Motivic L-functions

Sorry for taking such a long time for posting about the study group this term. The first talk already took place on the 13th of January. Minhyong Kim gave an overview of motivic L-functions and various conjectures. He has already posted the slides. Kevin Buzzard will give the next talk (at 2pm in Room 707 in UCL) on the Deligne conjectures. And Andreas Holmstrom has agreed to give the third talk (on the 27th of Jan.) on Beilinson conjectures – 1. A rough schedule is

20/1 Deligne conjectures -Kevin Buzzard

27/1 Beilinson conjectures 1 – Andreas Holmstrom

3/2 Cancelled

10/2 Beilinson conjectures – 2 -Andreas Holmstrom

17/2 Bloch Kato – 1 – Cecilia Busuioc

24/2 Bloch-Kato- 2 – Wansu Kim

3/3 Bloch-Kato – 3

10/3 Main conjecture (a la Kato)

17/3 Noncommutative main conjecture -1

24/3 (?) Noncommutative main conjecture -2

ICM satellite conference in Goa

John Coates, CS Dalawat, Anupam Saikia and Sujatha Ramadorai are organising ICM satellite conference on ‘Galois representations in arithmetic and geometry’ at The International centre in Goa, India, between the 10th and the 13th August, 2010.

Here is the link

Zeros and poles

Yesterday, we discussed the conjecture

for

and

Even though categories of motives are ill-defined, it’s useful to note that these conjectures can be loosely justified using a number of `standard conjectures.’ (And rather precisely justified using more careful foundations.) For example, we already pointed out that

is implied by the Hasse-Weil conjecture on the analytic continuation and functional equation. Now, suppose one believes also in a Galois-theoretic analogue, whereby

extensions in the category of Galois representations satisfying a Bloch-Kato style finiteness condition indicated by the subscript . Then the Fontaine-Mazur conjecture says any such extension is motivic, whence, should come from

On the other hand, the injectivity in

should be considered a conjecture of Tate type, whereby Galois-theoretic isomorphisms are induced by maps of motives.

Furthermore,

and this last order should be

by the obvious -adic analogue of the theorem on poles of Artin -functions. But the isomorphism

is nothing but Tate’s conjecture on algebraic cycles once again.

Motivic L-functions II

Here are the slides for the talk I gave today at the study group.

A correction to one silly thing I think I said: A pure system of realizations was defined to be a family of

*satisfying all the strong compatibility conditions*. So then, I believe the Fontaine Mazur conjecture should be saying in essence that they are all motivic.

L-functions

In preparation for my introductory lecture at the London number theory study group this Wednesday, I thought I’d upload the article I wrote for the IHES summer school on motives in 2006.

Within the mess

Following up on the topic of an earlier post , I discussed recently with philosopher David Corfield the shortcomings of optimality. This issue occurs in many versions, sometimes as simple as a very elegant lecture. At the recent workshop at Newton, Guy Henniart and Laurent Clozel gave two beautiful lectures on the Langlands programme. My impression is that the nature of such lectures has changed considerably in the last two decades or so, as the experts become more and more used to explaining their trade to the general audience. There are obvious benefits to this, especially from the standpoint of a fan like me. On the other hand, the messy and often essential portions are inevitably glossed over in such presentations, and Clozel and I agreed that something is lost in the process…

We discussed how such loss may occur also with entire theories. That is, one can imagine a rich but chaotic mass of mathematical objects, formulas, computations, theorems, lemma, and so forth, which eventually undergoes the process of being completely systematized, perhaps with the aid of several progressive breakthroughs and over several decades. At some point, one surveys the landscape and there seems no question that the community has reached the ‘right’ view of the matter, with all the harmony and unity expected of a great theory. Great advances have been made. Nevertheless, it’s hard to avoid the feeling that something is lost within the hegemony. The tidy, orderly theory is incapable of capturing the full strength and depth of the mathematical nature that was inherent in the mess.

I think I can come up with some good examples to illustrate such misgivings, but I thought I’d ask here for other suggestions as well. More precisely, I’m interested in examples of the following process:

Beginning point–Big mess
.
.
.
Grand theorizing and harmonizing, perhaps through the efforts of a whole school of Grothendiecks. ‘Right’ view of the subject and objects firmly established.
.
.
.
Someone’s re-examination of the mess, leading to entirely new insight.

Some examples will be more compelling than others and will certainly involve different scales of evolution. But I’d be interested in hearing of any view on the matter.

Seminar Jan-Mar 2010

From Andrei:

Dear all

The London Number theory seminar next term will be at UCL;
organised by me

Here is the list of speakers

January:

13 : Michael Schein (Bar-Ilan)

20 : Mathieu Florence (Paris)

27 : Fernando Villegas (Texas)

February

3 : Toby Gee (Harvard ?)

10 : Jonathan Pila (Bristol)

24 : Wansu Kim (Imperial)

17 : Frank Neumann (Leiceister)

March

3 : Lawrence Breen (Paris)

10 : Sarah Zerbes (Exeter)

17 : Cecile Armana (Paris/Barcelona)

Best wishes
Andrei

Taylor’s “Artin II” paper

I couldn’t follow one of the arguments in Taylor’s paper and eventually came up with my own proof of what he needed. Here’s what I was stuck on: the proof of Lemma 2.3. More precisely, let F_1 denote a totally real field, let Gamma be Gal(F_1-bar/F_1), and let rho-bar:Gamma–>PSL_2(Z/5Z) be a group homomorphism. The obstruction to lifting rho-bar to rho:Gamma–>SL_2(Z/5Z) is an element u in H^1(Gamma,+-1). If eps:Gamma–>{+-1} is a character then the obstruction to taking a square root of eps (that is, constructing s:Gamma–>mu_4 with s^2=eps) is, say, v, also an element of H^2(Gamma,+-1). Now assume u=v (and hence u+v=0 because everything has order 2). Taylor, if I’ve read it correctly, seems to imply (at the bottom of p13 of his MS) that the obstructions hence “cancel out” and so we can lift rho-bar to a map rho:Gamma–>GL_2(Z/5Z) with determinant eps. That sounds really plausible! Unfortunately I simply can’t dot the i’s and cross the t’s of that assertion. In the end I gave up and simply came up with a proof of the lifting in the case that we needed. Can anyone explain to me why I’m being dense and why Taylor’s argument is fine? If no-one does by Wednesday afternoon then they’ll have to see my own argument for the lifting in the case at hand (where we are given extra information about rho-bar which I have suppressed here but which you can read about in the statement of Taylor’s Lemma 2.3 and its proof).

Oxford Workshop

Alan Lauder and Kiran Kedlaya are organising a workshop on

Effective methods in p-adic cohomology

in Oxford. March 15-19, 2010.

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.