Addendum: the case when Serre weight is p

In my study group talk, I gave a proof of the existence of minimal lifting result (for mod p representation of G_Q) including the case when the Serre weight is p, which is not covered in Khare-Wintenberger. At first I thought that the assumption that Serre wt is not p is not used in their argument, but later I found one place in their argument which might not work if the Serre weight is p. (I believe that the proof I gave in the study group talk works even when Serre weight is p.) Please feel free to point out if I made any (mathematical or historical) mistake below.

From now on [KW] denotes Khare-Wintenberger’s “On Serre’s conjecture for 2-dimensional mod p representations of G_Q” (not the famous papers where they actually proved the Serre’s conjecture). The assumption that the Serre weight is not p appeared in the “potential modularity result” [KW, Thm 2.1], and the statement is stronger than the version I stated in the seminar talk: they needed the residual representation to be potentially modular be a Hilbert modular form of *tame level 1* and of weight 2 (so that it can be congruent to a Hilbert modular form of level 1 and of Serre weight). Note that [KW] used Fujiwara and Taylor’s patching so they needed to have the modular form to have the “minimal level” while I didn’t need such a strong control because I used Kisin’s patching.

I believe that [KW, Thm 2.1] without tame level assertion should work without excluding Serre weight p. (This is essentially done by Taylor, and the suitable references are given in the proof of [KW, Thm 2.1].) But when the residual representation is reducible, in order to get to the “level 1 case” [KW] used the following argument. First, they further extended the base field so that the residual representation becomes unramified outside p, and then applied level lowering using Skinner-Wiles. Here, one needs the residual representation is “p-distinguished”, which can be problematic when Serre weight is p.

I hope this clarifies.

Weil on algebra

Yesterday, I came by train from Paris to the beautiful city of Montpellier. Before boarding at Gare de Lyon, I bought a Le Monde magazine, which contained a series of sombre articles on civilization and its history. I found therein an amusing quote from Simone Weil:

‘Money, mechanization, and algebra: The three monsters of contemporary civilization.’

I seem to recall that she writes in her spiritual autobiography of feeling immensely oppressed as a child, to the point of contemplating suicide, by her older brother’s conspicuous talent.

The essay from her book ‘gravity and grace’ continues:

‘Algebra and money are essentially levellers, the first intellectually, the second, effectively…’

Graduate study group 25/11 and 02/12

Tomorrow (Wednesday 25th November), there won’t be a graduate study group talk, but Alexei Skorobogatov is giving a talk at 1 o’clock in Room 408 at Imperial, on connections between forms of algebraic groups and number theory.

On 2nd December, Carl will talk about elliptic curves with complex multiplication, at 12 in room 658 as usual.

Geometry

I’ve lifted the following quotation from an article by my friend John Baez:

Geometry enlightens the intellect and sets one’s mind right. All its proofs are very clear and orderly. It is hardly possible for errors to enter into geometric reasoning, because it is well arranged and orderly. Thus the mind that constantly applies itself to geometry is unlikely to fall into error….

It’s attributed to the remarkable North African scholar, Ibn Khaldun.

On education and difference

I wrote recently a short essay for the amusement for some friends on education and difference.

Even though it’s not about number theory, I thought I’d upload it here, in case someone else finds it amusing, or annoying.

Final workshop at Newton

I would like to encourage people to apply for participation in the final workshop at the Newton Institute from 14 December – 18 December. We have at present sufficient funds to be fairly generous with the cost of accommodation, and even for travel in reasonable circumstances. So please go ahead and apply.

Using PDF files for discussions

In case people would like to have serious mathematical discussions here in relation to Leopoldt’s conjecture or other topics, I thought I would suggest the option of using PDF files. That is, you can type latex into your comments normally by putting the script between two dollar signs, except you need to write ‘latex’ after the first dollar sign. But I’ve found this inconvenient as opposed to just creating a PDF file and putting in a link. So if you’d like to do that, you can either

-put the PDF file into your own public directory and create a link in the comment;

-or send your PDF file to James Haydon: ucahjhh@ucl.ac.uk

and put the url

http://www.ucl.ac.uk/~ucahjhh/pdfs

into the link.

For those of you entirely new to this kind of thing, what you need to do is create a PDF file with a name like ‘leopoldt.pdf’, send it to James, and then put into your comment something like:

[a href="http://www.ucl.ac.uk/~ucahjhh/pdfs/leopoldt.pdf "]Here are my comments [/a]

(You need to use pointy brackets wherever you see square brackets “[ ]” above, which I’ve used for obvious reasons.)

