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A (partial) explanation of the fundamental lemma and Ngo’s proof September 24, 2009

Posted by Joel Kamnitzer in Algebraic Geometry, Number theory, geometric Langlands, representation theory, things I don't understand.
2 comments

I would like to take Ben up on his challenge (especially since he seems to have solved the problem that I’ve been working on for the past four years) and try to explain something about the Fundamental Lemma and Ngo’s proof.  In doing so, I am aided by a two expository talks I’ve been to on the subject — by Laumon last year and by Arthur this week.

Before I begin, I should say that I am not an expert in this subject, so please don’t take what I write here too seriously and feel free to correct me in the comments.  Fortunately for me, even though the Fundamental Lemma is a statement about p-adic harmonic analysis, its proof involves objects that are much more familiar to me (and to Ben).  As we shall see, it involves understanding the summands occurring in a particular application of the decomposition theorem in perverse sheaves and then applying trace of Frobenius (stay tuned until the end for that!).

First of all I should begin with the notion of “endoscopy”.  Let G, G' be two reductive groups and let \hat{G}, \hat{G}' be there Langlands duals.  Then G' is called an endoscopic group for G if \hat{G}' is the fixed point subgroup of an automorphism of \hat{G} .  A good example of this is to take G = GL_{2n} , G' = SO_{2n+1} .  At first glance these groups having nothing to do with each other, but you can see they are endoscopic since their dual groups are GL_{2n} and Sp_{2n} and we have Sp_{2n} \hookrightarrow GL_{2n} .

As part of a more general conjecture called Langlands functoriality, we would like to relate the automorphic representations of G to the automorphic representations of all possible endoscopic groups G' .  Ngo’s proof of the Fundamental Lemma completes the proof of this relationship.

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The Fundamental Lemma September 13, 2009

Posted by Ben Webster in Algebraic Geometry, fields medals.
12 comments

So, Noah came up to New England a few days ago, and at some point over dinner, the topic of Fields Medal candidates came up. Neither of us had any good ideas (sorry, anonymous grad student) but I mentioned that I had heard Bao Châu Ngô’s name quite a bit. The conversation then went roughly like this: (more…)

Algebraic geometry without prime ideals August 6, 2009

Posted by Joel Kamnitzer in Algebraic Geometry, Anton Geraschenko, things I don't understand.
74 comments

The first definition in “Grothendieck-style” algebraic geometry is the affine scheme Spec R for any ring R. This is a topological space whose set of points in the set of prime ideals in R. Then one defines a scheme to be a locally ringed space locally isomorphic to an affine scheme.

The definition of Spec R goes against intuition since it involves prime ideals, not just maximal ideals. Maximal ideals are more natural, since if R = k[x_1, \dots, x_n]/I for some alg closed field k , then the set of maximal ideals of R is in bijection with the vanishing set in the affine space k^n of the ideal I . (Of course one can give a geometric meaning to the prime ideals in terms of subvarieties, but it is less natural.)

However, in Daniel Perrin’s text Algebraic geometry, an introduction, he states/implies that one can define affine schemes just using maximal ideals (at least for finitely-generated k algebras) and still get a good theory of schemes and varieties. Is this true?  If so why don’t we all learn it this way? (One answer to the this latter question could be that some people are interested in non-algebraically closed fields.)

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Quaternions and Tensor Categories July 30, 2009

Posted by David Speyer in Algebraic Geometry, Category Theory, things I don't understand.
27 comments

As you can tell from the title of this post, I am trying to drag John Baez over to our blog.

Let Q be the ring of quaternions, i.e., \mathbb{R} \langle i,j,k \rangle with the standard relations. Let Q-mod be the category of left Q-modules. This has an obvious tensor structure (including duals), inherited from the category of \mathbb{R} vector spaces. Actually, that structure doesn’t quite work; I’ll leave to you good folks to work out what I should have said.

Let q=a+bi+cj+dk be a quaternion. Anyone who works with quaternions knows that there are two notions of trace. The naive trace, 4a, is the trace of multiplication by a on any irreducible Q-module, using the obvious tensor structure. But there is a better notion, the reduced trace, which is equal to 2a. Similarly, there is a naive norm, (a^2+b^2+c^2+d^2)^2, and there is a reduced norm a^2+b^2+c^2+d^2.

This all makes me think that there is a subtle tensor category structure on Q-mod, other than the obvious one, for which these are the trace and norm in the categorical sense. Can someone spell out the details for me?

