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Matthew Emerton is smart and helpful April 22, 2009

Posted by David Speyer in Algebraic Geometry, characteristic p.
3 comments

In the comments to my previous post, John Mangual raised a number of questions about the relationship between etale and topological cohomology. Matthew Emerton has posted very thorough answers. If you are interested in the topic, and haven’t read his comments yet, you should.

More F_un November 30, 2008

Posted by Ben Webster in Algebraic Geometry, Number theory, characteristic p, crazy ideas.
9 comments

Incidentally, I hope you’ve all been reading F_un mathematics. Even if you aren’t all that interested in the field with one element, it’s a beautifully designed site and might give you some ideas about pushing Web 2.0 in mathematics a bit further than just blogs. While I like our blog, with all its messy diversity (as my collaborators can tell you, messy diversity is a core component of my mathematical style), F_un mathematics has a much more organized focused feel, which I think maybe more promising for getting actual mathematics done. I also think the division of the posts into “outreach,” “undergraduate,” “graduate,” and “research” has some interesting potential and sort of makes me feel like we should be doing a better job of indicating the background level for our post.

Uniform Position in Characteristic p March 8, 2008

Posted by David Speyer in Algebraic Geometry, characteristic p.
1 comment so far

Let X be an irreducible degree d curve in \mathbb{C} \mathbb{P}^r. Some hyperplanes will be tangent to X, or pass through singularities of X, or even perhaps contain X. But most hyperplanes won’t do any of these things. Let U be the space of hyperplanes which do not exhibit any of these bad behaviors; these hyperplanes are said to be transverse to X. Every hyperplane H in U meets X in d points. As we move H through U, these d points move around X. If H navigates a loop in U, before returning to itself, then these d points are permuted. Last week, in his class on Algebraic Curves, Joe Harris proved the following theorem.

The Uniform Position Principle: Let G and H be two hyperplanes in U, with G \cap X = \{ x_1, \ldots, x_d \} and H \cap X = \{ y_1, \ldots, y_d \}. Then, for any permutation \sigma \in S_d, there is a path from G to H within U such that traveling along this path takes x_i to y_{\sigma(i)}.

In this proof, it was crucial that we were working with a field of characteristic zero. The key lemma was that there was a little loop in U such that traveling around that loop swapped exactly two points. The proof, in sketch, is to find a hyperplane H_0 which is not transverse to U, but just barely; so that H_0 is tangent to X at one point and meets X transversely at d-2 other points. If we then wiggle H_0 in a little disc, then the tangency point of X and H_0 will separate into two distinct points, while the other d-2 points will wiggle around a little. The boundary of that disc will be a loop in U and (exercise!) as you travel around that loop, the two perturbations of the tangency point switch with each other.

In characteristic p, there are curves for which there are no such tangent planes H_0. As I’ll show you soon, there exist curves X where every single point x \in X is a flex, meaning that the tangent line to X at x touches X with degree \geq 3. So the proof falls apart. We got into a discussion in Harris’ class about whether the result also falls apart. I’ve done some computations and the answer is “yes”. Moreover, the monodromy groups we get are very pretty. (more…)

Representations of reductive groups in characteristic p February 12, 2008

Posted by Joel Kamnitzer in Algebraic Geometry, characteristic p, representation theory.
20 comments

I’ve been at a couple of interesting conferences lately and so I have a lot to talk about. I’ll start by summarizing an excellent expository talk by Jonathan Brundan which he gave at an MSRI introductory workshop last week.

Let G be a reductive group over an algebraically closed field k of characteristic p. The topic of this post is the algebraic representations of G. In other works, we want to study algebraic maps G \rightarrow GL(V) where V is a finite dimensional vector space over k. Over the years, a few people (Soroosh, Carl, Alex Ghitza) have asked me what I knew about this theory and I’m afraid that I always gave them very incomplete or inaccurate answers. Now, that I’ve been to Brundan’s talk I think that I understand what is going on much better and I’d like to summarize it. Of course there will be nothing “new” in this post — I think that all the theory was worked out 20 years ago.

