of-hilbert-modular-forms-of-partial-weight-one/): 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 […] - [Picard Groups of Moduli Stacks update](https://galoisrepresentations.org/2020/05/27/picard-groups-of-moduli-stacks-update/): 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 […] - [Panel Discussion: Mathematics Research Online](https://galoisrepresentations.org/2020/05/13/panel-discussion-mathematics-research-online/): 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 […] - [Picard Groups of Moduli Stacks](https://galoisrepresentations.org/2020/04/30/picard-groups-of-moduli-stacks/): 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 […] - [Chidambaram on genus two curves, II](https://galoisrepresentations.org/2020/04/24/chidambaram-on-genus-two-curves-ii/): We now continue a series of posts on the work of my student Shiva Chidambaram. (Click here for part I.) Today I would like to discuss another project with Shiva that was also joint with David Roberts (no, not David Roberts). We saw last time that the moduli spaces \(\mathcal{A}_2(\rho)\) and \(\mathcal{M}_2(\rho)\) are not in general rational over \(\mathbf{Q}\). On the other hand, the degree six cover \(\mathcal{M}^w_2(\rho)\) is always rational. So the next question is: what is an explicit parametrization? Slightly differently, start with a genus two curve with a Weierstrass point \(y^2 = x^5 + a x^3 + […] - [En Passant VIII](https://galoisrepresentations.org/2020/04/23/en-passant-viii/): I just found out that Lucien Szpiro recently passed away. I met him only once in late 2018 when I gave the joint NY number theory at CUNY. When I arrived at my hotel (the Gregory) around 10pm, I was somewhat shocked to find that nobody had made me a reservation, and the hotel was completely booked out. I was tempted to try the Langham on the opposite side of the street, but thought that might probably bankrupt the CUNY seminar budget for the next few years, so instead I started wandering the streets of NYC trying to find a […] - [En Passant VII](https://galoisrepresentations.org/2020/04/19/en-passant-vii/): Idle question which has surely been asked (and answered!). If \(X^{+}_{\mathrm{nsp}}(p)\) is the modular curve corresponding to the normalizer of the non-split Cartan, then one reason it is hard to find all rational points is that the all factors of the Jacobian have positive rank (probably contingent on BSD). Is the same true for \(X^{+}_{\mathrm{nsp}}(pq)\)? - [Chidambaram on genus two curves, I](https://galoisrepresentations.org/2020/04/15/chidambaram-on-genus-two-curves-i/): Before we start, just to alert you to a minor blogpage design change: all the posts (including this one) which talk about my students work can be accessed in one place by clicking the “work of my students” tab just below the picture on the top of this page. resume normal service. Those who study elliptic curves certainly know that if you start with an elliptic curve \(E/\mathbf{Q}\), the \(p\)-torsion gives rise to a Galois representation: \(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_p)\) with cyclotomic determinant. Conversely, if \(p = 2,3,5\) then the converse is true, that is, any such Galois representation comes […] - [How to reject a paper](https://galoisrepresentations.org/2020/03/31/how-to-reject-a-paper/): I just had the paper discussed in this post very quickly rejected. Since it was such a short paper, I thought it not too unreasonable to submit it to a strong journal, so I am not terribly disappointed. I can always say I chose such a journal for the benefit of my junior collaborator, of course. Rejections are almost always a bit of a kick in the gut, but I have to say that these were the nicest rejection letters I have ever received. It’s honestly more positive feedback than I receive on most of my papers which are accepted. […] - [You are welcome, Northwestern junior faculty](https://galoisrepresentations.org/2020/03/27/you-are-welcome-northwestern-junior-faculty/): Nobody has even accused Persiflage of being afraid to speak truth to power. After Chicago made their announcement that they were extending tenure by one year for all (qualified) faculty, I thought that, as a good local citizen of Evanston, I should put the pressure on Northwestern to do this same. Naturally, I first considered tweeting the official Northwestern account, or even better, the twitter feed of Morty Schapiro (Northwestern’s President). Alas, he didn’t seem to have a twitter feed, so I looked up the email address instead. The best I could find was nu-president@northwestern.edu, which I presumed would just […] - [More on Lehmer's Conjecture](https://galoisrepresentations.org/2020/03/21/more-on-lehmers-conjecture/): Lehmer said it was a “natural question” whether there existed an integer such that \(\tau(n)=0\) or not. I’ve wondered a little bit recently about how reasonable this is. (See this post.) The historical context is presumably related to the fact that, by the multiplicativity of coefficients, the vanishing of \(\tau(p)\) for one prime guarantees that a positive proportion of other coefficients vanish. From