I first met Zagier in 1993, during my last year of high school. He was the Mahler lecturer, a position which carries the responsibility of giving as many lectures as one can on mathematics (and number theory in particular) all around Australia. My brother encouraged me to play truant from school and sneak into a colloquium talk by Zagier, who talked (with characteristic enthusiasm) about Ramanujan’s Delta function and the Birch–Swinnerton-Dyer conjecture. My brother also introduced me (this very same day) to Matthew Emerton, who talked to me about math for three hours; in particular he talked about elliptic curves and Mazur’s theorem on the possible torsion subgroups over So it was a very auspicious day for me indeed! At the time, I was enthralled by Edwards’ book on the Riemann Zeta function and was expecting to become an analytic number theorist. But on that day, I completely abandoned those plans and decided to do algebraic number theory instead.

Zagier gave another lecture the next day (which I also skipped school to see). This time it was on volumes of hyperbolic manifolds and their relationship with the dilogarithm and the Bloch group. It is remarkably pleasing then to now — almost 24 years later — write a paper with Don and Stavros which is related to the theme of both those talks, namely the Bloch group and modularity.

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I finally get it. **Du musst Caligari werden!** Oh, and if you think I’m crazy, it’s not me; it’s you.

**Edit:** Found a frame from the English version:

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which was unramified at p and residually irreducible (and modular) was itself modular (in the Katz sense). Galois representations of this flavour are obviously something I’ve thought about (and worked on with David G) quite a lot since then. But I have never actually seen any examples of mod p^2 forms which didn’t lift to characteristic zero. I asked George Schaeffer about it once, but his computations were only set up to detect the primes for which non-liftable forms existed rather than to compute the precise structure of the torsion in H^1(X,omega). But just today I stumbled across an example in relation to a pairing I learned about from Akshay (which I will tell you all about some other time).

The particular form (or rather pair, since it comes with a twist by the nebentypus character) occurs at level and is defined over the ring It doesn’t lift to a weight one form mod 11^3. The nebentypus character is the only one it could be at this level and weight: the odd quadratic character of conductor 3. When I looked again at Schaeffer’s thesis, he does indeed single out this particular level as a context where computations suggested their might exist a mod p^2 form. (Literally, he says that a computation “seems to imply the existence” of such a form.) I guess this remark was not in any previous versions of the document I had, so I hadn’t seen it. Here are the first few terms of the q-expansion(s):

Some remarks. Note that the coefficients of g and f satisfy for all and where is the quadratic character of conductor 3 (the nebentypus character). On the other hand, at the prime 3, we have

and so the eigenvalue of U_3 is the image of Frobenius at 3 under or and hence satisfies the equality

I was temporarily confused about the fact that for the Steinberg prime rather than and thought for a while I had made an error or mathematics was wrong. But then I realized this was weight one not weight two, and so one should have instead that (note that ) And it just so happens that the equation

in a weird coincidence has a solution very close to 103 (this is a solution mod 11^3, in fact). It’s easy enough to see that the image of rho and its twist contains with index two, and so has degree 3513840. (At this level, the only real alternative is that the form is Eisenstein, which it isn’t.) The root discriminant is not particularly small, it is

Finally, the Frobenius eigenvalues at the prime p = 11 are distinct, which is easy enough to see because otherwise the coefficient of q^11 would have to be twice the squareroot of chi(11) = -1, which isn’t even a square mod 11.

Perhaps there’s not too much more to say about this particular example, but I was happy to come across it, nonetheless. Well, perhaps I should also say that I computed this example in SAGE, as I slowly wean myself off magma dependency.