Leopoldt’s conjecture

Last week, Preda Mihailescu gave a seminar talk at the DPMMS in Cambridge and offered many hours of discussion sessions at the Newton Institute on his paper

The and components of modules and Leopoldt’s conjecture.

Let be an algebraic number field. The theorem of Dirichlet says that the group of units in , the ring of algebraic integers in , has rank . Here, is the number of real embeddings of and .

For each non-zero prime ideal of , denote by the completion of at and the group of units in the integers of . For a rational prime , we define

There is then a diagonal embedding

analogous to the embedding

into the Euclidean completions. The main difference is that the image is not necessarily closed in the -adic topology, so that it is the closure that becomes relevant to many arithmetic considerations, such as class field theory. For example, we have a ‘pseudo-isomorphism,’ that is, a homomorphism

with finite kernel and cokernel, to the Galois group of the maximal abelian pro- extension of unramified away from the primes dividing . Now, from the formula for the usual -rank of , it is clear that the -rank of satisfies

.

In fact, it is common to write

and refer to as the *Leopoldt defect*.

The conjecture says simply

From the discussion above, it is clear that this statement is equivalent to

That is, we get a precise formula for the number of independent -extensions of .

For totally real, there is an interpretation in terms of a -adic regulator computed in a natural way by taking the determinant of a matrix formed out of the -adic logarithms of a basis

for the lattice . The conjecture then says that the determinant is non-zero.

Such difficult independence statements seem to come up in a number of different but related situations. One might think of transcendental number theory, for example, but also geometric statements like the Tate conjecture. There, one has a cycle map

on a smooth proper variety over a finitely-generated field. The best known part of the conjecture is a surjectivity statement, to the effect that the image of the cycle map generates the Galois-invariant subspace. However, there is also a portion that says

should be injective. That is, once again, one should not have -adic relations appearing between cycles that are -linearly independent modulo homological equivalence.

There is a curious principle relating surjectivity and injectivity statements for regulator maps or cycle maps. By and large, the surjectivity statements seem to have more direct impact, since they assert the existence of certain motives. However, in many specific circumstances, the surjectivity in one realm becomes dual to injectivity in another, rendering it unavoidable to consider them in conjunction. Although I don’t understand the formalism at all well, it should also be remarked that cycle maps and regulator maps are different portions of a single construction occurring in motivic homotopy theory.

There is a formulation of Leopoldt’s conjecture in terms of Galois cohomology that relates the conjecture to a broad range of investigations in arithmetic geometry. For this, we let be the Galois group of the maximal pro- extension of unramified outside of . Then Leopoldt’s conjecture is equivalent to the vanishing statement:

As some of you may know, several decades into its study, Galois cohomology has become no easier to understand than when it was first defined, in spite of its enormous importance. The main conjectures of Iwasawa theory, especially in the formulations of Kato and Perrin-Riou, are concerned with control theorems for Galois cohomology, while the finiteness conjecture for the Tate-Shafarevich groups of elliptic curves is essentially stating that we can compute their Mordell-Weil groups in terms of Galois cohomology. In my own humble work, the intractable nature of Galois cohomology underlies the complexity of Selmer varieties.

For an ambitious young arithmetician, a worthwhile goal might be to develop the kind of foundational theory that would render Galois cohomology as computable as the topological cohomology of two- or three-dimensional complexes. To some extent, the mysterious problem of ‘understanding’ the structure of Galois groups boils down to assertions about Galois cohomology, as might be seen most prominently in the work of Jannsen and Wingberg on the explicit structure of Galois groups in the local case. To put it a bit more philosophically, given a Galois group, its cohomology with coefficients in various representations may be viewed as the precise locus both of the structure theory *and* of its applications.

One the few general theorems we know of today is due to Soule. It says that for any finite set of primes containing the primes dividing ,

is finite for . Strangely enough, this result eventually relies heavily on harmonic analysis of the Laplacian on symmetric spaces! For , the corresponding cohomology lies between the class group and the Brauer group, while the groups for seem to be well-nigh intractable in general. This little bit of background should help you to see that Leopoldt's conjecture is both an incredibly vexing problem and of great conceptual importance.

The conjecture was proved in 1967 by Armand Brumer when is abelian:

On the units of algebraic number fields. Mathematika 14 1967 121–124,

using, in fact, the techniques of Diophantine approximation. I should probably read the proof, because I find the restriction rather surprising. That is, to my understanding, many assertions in Diophantine approximation don't care what number field we're working in, as in the height approach to Diophantine geometry.

The general case remained open, claiming the futile efforts of quite a few distinguished arithmeticians over the years.

Starting in May of this year, Preda Mihailescu posted a series of articles on the archive claiming to prove the conjecture in general, leading to his invitation to the Newton institute for a series of lectures.

