Category Archives: Algebraic Geometry

ABC conjecture rumor

There is a rumor circulating here in Japan, to the effect that S. Mochizuki has proved the ABC conjecture. My understanding is that blogs are for spreading such things.

Apparently, he had predicted some years ago that he would finish a proof in 2012, so I suppose this is an “on-time delivery”. It is certainly no secret that his research program has been aiming at the conjecture for several years.

Our very own Noah Snyder did some original work on the function field version of this conjecture, when he was a high-school student.

Update (Sept 4, 2012): This rumor seems to be true! You can find the four “Inter-universal Teichmuller Theory” papers on the very bottom of his papers page.

Fun with y^2=x^p-x

Here’s a basic example that comes up if you work with elliptic curves: Let p be a prime which is 3 \mod 4. Let E be the elliptic curve v^2=u^3-u over a field of characteristic p. Then E has an endomorphism F(u,v) = (u^p, v^p). It turns out that, in the group law on E, we have F^2 = [-p]. That is to say, F(F(u,v)) plus p copies of (u,v) is trivial.

I remember when I learned this trying to check it by hand, and being astonished at how out of reach the computation was. There are nice proofs using higher theory, but shouldn’t you just be able to write down an equation which had a pole at F(F(u,v)) and vanished to order p at (u,v)?

There is a nice way to check the prime 3 by hand. I’ll use \equiv for equivalence in the group law of E. Remember that the group law on E has -(u,v) \equiv (u,-v) and has (u_1,v_1)+(u_2,v_2)+(u_3,v_3) \equiv 0 whenever (u_1, v_1), (u_2, v_2) and (u_3, v_3) are collinear.

We first show that

\displaystyle{ F(u,v) \equiv (u-1, v) - (u+1, v) \quad (\dagger)}

Proof of (\dagger): We want to show that F(u,v), (u+1,-v) and (u-1,v) add up to zero in the group law of E. In other words, we want to show that these points are collinear. We just check:

\displaystyle{ \det \begin{pmatrix} 1 & u^3 & v^3 \\ 1 & u-1 & -v \\ 1 & u+1 & v \end{pmatrix} = 2 v (v^2-u^3+u) = 0}

as desired. \square.

Use of (\dagger): Let (u_0, v_0) be a point on E. Applying F twice, we get

\displaystyle{ F^2(u_0,v_0) \equiv F \left( (u_0-1,v_0) - (u_0+1,v_0) \right)}

\displaystyle{ \equiv (u_0-2,v_0) - 2 (u_0,v_0) + (u_0+2,v_0)}.

Now, the horizontal line v=v_0 crosses E at three points: (u_0, v_0), (u_0-2, v_0) and (u_0+2, v_0). (Of course, u_0 -2 =u_0+1, since we are in characteristic three.) So (u_0-2, v_0) + (u_0, v_0) + (u_0+1, v_0) \equiv 0 and we have

\displaystyle{F^2(u_0, v_0) \equiv -3 (u_0, v_0)}

as desired. \square.

I was reminded of this last year when Jared Weinstein visited Michigan and told me a stronger statement: In the Jacobian of y^2 = x^p-x, we have F^2 = [(-1)^{(p-1)/2} p], where F is once again the automorphism F(x,y) = (x^p, y^p).

Let me first note why this is related to the discussion of the elliptic curve above. (Please don’t run away just because that sentence contained the word Jacobian! It’s really a very concrete thing. I’ll explain more below.) Letting C be the curve y^2 = x^p-x, and letting p be 3 \mod 4, we have a map C \to E sending (x,y) \mapsto (y x^{(p-3)/4}, x^{(p-1)/2}), and this map commutes with F. I’m going to gloss over why checking F^2 = [(-1)^{(p-1)/2} p] on C will also check it on E, because I want to get on to playing with the curve C, but it does.

So, after talking to Jared, I was really curious why F^2 acted so nicely on the Jacobian of C. There are some nice conceptual proofs but, again, I wanted to actually see it. Now I do.

Continue reading →

Rationality of the zeta function mod p

Here’s a neat argument about counting points that you could present at the end of a second course in number theory. I’m sure it’s not original, but, hey, that’s what blogs are for!