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But first, it’s time for the current edition of **NAME AND SHAME**:

There are currently 105 universities which contribute a yearly amount of money to MSRI. Although the amount of money is not insubstantial (about $5000 per institution), it sends a clear message to those outside mathematics (= potential donors) that we as a community value the job that MSRI does. (As a certain dean at Harvard once explained to me, nobody wants to make large donations to an institution unless they feel part of a larger movement, which requires a high rate of participation amongst everyone else.) Since the main focus of MSRI is generally as a research institution, you would certainly expect that (say) that all the top 25 ranked graduate programs in the US would be sponsors of MSRI. But this is not true! How could this be, you ask? I imagine the reason is that this funding is at the discretion of the chair, and you (as a department member) might not even think to ask if your institution is an academic sponsor of MSRI. So who are the miscreants who have skirted their obligations? The guilty party: Brown! Why is Brown not an institutional member of MSRI? Is it because of their precipitous drop in the USA rankings as an undergraduate programme over the last few decades? Is it because — let’s face it — things haven’t been the same since the halcyon days of the (admittedly rather sexy) undergraduate class of 2001? Is it because their endowment is so low that their check to MSRI bounced? Let your disappointment be known by emailing the chair, Jeffrey Brock. (Oh, and NYU is not a member also.)

OK, now back to the meeting. I learned a few things. First (and it’s a little hard to determine exactly what this means given the fungibility of money), the summer graduate schools are exactly paid for by contributions from academic sponsors. In this light, the issue of fairness is even more acute than I previously realized. Second, there is already a committee in place which is very much aware of the issues relating to mixed levels of backgrounds and is trying to find ways to address it. They seem to have a few good ideas (in particular, making clear before hand what the expectations will be, in order for universities to self-select appropriate students as well as give students with weaker backgrounds information on what they should learn about before hand), but I agree there’s no simple fix.

Naturally, of course, you also want the update on what’s going in the Berkeley culinary scene, or at least what can be ascertained from by a casual and infrequent visitor.

Andronico’s (also known as Astrinomico’s) has been replaced by Safeway. The decline in quality is immediately apparent — although to be honest, this judgement is mostly based from the brief look I had at the current Champagne selections.

Cafe Rouge has closed! I think Dipankar introduced me to this store. I had a great cassoulet there once.

Babette still does a very good coffee and pastry, although there was a charm to the previous outdoor space which seems to have been a little lost in the move. The new Blue Bottle Cafe is very shiny and has the very good manners to open early (e.g. early enough to get two cortados before going up the hill to MSRI).

I also made it (as is my habit) to visit Blue Bottle on Mint Plaza on my way to the airport for breakfast for some aeropress coffee as well as another cordato:

Cortado at Babette

Aeropress at Blue Bottle

Cortado at Blue Bottle (these are both from the Mint Plaza store in SF)

Besides hitting the cheeseboard (more of a miss this time, too much potato) and Babette for lunch, I only really had one free evening, but I did manage to also visit Cesar’s, Gregoire’s, and Chez Panisse for a progressive dinner of fino Sherry, lamb pasta (delicious), and a pear galette with Madeira. Arthur Ogus also made a cameo appearance on his bike picking up takeout at Gregoire’s.

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There is one program at MSRI that (from my mostly second hand observations) seems as though it could do with some improvement. MSRI regularly holds a number of summer schools. For example, this summer, Kevin Buzzard is giving an introduction to the theory of automorphic forms. I would have loved to been able to go to such talks when I started out as an graduate student, and I really wanted to send some of my current students to this. The problem? Firstly, there are a severely limited number of places. I’m not entirely sure I understand this, but I can imagine a few reasons. However, MSRI (apparently) goes to extreme lengths to be as “equitable” as possible in admitting people from as many different sponsor schools as possible. The result? I am told (internally) that the University of Chicago will be at most able to send two or three students in total to the seven or so different programs available. The result is that the only people uchicago sends to these programs are basically advanced students who are about to graduate and aren’t in any sense the target audience. On the other hand, these programs also tend to admit students from schools with much weaker backgrounds who aren’t even comfortable with basic concepts from algebraic number theory. This seems to be a very stupid way to choose participants for any program, even if it is purely in the name of fairness. What does a Harvard number theory student have to do to be able to attend an introductory course on automorphic forms — prove the Sato-Tate conjecture?

Apparently at least one lecturer plans to give their 20 courses in the order 1,11,2,12,3,13,etc in order to try to please at least half the people half the time.

