User regenbogen - MathOverflow

Answer by Regenbogen for Quick proofs of hard theorems

2010-05-16T20:32:06Z

The fundamental theorem of algebra is a very easy consequence of Liouville's theorem in complex analysis.

There is also the (even simpler) proof due to Schep.

Conceptual understanding of the Gross-Zagier theorem.

2010-02-24T22:40:22Z

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more conceptual understanding.

The more conceptual attempts I know are the following:

$1$. The work of Kolyvagin on Birch-Swinnerton-Dyer conjecture, in which he re-proves part of Gross-Zagier using Euler systems. The problem with this is that some of the original Gross-Zagier is still needed for getting the results on BSD conjecture(if I understand things correctly. Please point out if I am wrong).

$2$. The volume of Darmon and Zhang published by MSRI, in which they attempt a $p$-adic theory. Again this is going away from the original complex analytic case. Again please correct me if I am wrong.

So I am wondering whether anybody published a more conceptual approach to the complex analytic Gross-Zagier theorem. I would be grateful for any references.

Embedding algebraic integers as a lattice

2010-03-22T15:55:20Z

I am sorry for this rather dumb-sounding question. But I am thinking of it for the last two days and am unable to find an answer.

Let $K, L$ be an algebraic number fields, ie a finite extensions of $\mathbb Q$. Let $F = KL$ be their compositum. Let $\mathcal O_K, \mathcal O_L, \mathcal O_F$ be the rings of integers of these fields. Let $n(K), n(L), n(F)$ be the degrees of these number fields.

Now, we have a triplet $(K, \mathcal O_K, i_{K} : \mathcal O_K \hookrightarrow \mathbb{R}^{n(K)})$, where $i_K$ is the embedding $x \mapsto (x^{(1)}, x^{(2)}, \ldots , x^{(n(K))})$, where $x^{(1)}, x^{(2)}, \ldots , x^{(n(K)}$ are the conjugates of $x$.

What bothers me in this situation is that the conjugates depend on the number field. To alleviate this situation, I go ahead as follows.

We consider the similar triplets $(L, \mathcal O_K, i_{L} : \mathcal O_L \hookrightarrow \mathbb{R}^{n(L)})$ and $(F, \mathcal O_F, i_{F} : \mathcal O_F \hookrightarrow \mathbb{R}^{n(F)})$. There is a natural embedding $\mathcal O_K \hookrightarrow \mathcal O_F$ and similarly $\mathcal O_L \hookrightarrow \mathcal O_F$ and these respect the inclusions of the number rings into their respective Euclidean spaces. So we can "include" the triplets for $K$ and $L$ into $F$.

Now, we order all number fields via inclusion. This is a directed set. And the set of all triplets considered above of the form $(K, \mathcal O_K, i_{K} : \mathcal O_K \hookrightarrow \mathbb{R}^{n(K)})$, is a directed system of such triplets. So we take the direct limit. The result should be some embedding of the ring of all algebraic integers into a countable dimension Euclidean space.

Question:

Does this embedding give a lattice?

I would be grateful for answers. Again I am sorry if this is a stupid question.

Avoiding Minkowski's theorem in algebraic number theory.

2010-03-22T15:00:54Z

For any course in algebraic number theory, one must prove the finiteness of class number and also Dirichlet's unit theorem. The standard proof uses Minkowski's theorem. Is there a way to avoid it?

The reasons I am asking this question are the following.

$1$. Minkowski lived long after Dirichlet and Dedekind(esp Dirichlet). So the original proof cannot likely have used Minkowski's theorem as such. If the original proof did use Minkowski's theorem, then it was of course found by someone else, most probably Dirichlet, and it is unfair to use the name Minkowski's theorem.

$2$. Even more importantly, the finiteness of classnumber and some version of unit theorem is true(at least I hope so) for all global fields. And there of course one cannot talk of Minkowski's theorem.

The objection I have for Minkowski's theorem is that it seems to be ad hoc, coming out of nowhere. And it seems that not much work is going on nowadays in the subject of geometry of numbers.

So it will be really nice to have a method which would feel more natural and is perhaps more general.