Let X be a smooth hypersurface in \mathbb{P}^{n}, over the field \mathbb{F}_p with p elements. The Weil conjectures are conjectures about the number of points of X over \mathbb{F}_{p^k}. Specifically, they say that there should be some matrix A such that

\displaystyle{ \# X(\mathbb{F}_{p^k}) = 1+p^k+p^{2k} + \cdots + p^{(n-1)k} + (-1)^{n-1} \mathrm{Tr} (A^k),}

and that the eigenvalues of A should be algebraic integers of norm p^{(n-1)/2}.
Here I am using the Lefschetz hyperplane theorem to know what H_{et}^i(X) is for i \neq n-1.

This is, of course, a famously hard theorem. The claim about the eigenvalues is the hardest part, but simply the existence of a matrix for which this formula holds is already quite hard; the first proof was due to Dwork.

What I am going to show you is that there is a much easier proof of the above formula modulo p; a proof of the sort that could be appear in Ireland and Rosen. Many of the terms above disappear mod p, so our goal is just to show that there is some matrix B such that

\displaystyle{ \# X(\mathbb{F}_{p^k}) \equiv 1 + (-1)^{n-1} \mathrm{Tr} (B^k) \mod p.}

Continue reading →

Diagrammatic and geometric categorification

In some previous posts, I’ve written about the relationship between categorification and geometry. I’m banging on this because I think it’s an underappreciated point, especially because a lot of people right now are coming at categorification with a background in topology and higher algebra, and thus aren’t as familiar with the geometric and representation theoretic techniques that actually underlie a lot of, say, what I do.

In particular, a lot of people (myself among them) are very excited about the categorification of quantum groups at the moment and there are two different ways of approaching said categorification:

  1. You can write down a bunch of algebras and categories in terms of generators and relations. This is the approach used in the work of Chuang and Rouquier and of Khovanov and Lauda. It has a lot of advantages: there are lots of parameters one can tweak (which of course, leads to the issue of different authors tweaking them differently), it’s very generally applicable, and it doesn’t really require any machinery one doesn’t acquire in a graduate algebra class, aside from a little terminology.
  2. You can take a geometric approach and work with varieties of quiver representations. This work was pioneered by Lusztig, though it has seen some very interesting developments in recent years in the work of Zheng and Li. This approach requires some very deep technology, involving perverse sheaves, etc. and is less flexible (in particular, it’s still not properly understood how to extend it to the case of non-symmetric Cartan matrix). On the other hand, geometry gives us powerful tools to show that things that are supposed to be non-zero or positive really are. In particular, certain properties of canonical bases require some quite deep geometric theorems.

The point I want to make in this post is that these approaches are the same. Continue reading →

My course begins

I just put up lecture notes for the first lecture from my course Algebraic Geometry II, a course on the complex approach to algebraic geometry, loosely taught out of Claire Voisin’s book.

The mathematical content of my opening lecture is something I have often considered as a blog topic: Seven ways of computing the cohomology of S^1. I think a lot of you will like it.

I am going to have my students take turns preparing electronic notes, which I will edit and post on the course website. Come read along!

Notes to date:
Differential forms
Sheaves
Sheaf cohomology
Sheaf cohomology II, de Rham cohomology
Complex differential forms
Dolbeault’s Lemma
Cohomology Vanishes on Polydiscs
deRham computation of topological cohomology of affine varieties — preview
Cartan’s Lemma
Gluing resolutions

Passage from compact Lie groups to complex reductive groups

Once again, I’m preparing to teach a class and needing some advice concerning an important point. I’m teaching a course of representation theory as a followup to an excellent course on compact Lie groups, taught this semester by Eckhard Meinrenken. In my class, I would like to explain transition from compact Lie groups to complex reductive groups, as a first step towards the Borel-Weil theorem.

A priori, compact connected Lie groups and complex reductive groups, seem to have little in common and live in different worlds. However, there is a 1-1 correspondence between these objects — for example U(n) and GL_n(\mathbb{C}) are related by this correspondence. Surprisingly, it is not that easy to realize this correspondence.

Let us imagine that we start with a compact connected Lie group K and want to find the corresponding complex algebraic group G. I will call this process complexification.

One approach to complexification is to first show that K is in fact the real points of a real reductive algebraic group. For any particular K this is obvious — for example S^1 = U(1) is described by the equation x^2 + y^2 = 1. But one might wonder how to prove this without invoking the classification of compact Lie groups. I believe that one way to do this is to consider the category of smooth finite-dimensional representation of the group and then applying a Tannakian reconstruction to produce an algebraic group. This is a pretty argument, but perhaps not the best one to explain in a first course. A slightly more explicit version would be to simply define G to be Spec (\oplus_{V} V \otimes V^*) where V ranges over the irreducible complex representations of K (the Hopf algebra structure here is slightly subtle).

