Bush, the Messiah (and Emma Kirkby)

The title of this post is, in part, a public service message to use the Oxford comma. However, there is a thread (in my mind) linking the three titular subjects. The connection between George Bush and the Messiah is not an obvious one, I admit, but hear me out. When I think of Bush, I think of the phrase “either you’re with us, or your against us.” I then always associate this phrase with “If God be for us, who can be against us?.” Was Bush consciously echoing the King James Bible? To me, of course, the latter phrase does not recall the Bible but rather Handel’s Messiah. All of which is a roundabout way of saying that this is another music post, with (who else) but Emma Kirkby performing with Christopher Hogwood and the Academy of Ancient Music (I have a recording on CD by the same ensemble which sounds to have been made contemporaneously with the video). The vintage of the haircuts is more H.W. than G., however.

Finiteness of the global deformation ring over local deformation rings

(This post is the result of a conversation I had with Matt). Suppose that

is a continuous mod- absolutely irreducible Galois representation. For now, let’s assume that is a CM field, and is essentially self-dual and odd. Associated to this representation is a global deformation ring (of essentially self-dual representations) consisting of representations with no local restriction at primes dividing and the condition of being unramified at primes away from . One also has a (collection of) local (unrestricted) deformation rings for the set of primes , combining to give a ring . Let us also assume that has suitably big image (for example, its restriction to is adequate). Then we have:

Proposition: The map is finite.

(Matt and Vytas prove this in the modular (odd) case when and , although I’m not sure whether the paper exists yet [actually, I'm pretty sure it doesn't]. Possibly if I was listening closer to Matt’s talk at Fields I might have remembered the argument, since I vaguely think it came up there, although possibly only briefly.)

Here one has to be a little careful defining deformation rings in the local case, of course (for those worried by such issues, simply choose suitable framings). To prove this, it suffices to prove the result after base change, so we may assume that is unramified at all primes, and completely trivial at all primes dividing . By Nakayama’s lemma, the problem above reduces to the following:

Proposition: Let be the maximal extension of unramified everywhere. Let be the Galois group of over . Then does not admit a continuous essentially self-dual representation:

such that is a complete local Notherian -algebra of positive dimension.

This is a special case of the generalization of the unramified Fontaine-Mazur conjecture due to Boston. Recall that the group may be infinite (Golod-Shafarevich), but that Fontaine-Mazur predicts that the image of any such representation into any characteristic zero -adic analytic group has finite image. Boston conjectured that the same finiteness would hold for homomorphisms of into for rings like . It turns out that even though the Fontaine-Mazur conjecture is hard, when has characteristic the conjecture is amenable to modularity lifting theorems by comparison to a new deformation ring in regular weight.

The proof is as follows:

Step 1: Using lifting theorems (Theorem 4.3.1 from BLGGT), we may assume, after a finite base change, that is potentially ordinarily modular of level one for some regular weight .

Step 2: Using minimal modularity theorems in the ordinary case (Section 10 from Thorne’s Jussieu paper, or Theorem 2.2.2 of BLGGT, both using work of Geraghty), deduce that the minimal weight ordinary deformation ring is finite over , and hence that is finite over . Strictly speaking, theorems of this kind are required to prove the previous result.

Step 3: Note that the minimal everywhere unramified deformations of (i.e., the deformations coming from ) of characteristic are all ordinary of weight , because everything unramified is ordinary, and in characteristic any two weights are the same. Hence is a quotient of , from which it follows from the finiteness of that is also finite.

While I am using the latest modularity lifting theorems here, weaker versions for with some local assumptions on follow from 90′s era technology (say Taylor’s Remarks on a conjecture of Fontaine and Mazur paper from 2000, or even earlier if one assumes residual modularity).