What geometric properties do properties of ell-adic Galois representations imply?

2010-03-12T20:33:16Z

This is the converse question to an earlier question. More precisely,

Let $X/K$ be a curve(or variety) over a global field $K$. We consider the Galois representation obtained by the absolute Galois group $G_K$ acting on $H_{et}^i(X_{/\bar K}, \mathbb{Q}_\ell)$.

Do properties of this representation, such as "unramified at a place $v$", semistable, de Rham, crystalline, Hodge-Tate, and so on and so forth, imply some geometric properties about $X_{/K}$? (I must confess that I know the proper definition of only the first property in this list, but I nevertheless put in the whole list from the original question for good measure.)

If so, please give examples.

Conjugacy classes in the absolute galois group

2010-03-10T17:02:05Z

We consider $G_{\mathbb Q} = Gal(\mathbb {\bar Q}/\mathbb Q)$. The Frobenius elements corresponding to each prime are well-studied. But these are really not elements; these are only defined as some conjugacy classes(upto inertia, etc..)

Question: Are these the only conjugacy classes in the absolute Galois group? If there are others, please give examples or methods to construct them.

The conjugacy classes are of course defined algebraically; this question is not asking for results of the form that the Frobenii form a dense set.

Algebraic number theory and applications to properties of the natural numbers.

2010-02-24T15:43:12Z

Please allow me, for the purposes of this question(but only here), to exaggerate matters and state two polemic definitions. Please forget these definitions after answering this question, and pardon my silly nitpicking.

Definition $1$: "Algebraic number theory" is the theory of algebraic numbers. We exclude arithmetic geometry and such.

Definition $2$: "Number theory" is the study of properties of natural numbers.

In the above sense, I seek examples of applications of algebraic number theory to number theory. I mean, those applications which throw light on "numbers" as we know them in primary school. There is of course enlightenment by looking at a bigger picture of so many number rings, but that is not what I mean. I have specifically the application to the down-to-earth integers in mind. What I know are the following:

$1$. The theorem that an odd prime is of the form $a^2 + b^2$ if and only if it is of the form $4n +1$, proved by looking at factorization in the Gaussian ring.

$2$. Pell's equation is solved with Dirichlet's unit theorem.

$3$. von Staudt–Clausen theorem on Bernoulli numbers, proved using cyclotomic theory.

$4$. Certain equations, like the Fermat equation $X^n + Y^n = Z^n$, may "split" in some extension field and thus it makes sense to go to bigger rings, to study diophantine equations. Here I mean the work of Kummer which started ideal theory, algebraic number theory, etc..

I exclude the following:

$5$. Arithmetic geometry can be used together with algebraic geometry, to study diophantine equations. Elliptic curves fall in here, when their geometry is used significantly(such as in the work of Katz-Mazur). That is "arithmetic geometry", for the purposes of this question. I am more interested in hearing about applications of "algebraic number theory", as defined above.

$6$. Again using algebraic geometry and also modular forms, conjectures such as the Ramanujan bound on the tau function can be proved. Here "modular forms" are "analytic", or "transcendental", and also "geometry is involved. So it goes beyond the "algebraic number theory"

$7$. Dirichlet's theorem on arithmetic progressions is "analytic number theory".

So I thus exclude any touch of "analytic number theory" and "arithmetic geometry", from "algebraic number theory" as defined above. But it can include Kummer theory, classfield theory, etc.. I do not know where to put in Dorian Goldfeld's results on the Gauss class number problem. It uses Gross-Zagier, which is significantly geometric, but gives a result expressible in terms of rational integers. Also I do not know whether Iwasawa theory is arithmetic geometry or not. Langlands theory etc., must be excluded, because it is even more abstract. I want only the "first course in algebraic number theory", "basic cyclotomic theory", "classfield theory" etc., in short only those things which are obviously the study of algebraic numbers.

So, question:

Are there other applications of "algebraic number theory" to "study of natural numbers", than the examples 1-4 above?

I tag this question "big-list" because I hope there are indeed quite a few.