the perspective of Galois representations, however, I’m a little confused as to whether we expect any sort of “automorphic” Lehmer’s conjecture to hold. To recall, we have \(\Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) […] - [The Hausdorff Trimester has been indefinitely postponed](https://galoisrepresentations.org/2020/03/18/the-hausdorff-trimester-has-been-indefinitely-postponed/): I’m not sure if this is 100% official yet, but the Hausdorff Trimester scheduled for this summer has been (unsurprisingly) postponed. This is probably no great surprise to many of you. We have hopes to reschedule it again (perhaps for 2022) if possible. Together with this update, this is the final unfortunate installment of this series of posts. On the other hand, mathematics departments have already started to hold online seminars. I was chatting with TG and GB last night and one of them pointed me towards the MIT NT seminar which held it’s first online talk yesterday. If you […] - [NSF Proposal, Graduate Fellowship Edition](https://galoisrepresentations.org/2020/03/17/nsf-proposal-graduate-fellowship-edition/): Note: I feel as a service to the number theory entertainment complex that I should blog more often in these times, even if it means being less coherent than usual. I might even try to get a few guest posts since I won’t be going to any conferences any time soon… I recently linked to my first NSF proposal here, but just today I stumbled upon my graduate NSF fellowship application from 1998. There is really only one page which involves any proposal (rather than a list of courses I took or references), and I include the mathematical portion here […] - [The Arbeitsgemeinschaft has been indefinitely postponed](https://galoisrepresentations.org/2020/03/11/the-arbeitsgemeinschaft-has-been-indefinitely-postponed/): An update on the last post: As you probably already know by now if you are a participant, the Arbeitsgemeinschaft has been indefinitely postponed. The recommendation to do so was made by the organizers as universities rapidly began to recommend the cancellation of all work related travel. I hope that as many of you as possible can cancel your travel plans and get fully reimbursed. For those of you who have difficulties with this, especially those to whom I committed travel funding, please stay tuned. The last message I received from our administration on this matter was the following: I […] - [Conferences New and Old: Coronavirus Edition](https://galoisrepresentations.org/2020/03/07/conferences-new-and-old-coronavirus-edition/): A number of people have asked me whether the various conferences and workshops I am organizing this summer are still running. I thought I would have a blogpost containing all the current information, which I can update when and if necessary. Question: Will the Arbeitsgemeinschaft and HIM trimester proceed as planned? Answer: The answer is that there are no current plans to cancel or postpone either of these events. The MFO has issued a statement here. That said, the situation may change. There was an upcoming conference in Darmstadt which was cancelled today (March 6) with participants receiving the following […] - [Counting solutions to a_p = λ, Part II](https://galoisrepresentations.org/2020/03/03/counting-solutions-to-a_p-%ce%bb-part-ii/): This is a sequel to this post where the problem of counting eigenforms with \(a_p = \lambda\) and \(\lambda \ne 0\) was considered. Here we report on recent progress in the case \(\lambda = 0\). It is a somewhat notorious conjecture attributed to Lehmer (who merely asked the question, naturally) that the coefficients of \( \Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum \tau(n) q^n = q-24q^2+252q^3+\ldots \) never vanish. One problem with this conjecture is that there really isn’t any compelling reason it should be true except (basically) on probability grounds given the growth of the coefficients. As […] - [Vale instantchess.com](https://galoisrepresentations.org/2020/02/27/vale-instantchess-com/): One of the few time wasting activities I still indulge in is speed chess. (1 minute per player for the entire game is the slowest time control I play online.) There are a number of excellent free online sites available, but one that wasn’t quite in that category was “instantchess.com.” One terrible aspect of this website was that your opponents were random, and in particular it completely disregarded ratings when assigning matches. Actually, it was slightly worse than this; it seemed to have a preference for setting up games between people who had played before, but the algorithm included games […] - [Vesselin Dimitrov on Schinzel--Zassenhaus](https://galoisrepresentations.org/2020/02/10/vesselin-dimitrov-on-schinzel-zassenhaus/): Suppose that \(P(x) \in \mathbf{Z}[x]\) is a monic polynomial. A well-known argument of Kronecker proves that if every complex root of \(P(x)\) has absolute value at most 1, then \(P(x)\) is cyclotomic. It trivially follows that, for a non-cyclotomic polynomial, the largest root \(\alpha\) in absolute value satisfies \(|\alpha| > 1\). Elementary considerations