Because I am far from an expert on these matters, it's hard for me to proclaim any judgment about the status of the proof. However, two things are very clear:

(1) Professor Mihailescu is very open to questions and discussions. He has been very generous with his time and energy this week in response to our diverse queries about the details of the argument as well as the prerequisites;

(2) Regardless of the eventual judgment, the *pedagogical value* of the manuscript seems indubitable. It certainly forces me to dwell on the intricate theory of cyclotomic fields with far greater intensity than I'm used to.

For these reasons, I thought I'd encourage young people especially to take a look, and work their ways through the arguments. If there are points that require serious clarification, of course it would be a great service as well to locate them precisely.

In reponse to my request, Professor Mihailescu has kindly provided an executive summary of the main line of argument. He tells me that the only prerequisites for the paper are the books of Lang and Washington, and then the foundational paper of Iwasawa:

Iwasawa, Kenkichi On -extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326,

especially for the discussion of the skew-symmetric pairing.

Maybe not everyone would agree, but I tend to regard very technical manuscripts as being highly instructive in a way very different from polished expositions. Deligne’s papers on the Weil conjectures, for example, with beautiful streamlined discussions to accompany the wealth of deep mathematics, obviously offer enormous value for every hour of effort devoted to their reading. The papers of Faltings, on the other hand, are notoriously difficult. I remember a discussion among students in the Harvard common room about his great paper on subvarieties of abelian varieties. When Tate overheard us complaining bitterly about the obscure points, he scolded: ‘They’re good! They make you guys work.’

Optimal proofs?

At the n-category cafe, we were having a discussion on an interview with Yuri Manin. I mentioned that coming up with a good programmatic proof of important results is a rather complicated process typically involving a large amount of piecemeal engineering. David Corfield then asked if mathematicians *eventually* work towards optimal proofs. I thought I’d reproduce my reply here, in case you can’t be bothered to follow the other thread:

——————————————–

The answer to this is a resounding ‘Of course!’ For the above-mentioned results in the Langlands program, this is happening at a fearful pace even as I write. There are so many other examples, I hardly know where to begin. A nice example with a classical flavor might be Hilbert’s twenty-first problem concerning the existence of regular singular differential equations with prescribed monodromy. I’m not sure how many times it’s been proved. In his summary of Deligne’s beautiful version, Nick Katz famously quipped that ‘what’s worth proving once is worth proving many times.’

The famous Index Theorem for elliptic operators has a very instructive history in this regard. In the paper explaining their second proof, Atiyah and Singer emphasize the philosophical importance of the new approach, which they refer to as an ‘embedding proof’ modeled on Grothendieck’s proof of the Riemann-Roch theorem. Eventually there were ‘local’ proofs, path integral proofs, supersymmetric proofs, ad infinitum, until one was left with the impression that the whole thing was gradually being reduced to a triviality. Such reduction is good in many ways, as in Whitehead’s remark about the progress of civilization. But a very interesting and relevant episode occurred in the early 70’s that profoundly influenced subsequent developments. This was the ‘heat equation proof’ for Dirac operators by Atiyah, Patodi, and Singer. Most of the later refined proofs follow the pattern of this one. It is remarked there that the ‘miraculous cancellations’ that occur in the asymptotic expansion of the heat kernel had first been calculated through a ‘tour de force’ earlier by Patodi alone (for the Laplace-Beltrami operator). It goes without saying that Patodi’s horribly messy calculations were a crucial step in the streamlining that eventually took place.

I sometimes flatter myself that I have been working for a while on a ‘philosophically satisfactory’ proof of the Mordell conjecture. (This sentiment will quite likely strike Professor Faltings as laughable.) But the story of the Mordell conjecture is itself a nice illustration of messy serendipity. To an outsider, Faltings may appear to be an abstract arithmetic geometer like any other, but his style has always been quite distinct from the Grothendieck school. Of course he’s incredibly well-educated in the Grothendieck machinery and uses it inccessantly, but there’s always been an aspect of his approach to mathematics that could also only be described as ‘pragmatic.’ That he managed to prove the Mordell conjecture that had eluded the enormously productive French school can be attributed to this pragmatism coupled to tremendous technical prowess. One way of describing the awkwardness of the Mordell conjecture is exactly that equations over number fields have none of the tidy patterns that govern finite fields a la Weil-Deligne. To our present day understanding, number fields display exactly the kind of order ‘at the edge of chaos’ that arithmeticians find so tantalizing, and which might have repulsed Grothendieck. The kind of things Faltings was good at that were not entirely mainstream in Orsay at the time included technology like Diophantine approximation, heights and metrized line bundles, and geometric invariant theory, to mention just a few salient items. And then there was the p-adic Hodge theory that had in fact been a preoccupation of Grothendieck’s, but which had remained largely unwieldy up to that point compared, say, to l-adic cohomology. Faltings’ success in applying it to something ‘practical’ was a driving force in yet another process of streamlining, so that it’s now standard machinery in too many theorems to name (including results of Wiles, Harris-Taylor, etc.). This particular kind of interaction between theory and problem-solving, even though it’s well-known to practitioners, has perhaps received insufficient attention from the commentators on the ‘two cultures.’ That is