Via the usual argument, this result also applies to even Galois representations with large image. In particular, the unramified deformation rings in these cases will be finite over , and there will be at most finitely many counter examples to the unramified Fontaine-Mazur conjecture in characteristic zero for a fixed residual representation. One can also apply it to many classes of higher dimensional non-self dual representations by taking irreducible summands of . For example, one can take any representation of whose image contains if is even, since then the associated -dimensional representation restricted to an auxiliary CM field is irreducible, odd, self-dual, and adequate for large enough . Similar remarks apply to representations over an arbitrary field with generic enough image by taking the tensor induction down to .

If one starts allowing ramification at auxiliary primes, things become a little harder. One fix is to build the auxiliary primes into the local deformation ring , although this might be considered cheating. The problem is that one cannot deduce (in general) that more general ordinary deformation rings are finite in the non-minimal situation. Although perhaps one can get by with the Taylor trick in some contexts. One should be OK with by Ihara’s Lemma.

Michael Pollan is not a scientist

Michael Pollan is popular because he is an engaging speaker who spins a narrative about food that dovetails with the political inclinations of his audience. He has a degree in English, and, as far as I know, no scientific training whatsoever, but yet, he commands an enourmous amount of space in the New York Times and other liberal media to pontificate about nutritional science. Why does anyone take him seriously?

I don’t see any reason why I should care what Pollan thinks I should be eating. Science reporting should consist of a reporter explaining the consensus opinion (or otherwise) of scientists, not a dilettante pedalling an Alice Waters based cult dressed up as homespun wisdom. Let me be clear that I am not claiming anything he says in particular is wrong, I’m just feel that most of his conclusions are not arrived at in any scientific way, and the reason he has such a following is that his voice resonates with the intuition of self-indulgent (relatively) highly paid and well educated liberal elites (a class which I include myself). I avoid processed food, I seek out organic produce [for certain foods when it makes an appreciable difference in taste] (well, to be honest, it’s not usually me who does the food shopping because when I’m in charge I usually forget half the ingredients), and I almost always eat home-cooked meals with relatively little meat and plenty of fresh vegetables; and I do this for reasons of culture, taste, socioeconomic status, and because my wife wants me to be healthy. I pretty much agree with a lot of Pollan says (in the brief interviews I’ve seen him give), but what’s to stop him deciding (if he hasn’t already) that genetically modified foods are rubbish based on his own oversimplified philosophy rather than what science has to say? Or that vaccines are dangerous because his grandmother didn’t get them?

Equidistribution of Heegner Points

I saw a nice talk by Matt Young recently (joint work with Sheng-Chi Liu and Riad Masri) on the following problem.

For a fundamental discriminant of an imaginary quadratic field , one has points in with complex multiplication by the ring of integers of . Choose a prime which splits in . One obtains a set of points in , given explicitly as follows:

for in the class group and one of the two primes above in . The complex points can be thought of as being tiled by copies of the fundamental domain in the upper half plane.

Problem: How large does have to be to guarantee that every one of the copies of contains one of the CM points by ?

This is the question that Young and his collaborators answer. Namely, one gets an upper bound of the shape (with some explicit , possibly 20), the point being that this is a polynomial bound. Note that this proof is not effective, since it trivially gives a lower bound on the order of the class group which is a power bound in the discriminant, and no such effective bounds are known.

I idly wondered during the talk about the following "mod-" version of this problem. To be concrete, suppose that (the general case will be similar). We now suppose that is chosen so that is inert in . Then all the points in are supersingular, which means that they all reduce to the same curve with -invariant . Now, as above, choose a prime which splits in . The pre-image of in consists of exactly points.

Problem: How large does have to be to ensure that these points all come from the reduction of one of the CM points by as above?

Since is supersingular, we know that is an order in the quaternion algebra ramified at and . In fact, it is equal to the integral Hamilton quaternions . If and are lifts of , then there is naturally a degree preserving injection:

The degree on the LHS is the degree of an isogeny, and it is the canonical norm on the RHS.
In particular, if and , then one obtains a natural map:

preserving norms. The norm map on is . The image of the natural isogeny is simply , whose image has norm . Hence the problem becomes:

Problem: If one considers all the -maps:

do the images of cover the elements of of norm ?