One wag suggested that schools from weaker programs should consider it in their best interest to only admit students from places like Harvard, since then at least the programs would be training their future professors ([Caveat: I heard this second hand, so it may well have been in jest]). Presumably the reason behind MSRI’s policy is that the spoils of MSRI programs should go equally to (students of) universities that fund MSRI (sssuming they contribute a similar amount.) However, there seems to me to be a very natural alternative solution to this problem, namely, to continue having “advanced” summer courses but also introduce some deliberately introductory courses tailored to people from schools with less background. Students from schools with more advanced programs could be barred from the lower level introductory programs (since they could be more easily reproduced locally) and then, when it comes to the more advanced topics, there wouldn’t be the restriction to admit at most one student from each school. In particular, the selection criteria should concentrate (in part) on who stands to get the most out of each program.

Please add any further suggestions or complaints below

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**Theorem [Agol]** Let be a compact hyperbolic 3-manifold. Then is unbounded as ranges over all finite covers

(There’s an analogous version for finite volume hyperbolic manifolds with cusps.) What is the corresponding conjecture in coherent cohomology? Here is a first attempt at such a question.

**Question:** Let be a proper smooth curve of genus defined over Let denote a line bundle such that As one ranges over all (finite etale) covers are the groups

of unbounded dimension?

One might ask the weaker question as to whether there is a cover where this space has dimension at least one (and in fact this is the first question which occurred to me). However, there are some parity issues. Namely, Mumford showed the dimension of is locally constant in , and this dimension is odd for precisely choices of (there are such choices and the choices are a torsor for 2-torsion in the corresponding Jacobian). But I think this means that one can always make effective for some degree 2 cover, and thus produce at least one dimensions worth of sections. For example, when then , and has global sections whereas the other square-roots correspond literally to 2-torsion points. But those sections become trivial after making the appropriate 2-isogeny.

Another subtlety about this question which is worth mentioning is that I think the result will have to be false over the complex numbers, hence the deliberate assumption that X was defined over or at least over a number field. Specifically, I think it should be a consequence of Brill-Noether theory that the set of X in such that

for any choice of and any cover of degree bounded by a fixed constant D will be a finite union of proper varieties of positive dimension. And now the usual argument shows that, as D increases, any countable union of varieties cannot exhaust But it *can*, of course, exhuast all the rational points, and even all the algebraic points.

There’s not really much evidence in favor of this question, beyond the following three very minor remarks.

- The only slightly non-trivial case one can say anything about is when is a Shimura curve over and then the answer is positive because there exist lots of weight one forms (which one can massage to have the right local structures after passing to a finite cover).
- The analogy between and is fairly compelling in the arithmetic case, so why not?
- There doesn’t seem to be any
*a priori*reason why the virtual Betti number conjeture itself was true, and it is certainly false in for related classes of groups (groups with the same number of generators and relations, word hyperbolic groups), so, by some meta-mathematical jiu-jitsu, one can view the lack of a good heuristic in the hyperbolic case as excusing any real heuristic in the coherent case.

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Consulting the table one immediately notices a number of beautiful facts, such as the fact that (Z/3Z)^3 does not occur as a class group. (Our knowledge of p-parts of class groups, following Gauss, Pierce, Helfgott, Venkatesh, and Ellenberg, is enough to show that (Z/2Z)^n and (Z/3Z)^n for varying n only occur finitely often [similarly these groups plus any fixed group A], but those results are not effective.) One also sees that D = – 5519 and D = -1842523 are the first and last IQF discriminants of class number 97. It’s the type of table that immediately bubbles up interesting questions which one can at least try to give heuristic guesstimates. For example, let mu(A) denote the number of imaginary quadratic fields with class group A. Can one give a plausible guess for the rough size of mu(A)? One roughly wants to combine the Cohen-Lenstra heuristics with the estimate To do this, I guess one would roughly want to have an estimate for

I wouldn’t be surprised if someone has already carried out this analysis (thought I don’t know any reference). As a specific example, what is the expected growth rate of mu(Z/qZ) over primes q? A related question: is there a finitely generated abelian group which provably does not occur as the first homology of a congruence arithmetic hyperbolic 3-manifold?

