## Category Archives: number theory

### Some points on an elliptic curve

The elliptic curve is $y^2=x^3-2$, which has the obvious point $P=(3,5)$. This file displays the denominators of the $x$-coordinates of the points $P, 2P, 3P, \ldots, 27P$. This was discussed in my course ‘Topics in Mathematics,’ Summer, 2016, at Ewha Womans University.

### 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

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

### Homomorphism of formal groups

Wei Yue asks the following question: In the definition of a homomorphism $f(T)$ between two formal groups $F$ and $G$, what is the reason we required $f\in TR[[T]]$? The short reason is that we want $f$ to preserve the origin, that is, $f(0)=0$. A more complicated reason is that in general, the expression $G(f(X), f(Y))$ may not make sense for a power series $G(X,Y)$ if $f$ has a non-zero constant term. This is a point we have been somewhat cavalier about: When can we substitute one power series in another and get a well-defined power series as the answer? You should ponder this for yourself a bit, and then try to formulate some conditions precisely using the definition

$R[[T]]= \lim_{\leftarrow} R[T]/(T^n)$

of the ring of power series with coefficients in $R$.

Here is an exercise worth trying out: Suppose $F(X,Y)$ is a polynomial that defines an algebraic group law on the field $\mathbb{C}$ with $0$ as the origin. Then $F(X,Y)=X+Y$ is the only choice.

$X+Y+XY$?

Well, you have to formulate a bit carefully what is meant by an algebraic group law. Anyways, the conclusion is that deforming the usual group structure in any reasonable sense requires us to move out of the realm of polynomials.

I mentioned in lecture something called the Hasse Principle, which holds for simple kinds of equations. You can read about this in the book ‘A course in arithmetic’ by J.-P. Serre. I highly recommend this book as an introduction to the theory of Diophantine equations with a different flavor from the present course.

Now, we’ve discussed numbers having square-roots in some ${\bf Q}_p$ (or ${\bf R}$) and not others. For example, I hope you can check that $\sqrt{-1}\in {\bf Q}_p$ if and only if $p \equiv 1 \mod 4$. But here is the quiz: Which rational numbers have square-roots in ${\bf R}$ and *all* ${\bf Q}_p$?

### Some obvious singularities

Wei Yue asked a question about an assertion that came up in lecture today. It was that if a curve $C$ in ${\bf P}^2$ has defining equation $F=0$, where $F=GH$ for two non-constant homogeneous polynomials $G$ and $H$, then $C$ is necessarily singular. The reason is the equation

$\nabla F=(\nabla G)H+G(\nabla H).$

So if $a$ is a point where $G(a)=0$ and $H(a)=0$, then $\nabla F(a)=0$. But the zero sets of $G$ and $H$ must meet (by Bezout’s theorem) and hence, the curve $C$ is singular.

$D: G=0$

and

$E: H=0$.

Since $F=GH$, we get

$C=D\cup E.$

But there must be a point $a\in D\cap E$, and this is a singular point of $C$. If you visualize $C$ as the union of two curves, you can imagine that these isn’t a single tangent line to $C$ along their intersection (and it *looks* singular there). For the easiest example, consider the union of two lines that meet at a point. (By the way, in ${\bf P}^2$ two lines *must* meet.)

Another question was about the factorization of such an $F$. That is, doesn’t $F$ factorize into linear factors anyways? The answer is no in general. The factorization we discussed in the lecture was for homogeneous polynomials in two variables. In three variables, many $F$ of large degree are *irreducible*. In fact, what we showed above is that if $F$ is reducible (in that case, we also say $C$ is reducible), then $C$ is necessarily singular.

### Some more comments on week 2

There are a few more points I forgot to emphasize in class, even though they’re written in the notes.

The zeroth fact is that $x\in {\bf Z}_p$ is a unit if and only if $x\mod p$ is non-zero in ${\bf F}_p$. This is easy to check, and I leave it to you.

The first fact is that a series

$\sum_{i=k}^{\infty}x_i$

is convergent in ${\bf Q}_p$ if and only if $|x_i|\rightarrow 0$. Thus convergence is very easy to test in ${\bf Q}_p$. This is one among many aspects of ${\bf Q}_p$ that are much easier than reals. So, for example, when we proved that every element of ${\bf Q}_p$ can be written uniquely as

$\sum_{i=N}^{\infty}a_ip^i$

for $a_i\in \{ 0, 1, \ldots, p-1\}$, the convergence, that is, the fact that the series defines an element of ${\bf Q}_p$, is essentially trivial.

Another point regarding the valuation is that while

$|x+y|_p \leq max (|x|_p, |y|_p)$

is always true, in fact, if the two valuations are different, $|x|_p\neq |y|_p$, then we have equality:

$|x+y|_p = max (|x|_p, |y|_p)$

This is easy to see if one is zero, so we assume both are non-zero. Then we write

$x=p^nu$

and

$y=p^mv$

with $u,v$ units.

If the valuations are different, then we may assume, without loss of generality, that $|x|_p>|y|_p$. But this is just saying that $n. Now,

$x+y=p^nu+p^m(v)=p^n(u+p^{m-n}v).$

But $u+p^{m-n}v\equiv u \mod p$, so $u+p^{m-n}v$ is also a unit. Hence,
$|x+y|_p=|p^n|_p|(u+p^{m-n}v|_p=p^{-n}=|x|_p.$

What might happen to the sum if the valuations are the same?

### Part C Elliptic Curves, Week 2

I mention here a few topics that came up in the class on Thursday, for the benefit of the students who take the Tuesday class.

1. We discussed the Hasse bound: If $E$ is an elliptic curve over a finite field ${\bf F}_q$ with $q$ elements, then the cardinality $|E({\bf F}_q)|$ of the set of points with coordinates in ${\bf F}_q$ satisfies the following inequalities:

$q+1-2\sqrt{q}\leq |E({\bf F}_q)|\leq q+1+2 \sqrt{q}.$

For example, an elliptic curve over ${\bf F}_{11}$ will have at most 18 points in ${\bf F}_{11}$. However, it is still an interesting problem to actually *find* elliptic curves with points near the upper bound. For example, can you write down an elliptic curve over ${\bf F}_3$ with 7 points? How about an elliptic curve over ${\bf F}_5$ with 10 points?

2. If you fix an elliptic curve $E: y^2=x^3+Ax+B$ with $A,B$ integers, it is interesting to consider the *distribution* of $|E({\bf F}_p)|$ as $p$ varies. For example, sheet 1 showed that those of the form $y^2=x^3+B$ have $p+1$ points in ${\bf F}_p$ for all $p\equiv 2 \mod 3$ such that $p$ doesn’t divide $3B$. (Why do we need this condition?). What happens for such elliptic curves when $p\equiv 1 \mod 3$?

This question for a general elliptic curve was the subject of the celebrated *Sato-Tate Conjecture*, resolved a few years ago by Laurent Clozel, Michael Harris, Nick Shepherd-Barron, and Richard Taylor.

3. For a point $P$ on an elliptic curve it might be interesting to search for points $Q$ such that $2Q=P$. How might one go about finding these?