Given a field in which is inert, it wasn’t obvious how to explicitly write down the maps , but this problem does start to look similar in flavour to the original one. Moreover, to make things even more similar, in the original formulation over one can replace modular curves by definite quaternion algebras ramified at (say) and , and then the Archimidean problem now also becomes a question of a class group surjecting onto a finite set of supersingular points. In fact, this Archimedean analogue may well be *equivalent* to the version I just described! Young told me that his collaborators had mentioned working with various quotients coming from quaternion algebras as considered by Gross, which I took to mean the finite quotients coming from definite quaternion algebras as above. Hence, with any luck, they will provide an answer this problem.

Exercise concerning quaternion algebras

Here’s a fun problem that came up in a talk by Jacob Tsimerman on Monday concerning some joint work with Andrew Snowden:

Problem: Let be a quaternion algebra such that the specialization splits for almost all . Then show that itself is split.

As a comparison, if you replace by , then although the condition that splits becomes empty, the conclusion is still true, by Tsen’s theorem.

This definitely *feels* like the type of question which should have a slick solution; can you find one?

Catalan’s Constant and periods

There is a 60th birthday conference in honour of Frits Beukers in Utrech in July; I’m hoping to swing by there on the way to Oberwolfach. Thinking about matters Beukers made me reconsider an question that I’ve had for while.

There is a fairly well known explanation of why should be irrational (and linearly independent of ) in terms of Motives. There is also a fairly good proof that in terms of the non-vanishinjg of Borel’s regulator map on . (I guess there are also more elementary proofs of this fact.) A problem I would love to solve, however, is to show that, for all primes , the Kubota-Leopoldt -adic zeta function is non-zero. Indeed, this is equivalent to the injectivity of Soule’s regulator map

(Both these groups have rank one, and the cokernel is (at least for ) equal to by the main conjecture of Iwasawa theory.) It is somewhat of a scandal that we can’t prove that is zero or not; it rather makes a mockery out of the idea that the “main conjecture” allows us to “compute” eigenspaces of class groups, since one can’t even determine if there exists an unramified non-split extension

or not. Well, this post is about something related to this but a little different. Namely, it is about the vaguely formed following question:

What is the relationship between a real period and its -adic analogue?

Since one number is (presumably) in and the other in , it’s not entirely clear what is meant by this. So let me give an example of what I would like to understand. One could probably do this example with , but I would prefer to consider the “simpler” example of Catalan’s constant. Here

is the real Catalan’s constant, and

is the -adic analogue. (There actual definition of the Kubota-Leopoldt zeta function involves an unnatural twist so that one could conceivably say that and that the non-zero number is , but this is morally wrong, as the examples below will hopefully demonstrate. Morally, of course, they both relate to the motive .)

So what do I mean is the “relation” between and . Let me give two relations. The first is as follows. Consider the recurrence relation (think Apéry/Beukers):

It has two linearly independent solutions with and , and and . One fact concerning these solutions is that , and Moreover one has that:

The convergence is very fast, indeed fast enough to show that . What about convergence in , does it converge to the real Catalan constant? Well, a numerical test is not very promising; for example, when one gets , which isn’t anything like ; for contrast, for this value of one has , which is pretty small. There are, however, two linearly independent solutions over given analytically by

from which one can see that must converge very slowly, and indeed, one has (caveat: I have some idea on how to prove this but I’m not sure if it works or not):

So one has a naturally occurring sequence which converges to in and in . So that is some sort of “relationship” alluded to in the original question. Here’s another connection. Wadim Zudilin pointed out to me the following equality of Ramanujan:

This sum also converges -adically. So, one can naturally ask whether

(It seems to be so to very high precision.) These are not random sums at all. Indeed, they are equal to

at . Presumably, both of these connections between and must be the same, and must be related to the Picard-Fuchs equation/Gauss-Manin connection for . This reminds me of another result of Beukers in which one compares values of hypergeometric functions related to Gauss-Manin connections and elliptic curves, and finds that they converge in and for various to algebraic (although sometimes different!) values. Of course, things are a little different here, since the values are (presumably) both transcendental. Yet it would be nice to understand this better, and see to what extent there is a geometric interpretation of (say) the non-vanishing of for some odd quadratic character . Of course, one always has to be careful not to accidentally prove Leopoldt’s conjecture in these circumstances.