imply that this can be improved to \(|\alpha| > 1 + c_n\) for some real constant \(c_n > 0\) that only depends on the degree. What is the true rate of decay of this parameter as the degree increases? By considering the example \(x^n – 2\), the […] - [Job Dedication](https://galoisrepresentations.org/2020/01/25/job-dedication/): Earlier last quarter, I had suffered a fairly poor night from some sort of stomach bug. Unfortunately, I made the ill-advised decision not to cancel my classes and went to work, which involves driving from Evanston to Hyde Park. Things did not go well; I was feeling so grim during honours group theory that for a 20-minute period I had to sit down with my head slumped on the table, occasionally able to utter a few sentences explaining the Orbit-Stabilizer theorem (I think I did a pretty good job in the circumstances). I somehow managed to survive through the full […] - [Inter-universal Teichmüller theory explained](https://galoisrepresentations.org/2020/01/17/inter-universal-teichmuller-theory-explained/): Normally a message such as the one below would go immediately to the rubbish bin. Fortunately for me, I happened to open it accidentally and thereupon discovered the most cogent explanation to date of IUT. I hereby share with you (in its entirety) the following email which was sent to me by a gentleman going by the name Samarium Beesix. PRIVATE AND CONFIDENTIAL: 16.01.20 DEAR SIR, PLEASE FORGIVE THIS INTRODUCTION. THIS MORNING I WAS LOOKING AT IMAGES FOR INTER UNIVERSAL TEICHMULLER THEORY. I DISCOVERED ONE PARTICULAR IMAGE, A SKETCH FROM SOMEONE AT MSRI, WHICH CONTAINS A FAMILIAR LOOKING GEOMETRY. IN […] - [The last seven words of Kedlaya-Medvedovsky](https://galoisrepresentations.org/2020/01/14/the-last-seven-words-of-kedlaya-medvedovsky/): New paper by my student Noah Taylor! It addresses some conjectures raised by Kedlaya and Medvedovsky in this paper. Let \(\mathbf{T}\) denote the Hecke algebra acting on modular forms of weight two and prime level \(N\) generated by Hecke operators \(T_p\) for \(p\) prime to \(N\) and \(2\) (the so-called “anemic” Hecke algebra). If \(\mathfrak{m}\) is a maximal ideal of \(\mathbf{T}\) of residue characteristic two, and \(\mathbf{T}/\mathfrak{m} = k\), there exists a corresponding Galois representation: \( \overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{T}/\mathfrak{m}) = \mathrm{GL}_2(k).\) If \(S\) denotes the space of modular forms modulo \(2\), then certainly \(\mathrm{dim}_{k}(S[\mathfrak{m}]) \ge 1\). Since there can […] - [New Results in modularity, Christmas Update II](https://galoisrepresentations.org/2019/12/30/new-results-in-modularity-christmas-update-ii/): Just like last year, once again saint Nick has brought us a bounty of treasures related to Galois representations and automorphic forms in the final week of the year. First there was this paper by Newton and Thorne, proving, among other things, the modularity of symmetric powers for a large range of holomorphic modular forms, including \(\Delta\) and any newform associated to a semistable elliptic curve. There is a lot to enjoy about this paper, not least of which is the nice application of an old computation of Buzzard and Kilford. But there are also some very nice new results […] - [Background for the Hausdorff Summer School](https://galoisrepresentations.org/2019/12/20/background-for-the-hausdorff-summer-school/): For those attending the Haussdorf Summer School previously mentioned here, I followed up with the speakers to ask them a little about what background was optimal for getting the most out of their lectures. In particular, I asked them to send me a sentence along the following lines: it would be useful for participants in this course to know X and to have some familiarity with Y, but no knowledge of Z is assumed for various (set) values of X,Y, and Z. Here are the responses, which I hope will be useful for some of you. (Some light editing has […] - [Mathjobs Application Tips Update](https://galoisrepresentations.org/2019/11/05/mathjobs-application-tips-update/): Previously I wrote about what I considered a “bug” in mathjobs: when letter writers submit their letter, the default time those letters are available is 18 months. But this leads to the following chain of events: An applicant applies for a job. Perhaps because of the vagaries of the market (or because they only apply to a limited number of places) they do not get an offer. The same applicant applies (perhaps more broadly) the next year. Because the letters have 18 month expiry dates, the applications all list THE OLD LETTERS as well as the new letters. Because letter […] - [En Passant VI](https://galoisrepresentations.org/2019/10/29/en-passant-vi/): I just learnt (from a comment on this blog) that Pierre Colmez hosts a wonderful page on Fontaine and Wintenberger here. I particularly recommend reading both the personal recollections of their friends and collaborators (sample quote from Mark: These \(p\)-adic Hodge theorists seemed to me like an order of monks, who were able to reveal the hidden design of a tapestry by examining it one thread