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The Cheewhye Diagram December 14, 2007

Posted by Scott Carnahan in Number theory, characteristic p, representation theory, things I don't understand.
17 comments

About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GL_r over function fields, building up to a description of Lafforgue’s proof of the theorem. In the beginning, he drew a diagram that captured the general architecture of the proof, and I liked it so much that I stole it for a talk I gave at Talbot in 2005. It seemed to get a good reception, and Mark Behrens pointed out that the Eichler-Shimura correspondence also fits into the picture with minimal alteration.

If we remove all of the explanatory text, the diagram looks like this:

. \qquad \quad \text{Geometric}
. \qquad \nearrow \qquad \qquad \searrow
\text{Spectral} \qquad \qquad \text{Algebraic}
. \qquad \nwarrow \qquad \qquad \swarrow
. \qquad \quad \text{Analytic}

I was a bit hesitant to draw this, because my advisor once told me, “If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.” Anyway, I’ll explain roughly how it works.

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Topology in prime characteristic December 2, 2007

Posted by Joel Kamnitzer in Algebraic Geometry, characteristic p.
17 comments

This semester in Berkeley, we have been running a learning seminar on Weil conjecture and perverse sheaves following the book by Kiehl and Weissauer. All of this business with “weights” used to scare me a great deal, but after this semester, I feel I understand a bit. I thought I would try to share with everyone a bit of what we have been learning. It will also give me an opportunity to make sure that I understand this stuff.

Let X_0 denote a variety over a finite field \mathbb{F}_q and let X denote its base change to the algebraic closure.

The first important point is that if X is such a variety, then there is a well behaved etale cohomology of X, H^*(X, \mathbb{C}) . This cohomology behaves like you would expect it too — in particular, if there is an obvious version of X over the complex numbers, then typically this etale cohomology will agree with the ordinary cohomology of the complex points of X. The development of etale cohomology is actually quite complicated and was not the subject of our seminar and will not be the subject of this post.

Since X comes from X_0 by base change, X comes with a Frobenius map F : X \rightarrow X . For example if X_0 = \mathbb{A}_1 , then F(x) = x^q . Now, this Frobenius map gives us a endomorphism of the etale cohomology F : H^*(X, \mathbb{C}) \rightarrow H^*(X, \mathbb{C}) .

If X is smooth and proper, then the following amazing theorem is true: The eigenvalues of F acting on H^i(X, \mathbb{C}) are all algebraic integers of size q^{i/2} (this is the “Riemann hypothesis” part of the Weil conjecture).

(There are some notes online of a talk by Beilinson which give some complex geometry motivation for this statement.) (more…)

Small finite sets October 27, 2007

Posted by Scott Carnahan in Algebraic Geometry, Number theory, characteristic p, small examples.
10 comments

Serre has been giving a series of lectures at Harvard for the last month, on finite groups in number theory. It started off with some ideas revolving around Chebotarev density, and recently moved into fusion (meaning conjugacy classes, not monoidal categories) and mod p representations. In between, he gave a neat self-contained talk about small finite groups, which really meant canonical structures on small finite sets.

He started by writing the numbers 2,3,4,5,6,7,8, indicating the sizes of the sets to be discussed, and then he tackled them in order.

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Berkovich spaces I September 18, 2007

Posted by Scott Carnahan in Number theory, characteristic p, representation theory.
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Around 1990, Berkovich discovered a rather elegant refinement of the notion of scheme that is particularly well-suited to non-archimedean analytic geometry. I feel that the ideas here deserve more exposure, although I don’t have any specific applications in mind. Note: this post is really long.

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Frobenius splitting September 9, 2007

Posted by Joel Kamnitzer in Algebraic Geometry, characteristic p, talks.
11 comments

After Ben’s post on “terrible” algebraic varieties, I would like to write about some nice ones. In particular, here is a nice property for an algebraic variety X:

(*) For all ample line bundles L on X, H^i(X, L) = 0 for all i > 0 .

On Tuesday, I went to a talk by Sam Payne in which he explained (among other things) how you could prove such a property using Frobenius splitting. I’d heard of this Frobenius splitting years ago from my advisor, but had no idea how it worked before.

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