In fact, not only is every compact Lie group real algebraic, but every smooth map of compact Lie groups is actually algebraic. So the
the category of compact Lie groups embeds into the category of real algebraic groups. For a precise statement along these lines, see this very well written
MO answer by BCnrd.

A different approach to complexification is pursued in
Allen Knutson’s notes and in Sepanski’s book. Here the complexification of K is defined to be any G such that there is an embedding K \subset G(\mathbb{C}) , such that on Lie algebras \mathfrak{g} = \mathfrak{k} \otimes_{\mathbb{R}} \mathbb{C} . (Actually, this is Knutson’s definition, in Sepanski’s definition we first embed K into U(n) .) This definition is more hands-on, but it is not very obvious why such G is unique, without some structural theorems describing the different groups G with Lie algebra \mathfrak{g} .

At the moment, I don’t have any definite opinion on which approach is more mathematically/pedagogically sound. I just wanted to point out something which I have accepted all my mathematical life, but which is still somewhat mysterious to me. Can anyone suggest any more a priori reasons for complexification?

Some questions about motives

As I tried to read up on motives in preparation for my last post, I thought of some questions that seemed natural to me, but weren’t addressed in the sources I was reading. So here they are, in the hope that experts will find them obvious. One set of questions concerns motives for nonproper varieties, the other concerns higher categorification.

Continue reading →

Motive-ating the Weil Conjecture Proof

This post concludes a series of posts I’ve been writing on the attempt to prove the Weil Conjectures through the Standard Conjectures. (Parts 1, 2, 3, 4, 5.) In this post, I want to explain the idea of the category of motives. In the modern formulation of algebraic topology, cohomology theories are functors from some category of spaces to the category of abelian groups. The category of motives is meant to be a universal category through which any such functor should factor, when the source space is the category of algebraic varieties. At least in the early days of the subject, the gold test of this theory was the question of whether the Weil Conjectures could be proved entirely in this universal setting. Nowadays, this question is still open, but the use of motives has grown. To my limited understanding, this growth has two reasons: among number theorists, it has become clear that motivic language is an excellent way to formulate results on Galois representation theory; among birational geometers and string theorists, many applications have been found for motivic integration. There will be a bunch of category theory in this post, which I hope will make it more attractive to the tensor category crowd.

I am much less comfortable with this topic than the other posts in this series; my understanding doesn’t go much further than Milne’s survey article. So I’m going to make this post a pretty short introduction to the main ideas. That will be the end of my expository posts; I also want to write one more post raising some questions about motives that seem natural to me.

Continue reading →

The Weil Conjectures: The approach via the Standard Conjectures

The aim of this post is to outline a strategy for proving the Weil conjectures, proposed by Grothendieck and others. This strategy is incomplete; at various stages, we will need to assume conjectures which are still open today.

Our aim is to prove:

Theorem:
Let X be a smooth projective variety, over a field of any characteristic. Let H^* be a “reasonable” cohomology theory. Let \omega \in H^2(X) be the hyperplane class for a projective embedding of X. Let F be an automorphism of X, such that F^* \omega = q \omega. Then the eigenvalues of F^*: H^r(X) \to H^r(X) are algebraic numbers and, when interpreted as elements of \mathbb{C}, have norm q^{r/2}.

In previous posts, we gave for a proof in characteristic zero and a proof in the case that X is a curve. I also explained why I need to say how I am embedding these eigenvalues into \mathbb{C}. Our proof requires all the ideas of these previous posts, plus some new ones.
Continue reading →

Equivariant cohomology wiki

A few weeks ago, I went to a superb conference at AIM on Localization Techniques in Equivariant Cohomology. A frequent topic at that workshop was the lack of good surveys and references. Results from the 70s were being rediscovered, and no one knew where to start reading.

As a partial solution, the conference organizers have set up a wiki. There is a list of references, particularly on Schubert calculus and related problems, and of open problems. They are keeping editing restricted to selected users, but I imagine many of our readers would make excellent editors. Please take a look and, if you have anything to add, e-mail Rebecca Goldin or Julianna Tymoczko to become an editor.