at a time), as well as this article by Colmez which gives a beautiful introduction to Fontaine’s work (rather than my own somewhat superficial summary). One can’t mention the early work of Fontaine in […] - [Appropriate Citations](https://galoisrepresentations.org/2019/10/27/appropriate-citations/): Once I wrote a paper (two, in fact) on even Galois representations. The second paper in particular proved what I thought was a fairly definitive result ruling out the existence of a wide class of even de Rham representations with distinct Hodge-Tate weights. It turns out that almost nobody seems to cite these results, probably because they aren’t particularly useful — at least in any obvious sense. On the other hand, almost everyone who does cite the paper seems to cite it for a specific proposition (3.2) which is an easy consequence of the results of Moret-Bailly. The proposition, more […] - [A homework exercise for Oaxaca](https://galoisrepresentations.org/2019/10/12/a-homework-exercise-for-oaxaca/): Here’s a homework problem for those coming to Oaxaca who have a facility for working with Breuil-Kisin modules and finite flat group schemes. Let \(\mathbf{F}\) be a finite field of characteristic \(p\), and consider a Galois representation: \(\rho: G_{\mathbf{Q}_p} \rightarrow \mathbf{GL}_2(\mathbf{F}).\) which (one should imagine) is the local restriction of a global representation coming from a modular form. By a standard global argument, one can find a congruent form in weight \(2\), and thus a lift to a representation which is de Rham with Hodge-Tate weights \([0,1]\). For almost all such representations one can ensure that lift is potentially crystalline […] - [Read my NSF proposal](https://galoisrepresentations.org/2019/10/09/read-my-nsf-proposal/): Since this is NSF season, I took the opportunity to go back and look at some of my old proposals. I am definitely too shy to put my *most recent* proposal online, but I thought it might be interesting to share the very first proposal I ever submitted back in 2006. You can find it here. Honestly, it’s not as bad as I might have imagined. Here are some first impressions: The first thing that strikes me is that there is no “results from prior support section.” In particular, there is a pretty limited discussion of my previous work. It […] - [I asked... and you responded!](https://galoisrepresentations.org/2019/09/30/i-asked-and-you-responded/): I often ask mathematical questions on this blog that I don’t know how to answer. Sometimes my smart readers are able to answer the questions I ask. Surely they deserve some recognition for this? Here are two such occasions which come to mind (one very recent): In this post, I asked whether there are infinitely many integers \(n\) such that all the odd divisors of \((n^2 + 1)\) *not* of the form \(1 \bmod 2^m\) for large enough fixed \(m\), and asked whether that was an open problem. The answer: it was then, but no longer! It has now been […] - [Hausdorff Trimester Summer School, May 11-15, 2020](https://galoisrepresentations.org/2019/09/26/hausdorff-trimester-summer-school-may-11-15-2020/): This post is to encourage both PhD students and any junior researchers who are interested to consider applying to a summer school on the arithmetic of the Langlands Program. (Some financial support will be available.) This is the first event of the Haussdorff Trimester mentioned previously on this blog. A great lineup of speakers has agreed to give courses, namely: Arthur-Cesar le Bras and Gabriel Dospinescu on p-adic geometry, George Boxer and Vincent Pilloni on Higher Hida theory, Patrick Allen and James Newton on Automorphy lifting, Eva Viehmann and Cong Xue on Shtukas, Sophie Morel and Timo Richarz on Geometric […] - [NSF Application Tips: LaTeX edition](https://galoisrepresentations.org/2019/09/06/nsf-application-tips-latex-edition/): I’ve previously written about applying for an NSF grant here. But for those applying this year, I have a few further technical tips concerning the technical specifications of your LaTeX document. If you are the type of person who uploads all their files at the last moment (not me), then you could be in for a rude shock if you haven’t written your proposal up to code — rules are being checked by computer and are much more stringent this year. The first requirement is that the various subject headings “Intellectual Merit,” “Broader Impacts,” etc. need to be on separate […] - [Administrative Note](https://galoisrepresentations.org/2019/09/05/administrative-note/): Ideally nobody will really notice, but this blog has moved from www.galoisrepresentations.wordpress.com to www.galoisrepresentations.com. This blog still runs on the (open source) wordpress system, the only difference is that it is longer hosted by wordpress. The reason for hosting this blog elsewhere is that I now have access to plugins without having to buy a very expensive business plan from wordpress. And the particular plugin that I have been wanting to use is \(\LaTeX\). There was LaTeX functionality previously built in, but it was pretty poorly integrated and looked pretty bad. The