theory–> resolution of important problems

is the subject of many conspicuous arguments, but the arrow

resolution of important problems–> mature development of theories

is equally important. Obviously, a more accurate picture would be a complicated graph with different amalgams along the nodes and multiple edges.

The collective arsenal of the mathematical community includes grand visions, brute force, wishful thinking, and even wrong proofs that lead to important developments. Some other essential ingredients that may go unnoticed include the ‘misguided program’ and the *accident.* My advisor Serge Lang has sometimes been accused of pointless generality. This is supposed to occur, for example, in his (great) book on Diophantine geometry. Part of the reason for this impression has also to do with his somewhat awkward stance vis-a-vis the Grothendieck school. Lang was a great fan of algebraic geometry in the style of Grothendieck, but never really mastered it well enough to use it with fluency. This imparted something of an archaic flavor to his mathematical style, even when he was theorizing in a grand fashion. The statement of the Mordell-Lang conjecture, for example, will strike many as rather odd, and meaninglessly general. However, I can’t resist recounting a story that combines into a single narrative the many different faces of accidental progress and quixotic programs. In the early 90’s I was briefly a colleague of the logician Udi Hrushovski. He was a person with very lively interests in all aspects of mathematics, and would frequently approach me with an array of questions about number theory, including the Weil conjectures and etale cohomology. However, we once had a conversation about the work of Angus Macintyre, strikingly ironical in retrospect. Macintyre was still on the faculty of Yale when I was contemplating studies there, and I had some interest in him as a possible advisor. So he happened be the only logician whose work I knew something about. When I inquired about Hrushovski’s opinion of it, the reponse was something like ‘It’s very nice, but somehow too applied. Macintyre always thinks about applications to algebra, whereas I’m much more of a pure model-theorist.’ (Even these days, Macintyre constantly urges young logicians to educate themselves in mainstream areas of mathematics.) Pure model theory here refers roughly to the classification of a more extensive collection of structures than most self-respecting mathematicians at the time would have cared a hoot about. One lazy afternoon, I had been browsing through a paper by Alexandru Buium in the common room, in which he used differential algebras for the proof of some geometric version of Mordell-Lang. (Such proofs, by the way, had evolved out of Manin’s slightly incorrect proof of the geometric Mordell conjecture.) At some point, I absent-mindedly dropped it on the table as I left the room. Hrushovski told me that he picked it up soon afterwards and suddenly realized that a significant portion of the structures there were familiar to him. In particular, the differentially closed fields used by Buium, somewhat exotic to geometers, were standard among model-theorists and the ‘too general’ statement of the Mordell-Lang conjecture was strongly reminiscent of definable subsets of certain groups of finite Morley rank. Some of you will know that Hrushovski rapidly came up with a model-theoretic proof of the Mordell-Lang conjecture over function fields of positive characteristic, and ushered in a really new era of interaction between arithmetic geometry and model theory. In short, spectacular ‘applications,’ a great deal of entirely new theorizing, and freshly uncovered bits of harmony. Incidentally, Hrushovski’s proof, influential as it was, is still very hard to understand after nearly two decades of collective effort. My friend Anand Pillay has been devoting an inordinate amount of time to coming up with a more ‘philosophically satisfactory’ version, even though he dislikes philosophy on the whole (except perhaps in its Marxist incarnations).

My own feeling is that there is an abiding logic in all this, but at a much higher level than can be captured by familiar dichotomies.

Seminar Oct-Dec 2009

It took me until November to actually get round to posting the seminars on the blog. Apologies.

The London Number Theory Seminar this term is at Imperial College,
room 658, the Huxley Building, at 4 pm. Tea is served in the Common
Room on the 5th floor at 3.30.

4 November. Roger Heath-Brown (Oxford) “Counting points on cubic curves”

11 November. Don Blasius (UCLA) “Asymptotic Fullness of Automorphic Galois
Representations”

18 November. Herbert Gangl (Durham) “Double zeta values and periods of modular forms”

25 November. Fabien Trihan (Nottingham)

2 December. Behrang Noohi (King’s) “Galois cohomology of crossed-modules and
cohomology of reductive groups”

9 December. Javier Lopez (Queen Mary)