By the way, a note about why I am asking. I am reading Milne’s excellent notes on motives, and I therefore want to understand the notion of a non-neutral Tannakian category (page 10). As I understand it, this notion allows us to evade some of the standard problems in defining characteristic p cohomology; one of which is the issue above about traces in quaternion algebras.

Topology that Algebra can’t see July 28, 2009

Posted by David Speyer in Algebraic Geometry, Number theory.
15 comments

Let X be an algebraic variety over \mathbb{C}; that is to say, the zero locus of a bunch of polynomials with complex coefficients. We will consider this zero locus as a topological space using the ordinary topology on \mathbb{C}. One of the main themes of algebraic geometry in the last century has been learning how to study the topology of X in terms of the algebraic properties of the defining equations.

In this post, I will explain that there are intrinsic limits to this approach; things that cannot be computed algebraically. In particular, I want to explain how from a categorical point of view, we can’t even compute the homology H_1( \ , \mathbb{Z}). And, even if you don’t believe in categories, you’ll still have to concede that we can’t compute \pi_1( \ ). This is a very pretty example and it should be more widely known.

Absolutely none of the ideas in this post are original; I think most of them are due to Serre. (Thanks to Attila Smith in comments for the reference.)

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Continued Fractions and Hyperelliptic Curves July 2, 2009

Posted by David Speyer in Algebraic Geometry, Number theory.
4 comments

I recently read a charming little paper: Quasi-elliptic integrals and periodic continued fractions, by van der Poorten and Tran. Most of us who have taken a number theory course of some kind learned how to solve Pell’s equation: x^2 - D y^2 =1 where D is a nonsquare positive integer. The usual method is to compute the continued fraction
\displaystyle{\sqrt{D} = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots}}}}.
One then defines the convergents of \sqrt{D} by
\displaystyle{x_0/y_0 = a_0}
\displaystyle{x_1/y_1 = a_0 + \frac{1}{a_1}}
\displaystyle{x_2/y_2 = a_0 + \frac{1}{a_1+\frac{1}{a_2}}} etcetera.

Then x_i^2 - D y_i^2 tends to be very small and, if you compute long enough, for some i you will have x_i^2 - D y_i^2=1.

What van der Poorten and Tran do is to ask what happens if D is not an integer, but a polynomial D(t) = t^{2g+2} + d_{2g+1} t^{2g+1} + \cdots + d_1 t + d_0. Before I get into details, I want to tell you about something gorgeous that I won’t explain at all. Using the methods in their paper, van der Poorten and Trap can discover identities like
\displaystyle{ \int \frac{3 x dx}{\sqrt{x^4+2x}} = \log \left( x^3+1+x \sqrt{x^4+2x} \right)}.
Isn’t that pretty?

It turns out that the continued fraction algorithm for \sqrt{D(t)} is actually much prettier than for integers. Everything should be understood in terms of the curve C cut out by y^2 = D(t). This is a curve of genus g, with two points at infinity. (One of these points is the limit of (t, \sqrt{D(t)}) and the other is the limit of (t, -\sqrt{D(t)}).) I’ll call these two points \infty_{+} and \infty_{-}. The theory is controlled by the line bundles \mathcal{O}(k \infty_+ + \ell \infty_-). In particular, there are nontrivial solutions to x(t)^2 - D(t) y(t)^2 =1 if and only if the continued fraction is periodic, if and only if \mathcal{O}(k \infty_+) = \mathcal{O}(k \infty_-) for some a >0.

Below the fold, I’ll explain what is meant by the continued fraction algorithm for an algebraic function, and tell you some of the other nice results from the paper.

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Local Systems: The connection perspective June 15, 2009

Posted by David Speyer in Algebraic Geometry, D-modules.
5 comments

Welcome to the next installation of my series on local systems. In this post I’ll be talking about connections. This post should require less sophistication than the last few — no schemes, no functors — I’ll almost be coming at the subject afresh. There will be another post later, explaining how you might get to connections if you started out thinking about the infinitesimal site.

To start out with, let’s talk about derivatives; ordinary, single variable calculus derivatives. We have a function f of a variable x. Then the derivative of f is the function f'(x) := \lim_{h \to 0} (f(x+h) - f(x))/h. There are two directions in which we might want to generalize this idea. The first is to work with functions on a manifold, on a space which has no inherent coordinate system. This is the subject of your standard Calculus on Manifolds course, and I am going to assume that my readers are at least vaguely familiar with it. The second is to work, not with functions, but with sections of vector bundles. That’s our subject in this post.