old blog automatically links to here, and should […] - [Arbeitsgemeinschaft 2020](https://galoisrepresentations.org/2019/08/17/arbeitsgemeinschaft-2020/): The April 2020 Oberwolfach Arbeitsgemeinschaft will be on derived Galois deformation rings and the cohomology of arithmetic groups! For those who don’t know, the Arbeitsgemeinschaft (“study group”) is different from usual Oberwolfach workshops (or workshops more generally) — the idea is that the participants learn the material and then teach it to each other. I have never actually been to one (please leave a comment on your experience if you have), so I’m not sure that I can describe it better than reproducing the official blurb here: The Arbeitsgemeinschaften mainly address to non-specialists who want to broaden their outlook on […] - [Mathematische Zeitschrift (Part II: for authors)](https://galoisrepresentations.org/2019/08/05/mathematische-zeitschrift-part-ii-for-authors/): In this post, I give some tips for authors considering submitting to Math Zeitschrift, especially a paper in algebraic number theory. The first suggestion is to read Part I. This should give you a good sense of the standards required. (Of course, it’s always hard to judge your own work without bias.) I do, however, have a few more specific tips for authors: Submit the paper to the journal rather than email me directly: It’s certainly not a faux pas to send it to me directly, it’s just that it’s easier for me for various administrative reasons if you send […] - [Mathematische Zeitschrift (Part I: for reviewers)](https://galoisrepresentations.org/2019/08/03/mathematische-zeitschrift-part-i-for-reviewers/): I am an associate editor for Math Zeitschrift. I thought that here could be a good place to record a few useful comments that I often pass on to reviewers. It is my intention for future referee requests to include a link to this post. I have tried to keep to issues specifically related to Math Zeitschrift rather than issues common to all reviews, although much of this advice could be applied more broadly. In particular, I have avoided discussing problems like “how much of the proof am I supposed to check” because that is not really an answerable question […] - [The Ramanujan Machine is all hype](https://galoisrepresentations.org/2019/07/17/the-ramanujan-machine-is-an-intellectual-fraud/): Edit (17/02/21) I changed the title of this post which was unnecessarily incendiary. There’s a lot that I like about how mathematics operates as a social discipline. We have a great respect for the history of the subject, which in particular includes acknowledging the work that has come before us. In the end, we ultimately agree that it is the mathematics which does the talking. Each of us has our own tastes (of course) and some of us are more prone to be excited about our own work than others, but we are remarkably free from bullshit (about the actual […] - [En Passant V](https://galoisrepresentations.org/2019/07/07/en-passant-v/): (warning: today’s persiflage comes with possible extra snark due to sleep deprivation) The Ramanujan Machine: I learnt from John Baez on twitter about The Ramanujan Machine, a project designed to “help reveal [the] underlying structure” of the “fundamental constants” of mathematics. It seems that more effort has been spent on hype rather than on learning anything about continued fractions, and there is nothing there that would be surprising to Gauss let alone Ramanujan. Despite the overblown rhetoric (sample nonsense from the website: Suggest a proof to any of the conjectures that were discovered by the Ramanujan Machine. Have a formula […] - [The stable cohomology of SL(F_p)](https://galoisrepresentations.org/2019/06/19/the-stable-cohomology-of-slf_p/): Back by popular demand: an actual mathematics post! Today’s problem is the following: compute the cohomology of \(\mathrm{SL}(\mathbf{F}_p)\) for a (mod-p) algebraic representation. Step 0 is to say what this problem actually is. It makes sense to talk about certain algebraic representations of \(\mathrm{SL}_n(\mathbf{F}_p)\) as n varies (for example, the standard representation or the adjoint representation, etc.). For such representations, one can prove stability phenomena for the corresponding cohomology groups. But my question is whether one can actually compute these groups concretely. The simplest case is the representation \(\mathbf{L} = \mathbf{F}_p\) and here one has a complete answer: these cohomology […] ## Pages - [About](https://galoisrepresentations.org/about/): A number theorist blogs (sometimes) about math. I am inspired by the blogs I have linked to, but I do not aspire to be similar to them in any particular way. I am a professor of mathematics at the University of Chicago. The (faux) anonymity is merely to confuse google, though I suspect that if you can’t work out who I am then you probably won’t get much out of my posts. This blog will neither contain the fine photography nor the interesting diversions on language in EK’s blog; it will not have the disciplined academic focus of DC’s blog; […] [comment]: # (Generated by Hostinger Tools Plugin)