Origins of functional field arithmetic

2010-02-25T01:44:59Z

Background: By function field, we mean a finite extension of the field of rational functions of one variable over a finite field with $p$ elements. Classfield theory for function fields was established by Chevalley in an Annals paper. An axiomatic characterization for number fields and function fields was established by Artin and Whaples, thus finally putting on firm ground the analogy between function fields and number fields. I have seen allusions that the germ of the idea was coming from Gauss. However since fields were not defined then, this was not a definitive statement.

Question: When was a definitive conjecture first made in mathematical history that there is a major analogy between algebraic number fields and function fields over finite fields?

Applications of étale cohomology

2010-02-23T21:48:47Z

It is well-known that étale cohomology is used in the proof of Weil conjectures and that SGA 4.5 is devoted to it. Also it seems(from a brief perusal of Milne's notes) that it is a kind of Galois Cohomology/Kummer theory for arbitrary algebraic varieties.

However I have heard a lot of people praising it, and this leads me to suspect that it must have applications beyond proving the Weil conjectures. I would be grateful if some of these can be given. I am sorry if this is a stupid question. The wikipedia page, Milne's article, etc., did not give any extra applications and so I hope asking people is more sensible. Please provide references also if available.

Comment by Regenbogen

2010-05-17T23:43:10Z

I was meaning just the conjecture that homotopy spheres are homeomorphic to S^n, not the conjecture that they are diffeomorphic.

Comment by Regenbogen

2010-05-17T22:09:09Z

Well, one way could be as here: abstrusegoose.com/26

Comment by Regenbogen

2010-05-17T14:13:37Z

The colorful language is in the usage of leftist political phrases like "party line". If you are asking about the mathematical part, then it is hard to explain within the short space of a comment. I suggest that you have a look at Vol 1, last two pates of Chap. 4.. Here I will limit myself to saying that he is pleased with the usage of bundles in general and seeing vector fields as their sections, rather than defining them as a certain equivalence class of functions.

Comment by Regenbogen

2010-05-17T13:12:42Z

You could perhaps request the library to acquire it? In any case iff you want to develop a sheaf-theoretic thinking in differential topology, it would be great if you imbibe a little bit of algebraic geometry, in which they are heavily used. After you see the approach over there, you would tend to develop similar modes of thinking in differential topology and geometry too. I notice that you are in theoretical physics. Whether this is beneficial or not, perhaps depends on whether you will use algebraic geometry or not.

Comment by Regenbogen

2010-05-15T20:38:11Z

If you first of all define the logarithm, then you can define Tanh^-1. You can do this in a good way in a simply connected domain not containing zero .. You choose the domain, and define log z as the integral of dt/t from 1 to z, etc..

Comment by Regenbogen

2010-05-15T20:24:19Z

Greasemonkey is built into chrome. So installing/not installing it is not an issue.

Comment by Regenbogen

2010-05-15T20:23:14Z

The display looks ok for me, though I must say that it has a few drawbacks. I had imagined that it had worked perfectly as you intended. I am unable to check with firefox how it looks there and whether it is any better over there, since I had thrown out all browsers except chrome.

Comment by Regenbogen

2010-05-15T19:17:21Z

VA, I just note that you can avoid step 1 if you use Google chrome. In any case, check out chrome; it is amazing and ultra-fast.

Comment by Regenbogen

2010-05-15T17:08:28Z

According to wikipedia: Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name.

Comment by Regenbogen

2010-05-15T14:23:39Z

But you don't have to read all these leading hundreds of pages to read the particular proof you want. .

Comment by Regenbogen

2010-05-15T14:17:54Z

But both are relevant for people with analytic interest in "reciprocity".

Comment by Regenbogen

2010-05-15T13:49:23Z

Thanks Steve D.

Comment by Regenbogen

2010-05-15T13:23:36Z

Sorry, I it turns out that the book with classfield theory is his Zahlbericht and unfortunately it does not seem to be published in English. And, the books are in no particular order.

Comment by Regenbogen

2010-05-15T10:22:07Z

Should be community wiki.

Comment by Regenbogen

2010-05-15T10:17:36Z

@Reid Barton: Could you please provide a counterexample?