At any rate, this is a result that Gauss would have appreciated. Curiously enough, this paper was recently posted as an answer on to the (typically ridiculous as usual) MO question *Which results from the last 30 years, in any area of mathematics, do you think are the most important ones?* While I wouldn’t quite put it in that class, I do find it curious that it this answer (at the time of writing) has -4 votes on mathoverflow. Given the enormous crap that does receive positive votes, I suppose that such minus votes are not to be taken too seriously. I would, however, make the following claim: Watkins’ result is as least as interesting as any original number theory research that has appeared on MO (at least as far as anything I have seen).

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I still think it’s an interesting problem to determine which extensions of by cyclic groups occur as the Galois groups of *minimally ramified up to twist* extensions, but that is not the same as the inverse Galois problem.

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Letter from Tate to Serre, Dec 8, 1958:Are you aware that the class number of the field of 97th roots of 1 is divisible by 3457 and 118982593? And that 3457 = 36 * 96 + 1 and 118982593 = 1239402 * 96 + 1?

If reading that doesn’t give you just a little thrill, then you have no soul. Does it have any significance mathematically? The class number is large, of course, which relates to the fact (proved by Odlyzko) that there are only finitely many Galois CM fields with bounded class number. (The reason why one can access class numbers of CM fields F/F+ is that the unit group of F and F^+ are the same up to finite index, so the *ratio* of zeta values is directly related to the minus part of the class group uncoupled from any regulator term, so one can access this analytically.) Alternatively, one might be interested in the congruences of the primes q dividing the class number. In this case, we see a reflection of the conjectures of Cohen and Lenstra. Namely, we expect that there is a strong preference for the class group to be “more cyclic,” especially for larger primes. The class group also has an action of which is cyclic of order 96. Since one expects the plus part to be very small (and indeed in this case it is trivial), this means that complex conjugation should act non-trivially, which means that the group of order 96 should (at least) act through a quotient of order at least 32. So if the class group is actually cyclic, this forces the prime divisors q of h_F to be 1 mod 32, and even 1 mod 96 if the class group of F doesn’t secretly come from the degree 32 subfield of F (which it doesn’t). (Not entirely irrelevant is Rene Schoof’s nice paper on computing class groups of real cyclotomic fields.)

Both Serre and Tate are unfailingly polite to each other. As a running joke, the expression “talking through one’s hat” occurs frequently, as for example the letter of Nov 14, 1961, where the subtle issue of the failure of is discussed. (Another amusing snippet from that letter “Even G. himself makes mistakes when he thinks causally.”) The correspondence is also fascinating from the perspective of mathematical history — one sees the progress of many ideas as they are created, including the Honda-Tate theorem and the Tate conjecture over finite fields. The first time the latter appears (as a very special case) it actually turns out to be an argument of Mumford, who shows Tate an argument (using Deuring) why when two elliptic curves have the same zeta function they are isogenous. This elicits the following reaction from Tate:

Letter from Tate to Serre, May 9, 1962:“Damn! The result is certainly new to me, and it frankly makes me mad that I never noticed it”

We have all been there, although, to be fair, most of us have the excuse of not being Tate!

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Here is how I plan to celebrate: I will spare you with the complete 12 hour 23 minute breakdown of my carefully curated iTunes play list, but suffice to say it is both suitably seasonal and indulgent. A few highlights: Spem in Alium, Lupu and Perahia playing Schubert’s Fantasia in F minor, Bach Cello Suites, A Musical Offering, and plenty of carols from King’s College Cambridge. There will be a trip to Intelligentsia (by Uber — it is -17 C outside) and then a wander around the Chicago Art Institute. For lunch, smoked salmon and smoked eel (hat tip to Bao who pointed out the existence of a fine food store in duty free at Schiphol airport!) with Champagne, followed by left-over home made Coq au vin with orzo. For dinner, Foie Gras mi-cuit with a side of poached apples, together with Sauternes. And then maybe some vegetable and tofu stir fry to balance things out slightly, in order to justify finishing off the evening with some delicious moist fruit cake. The final indulgence: I plan to spend the afternoon at home drinking tea, reading the Serre-Tate correspondence, and looking out my window onto the lake (which right now has a beautiful cover of fog which my poor photography isn’t quite able to capture):

All of this, of course, with the best possible company imaginable for a quiet day of self-indulgence. Happy holidays!

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