Exposition is not underrated

It seems to be the conventional wisdom (for example, some of the comments here) that exposition is undervalued in our profession. I disagree. To cast things in economic terms, let’s take “valued” to mean one of two things: increased salary (cash) or increased recognition by peers (ego). First, I think it is unquestionably the case that a reputation as a good expositor is more likely to lead to invitations to conferences, to give colloquiua, and to give invited addresses, all of which also affect one’s career in a positive way. Second, a well written paper is more likely to be accepted by a higher ranked journal, and is also more likely to be cited by others – factors which also have a direct impact on one’s career (the effect here may be marginal, but is, I think, non-trivial.) Third, I think that certain forms of expository writing — such as graduate texts (think “the Arithmetic of Elliptic Curves”) — are widely known and (deservedly) widely praised. So what is the complaint?

I think the key point here is to distinguish between several flavours of expository writing. The first concerns articles which might once have appeared as short articles in the Monthly. Here a highlight of the form is something like Elkies on Pythagorean triples and Hilbert’s Theorem 90. This is the amuse-bouche of the exposition world.
Second is the account of a known result whose proof is not readily available in the literature, perhaps something like A proof that Euler missed. Third is an attempt to come to terms with some body of work by either filling in details, giving plenty of examples, or offering a slightly different perspective; let’s say Vakil’s algebraic geometry notes. Finally, there is the survey/overview style paper which seeks to convey a vision of the field and its connections to mathematics, pehaps something like Mazur’s paper “The theme of p-adic Variation” (a title that is both poetic and yet almost a pun).

The majority of expository writing falls in the third class. It usually takes the form of notes for a graduate class that someone posts on their webpage. The “level” of mathematics is usually that of a graduate class, or an advanced undergraduate class. Let me freely admit that it is wonderful to have such sources freely available, and that they can be useful. They play an important educational role. But how much of a contribution do they make to the advancement of mathematics? I think the level is relevant here. An exposition of Dirichlet’s theorem on arithmetic progression is essentially worthless — it is a topic covered well in an endless numbers of textbooks. And let me pass on without mentioning (apophasis alert) any article concerning the discrete geometry of Chicken McNuggets. Then, as the difficulty of the subject matter becomes higher, and the number of available resources become scarcer, the utility of such notes are increased. However, there’s a catch. The most inspiring, fundamental, insightful, and useful expository pieces can only possibly be written by a very few people. This is due to two obvious restrictions: few people can write well, and few people have interesting and deep things to say. Take, for example, the topic of recent progress in the Langlands programme. It’s perfectly possible for many people to give an anodyne talk to a broad audience on the latest developments. With some effort, a smaller number of people can also present some intuition for some of the core ideas. But anyone qualified to give a detailed exposition of the latest modularity lifting theorems is more interested in proving new theorems themselves.

By all means encourage good exposition, by all means cherish it when the masters commit their intuition to paper, by all means enjoy the wealth of expository notes available on the web, by all means encourage (through the reviewing process) authors to write clearer papers and describe their intuition, by all means use NSF money to fund instructional workshops. But don’t, as Cathy O’Neil suggested (update: I heard this suggestion from Cathy in person, but it was pointed out to me that she says something similar here), pay good mathematicians to spend six months learning topic X in order to produce a purely expository treatment of some important piece of mathematics; either they won’t be up for the task, they will have better things to do, or they would have done it naturally out of their own accord and inclination.