So, let’s think about a vector bundle V on the line \mathbb{R}, and let \sigma be a section of V. If we want to define \sigma', we need to subtract \sigma(x+h) and \sigma(x), two vectors which live in different fibers. To think of it another way, we need to distinguish between f(x), the point in the fiber over x, and f(x), the constant function which assigns the same value at every point. Suppose that, for any v \in V_x, we had a local section c_v of V with c_v(x)=v; we think of c_v as a constant function. Then we could define \sigma'(x) = \lim_{h \to 0} \left( \sigma(x+h) - c_{\sigma(x)}(x+h) \right)/h.

A local system gives us the constant functions c_v. (Indeed, in definitions A.2 and B.6, we took a local system to be the constant functions, along with the data of certain maps between them.) Today, we will take the fundamental object to be the operation of derivation, and see how to build everything else from it.

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Bleg: Independence of points in Picard group May 27, 2009

Posted by David Speyer in Algebraic Geometry.
2 comments

For reasons discussed in my post below, I would like to construct a curve \overline{S} (over \mathbb{C}) and a finite map \overline{F}: S \to \mathbb{C}) with the following properties:

(1) The map \overline{F} is ramified somewhere. (This will be automatic if \overline{F} is not an isomorphism.) Let R \subset \overline{S} be the ramification locus and let S be \overline{S} \setminus R.

(2) The coordinate ring of S has no nontrivial units. This can be thought of as saying that R is not too large.

I think I have such a construction. But my argument that S has no units is basically that I need a bunch of elements of a Picard group to be linearly independent, and I can’t see any relations between them. Obviously, this needs some help! Details follow:

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How not to prove the Jacobian conjecture May 27, 2009

Posted by David Speyer in Algebraic Geometry.
5 comments

A few years ago, I got curious to see how people attack the Jacobian conjecture. The Jacobian conjecture says the following:

Let k be a field of characteristic zero, and let F: k^n \to k^n be an unramified map. Then F is an isomorphism.

The conjecture is usually stated algebraically, not geometrically. That statement goes as follows: Let the map F be given by \displaystyle{(x_1, \ldots, x_n) \mapsto \left( f_1(x_1, \ldots, x_n),\ \ldots,\ f_n(x_1, \ldots, x_n) \right)}.
Define J(F), the Jacobian of F, to be \det(\partial f_i/\partial x_j). Then the algebraic statement is

Let f_1,f_2, …, f_n are n polynomials in n variables such that J(f_1, \ldots, f_n) is in k^*. Then the f_i generate the ring k[x_1, \ldots, x_n].

The geometric meaning of J(F) is that J(F) vanishes at the points where F is ramified1.

I have noticed a pattern in many of the false proofs. Today, I’m going to tell you how to spot these proofs. The warning sign is the phrase “but k[x_1, \ldots, x_n] has no nontrivial units.”

Disclaimer: I am not an expert on the Jacobian conjecture. I will not referee manuscripts on the subject unless they are relevant to my published work in some specific way. Please do not send me preprints on the subject (unless you are a personal friend).

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The Grothendieck trace formula as categorification, II: the function-sheaf correspondence May 21, 2009

Posted by Ben Webster in Algebraic Geometry.
12 comments

So, I started this series a while back with this post, by talking about some sheaves on an algebraic variety.  This post will probably not make a lick of sense to those who haven’t read the first part yet, though you’re welcome to try.  The upshot of that was that one has a machine into which one can can feed in

  1. a variety X defined over \mathbb Z (i.e. defined by polynomials with integer coefficients)
  2. a choice of stratification of X (a good example might be the orbits of an algebraic group with finitely many orbits, so for example, the Schubert cells on a Grassmannian)
  3. a set of acceptable local systems valued in your favorite field k (though, if want to do things properly, your favorite field should probably be \mathbb Q_\ell).

and receive out the other end a category.  The cool fact about this category is that you can think of it as sheaves on the complex variety X_{\mathbb{C}} or its analogue X_{\bar{\mathbb{F}}_p} over characteristic p (let p be your favorite prime) and you will get the same answer.  The former category has the advantage of involving geometry that you probably care about, like the cohomology of smooth varieties, and the latter has the advantage that there is a Frobenius acting.

What I’d like to explain in this post how to analyze the structure of this category, and what that has to do with categorification.   This will require a bit of machinery, but believe me, the result will justify it.

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