Category Archives: topology

SNU Topology of number fields, week of 15 July

I was looking over the notes and saw a few items omitted by Milne.

First, in the proof of Hilbert’s theorem 90, there is no proof of the linear independence of distinct characters on a group. You can try to figure it out yourself (quite easy) or look at the notes with that title Keith Conrad’s webpage. (For some strange reason, I can’t create a direct link to the paper.)

Secondly, Milne has a cryptic remark without definition or proof about a ‘Verlagerung’ (transfer map) in Proposition 3.2. This is reference to the fact that the restriction map for Tate cohomology in degree minus 2 corresponds to a classical formula for a map between abelianizations. This proof is not so straightforward, and can be found in Prop.11.12 of these nice notes by Holden Lee. It seems they’re from a reading course he took with Professor Shin Sug Woo.

If some of you would like to read a more leisurely exposition of group cohomology, I found online these notes by Kenneth Brown.

Finally, this old paper of Oort contains a discussion of extensions and cup products.

SNU Topology of Number Fields

Here are some documents to help you with the course.

1. General Motivation:

Lecture at the Cambridge workshop on non-abelian fundamental groups in arithmetic geometry (2009)

Lecture at Bordeaux conference in honor of Martin Taylor’s 60th birthday (2012)

Lecture at British Mathematical Colloquium (2011)

Lecture at AMC, Busan (2013)

2. James Milne’s lecture notes:

Class Field Theory

Algebraic Number Theory

Arithmetic Duality Theorems

3. Langlands programme:

Langlands’s lecture at Helsinki ICM

Report on work of Bao Chau Ngo (with Sugwoo Shin)

4. Topology of number fields:

Mazur’s paper on etale cohomology of number fields

Morishita’s paper on knots and primes

Furusho’s paper on Galois action on knots

Kings lecture

Since arriving in the UK, I gave colloquium lectures at QMUL, Leeds, Durham, and Exeter. A `colloquium’ is when you lecture for about an hour to the whole mathematics faculty on the topic of your research. It’s considerably harder to prepare than specialized seminars (for example, the London-Paris number theory seminar) where you are speaking to people whose expertise is typically similar to your own. That is to say, the background of the audience is much more varied for a colloquium. Anyways, I acquired this year the bad habit of preparing my lecture at the last minute on the train. Probably because of this, the four colloquium talks turned out disastrously. I couldn’t stay focussed, and I think I lost everyone pretty soon after I started. This is in spite of the fact that I was constantly recycling material from the previous colloquium. Anyways, today I delivered a colloquium lecture at Kings, and my impression was that it was finally tolerable on the fifth try. So I’m putting a link here. Because these are just summaries of what was said, it probably looks no different from the previous lectures if you weren’t there, and maybe even if you were.

Kazuya Kato

The great arithmetician Kazuya Kato visited twice over the last few weeks, so I thought I’d use the occasion to recommend some writings. An undergraduate level textbook on number theory is

Number Theory I: Fermat’s Dream

published by the American Mathematical Society. It is short and covers fairly standard material, but contains many unusual insights. A research article that represents quite well Kato’s vision of number theory is

Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. Part I

available online if your institution subscribes to Springer. The background required for a genuine reading of the paper is rather extensive, but even without it, you can enjoy the introduction, the first few sections, and the closing remarks.

A main theme of the work is that the somewhat mysterious p-adic zeta functions and L-functions are the objects with direct relevance to the important problems of arithmetic geometry, while the usual complex functions are a sort of intermediary. It took me a long time to come to terms with this view, especially since I still don’t understand these functions (actually elements of some non-commutative algebra of measures) at all well, but it is eventually an essential component of my own thoughts about Diophantine geometry.


Fundamental groups and Diophantine geometry

A few weeks ago, I gave a colloquium lecture at Leeds university and subsequently wrote up an exposition based on it. It’s still not entirely `popular,’ but may give a somewhat better sense than my previous remarks of at least a few ideas.

Let me know if there are some points on which you would like elaboration.


Non-commutative constructions

In June of next year, there will be a small workshop at UCL with the title

`non-commutative constructions in arithmetic and geometry.’

I will write later in more detail about the goals of the meeting, but here is a link to a preliminary web page.


Fundamental groups

Yesterday, at the last algebraic number theory lecture of the term, I defined the fundamental group \pi_1(O_K) of the ring of algebraic integers O_K in an algebraic number field K as the Galois group Gal(F/K), where F is the compositum of all the unramified extensions of K. And then, it was stated that the ideal class group Cl_K that we’ve been studying is just the abelianization of \pi_1(O_K). The point is that it seems reasonable now to start making computations of the whole \pi_1 rather than just the class group, if we want to capture finer invariants reflecting the complexity of a field. Of course, I’m not sure yet how to begin! But that’s what the best problems are like. Incidentally, the definition I gave isn’t quite right. One can give a better definition of \pi_1(O_k), closer to topological loops, after which the isomorphism with the Galois group becomes a theorem.

One of these days, I really will write some expository introduction at the undergraduate level. In the meanwhile the connection to topological fundamental groups is alluded to in the London-Paris lecture mentioned in an earlier post.

The basic reference in the field explaining this view is the great work Séminaire de Géométrie Algébrique, Vol. I by Alexandre Grothendieck and his students. Of course it’s hard to read if you’ve only now taken my course. But it’s good to get a sense of the classics even at this stage.


Spec(Z) and three-manifolds

The following question comes from an old friend, John Baez. John is a renowned mathematical physicist and internet guru. You should look at his website and his blog for an enormously informative and entertaining survey of mathematics and physics.

My reply is here.


Hi! Why is Spec(Z) 3-dimensional?

I think I understand what the number theorist James wrote in response to “week257” of This Week’s Finds:

“Instead of considering Z, let’s consider F[x], where F is a finite field. They are both principal ideal domains with finite residue fields, and this makes them behave very similarly, even on a deep level. I’ll explain why F[x] is three-dimensional, and then by analogy we can hope Z is, too. Now F[x] is an F-algebra. In other words, X = Spec(F[x]) is a space mapping to S = Spec(F). I already explained why S is a circle from the point of view of the etale topology. So, if X is supposed to,be three-dimensional, the fibers of this map better be two-dimensional. What are the fibers of this map? Well, what are the points of S? A point in the etale topology is Spec of some field with a trivial absolute Galois group, or in other words, an algebraically closed field (even better, a separably closed one). Therefore a etale point of S is the same thing as Spec of an algebraic closure Fbar of F. What then is the fiber of X over this point? It’s Spec of the ring Fbar[x]. Now, *this* is just the affine line over an algebraically closed field, so we can figure out its cohomological dimension. The affine line over the complex numbers, another algebraically closed field, is a plane and therefore has cohomological dimension 2. Since etale cohomology is kind of the same as usual singular cohomology, the etale cohomological dimension of Spec(Fbar[x]) ought to be 2.

Therefore X looks like a 3-manifold fibered in 2-manifolds over Spec(F), which looks like a circle. Back to Spec(Z), we analogously expect it to look like a 3-manifold, but absent a (non-formal) theory of the field with one element, Z is not an algebra over anything. Therefore we expect Spec(Z) to be a 3-manifold, but not fibered over anything.”

But, it would be nice to go a bit deeper than mere “reasoning by analogy with Fbar[x]”, without yet diving into a full-blown calculation of the etale cohomology of Spec(Z). Can you say anything that’s helpful and fun? related question is this. Nowadays various people are making serious attempts to define the field with one element and do algebraic geometry over this field. With these, Z *is* an algebra over the field with one element. How much does this help?