algebraic geometry « Bloomsbury Journal

Category Archives: algebraic geometry

Homomorphism of formal groups

February 13, 2013 – 6:08 pm

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.
Hang on, what about

$X+Y+XY$ ?

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.

By minhyong kim | Also posted in number theory | Comments (0)

Some obvious singularities

January 30, 2013 – 9:55 pm

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.

Another way of thinking about this is to explicitly consider the zero sets

$D: G=0$

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.

By minhyong kim | Also posted in number theory | Comments (0)

Some more comments on week 2

January 28, 2013 – 11:33 am

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 $n<m$ . 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?

By minhyong kim | Also posted in number theory | Comments (0)

Part C Elliptic Curves, Week 2

January 28, 2013 – 12:29 am

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?

By minhyong kim | Also posted in number theory | Comments (0)

Part C Elliptic Curves, Week 1

January 19, 2013 – 10:01 pm

As we progress through the course in Hilary, 2013, I will be posting comments on the lectures at semi-regular intervals. Here is the first installment.

By minhyong kim | Also posted in number theory | Comments (0)

Non-examinable material

May 16, 2011 – 9:56 pm

I received the following message:

—————————————-
Dear Professor,

In the list of non-examinable material in the book you put sections 1.1.3- 1.1.5. However section 1.1.6. seems to rely on those sections ruled out. I wasn’t sure if it was the case that you forgot to list it, or that this section is indeed examinable?
——————————————–

I’m at home right now without the textbook, but since the examination is near, I thought I’d send a quick generic clarification. If material in section B depends on section A and the latter is listed as ‘non-examinable,’ this means that the results needed from section A should simply be assumed when reading section B. I’m sorry if that wasn’t clear.

I also apologize that I didn’t manage to schedule another revision session.

By minhyong kim | Comments (0)

Roots and coefficients of polynomials

April 20, 2011 – 3:57 pm

A question came up in the algebraic geometry revision session about the relation between the roots and coefficients of a polynomial. Here is a note that summarizes the statement.

By minhyong kim | Comments (0)

Remark on the textbook

April 20, 2011 – 3:29 pm

During the algebraic geometry revision session today, a question arose about the portions of the textbook that were not discussed carefully on class. So here is a list of the parts that are not examinable:

section 1.1.3
section 1.1.4
section 1.1.5
section 1.3.2

chapter 5
chapter 6

If there is a result in some of the other parts that depend on the results in the these sections, the *statement* used should still be regarded as important, even if the proof will not be examinable.

The introduction should be read, even if it is not examinable in a strict sense (except the ideas that reappear elsewhere).

By minhyong kim | Comments (0)

M211 review

March 23, 2011 – 3:36 pm

Here are some questions to give you general orientation on the material you should know for the examination in algebraic geometry.

By minhyong kim | Comments (0)

Revision sessions, 3704 and M211

March 23, 2011 – 11:08 am

There will be revisions sessions for Math 3704 and Math M211 as follows:

3704
Wednesday, 20 April, 10:00-12:00 in Math 706

M211
Wednesday, 20 April, 14:00-16:00 in Math 706.

Because of the many holidays towards the end of the month, it was hard to find any other days possible. I hope many of you can come.

By minhyong kim | Also posted in 3704, number theory | Comments (0)

Bloomsbury Journal

Category Archives: algebraic geometry

Homomorphism of formal groups

Some obvious singularities

Some more comments on week 2

Part C Elliptic Curves, Week 2

Part C Elliptic Curves, Week 1

Non-examinable material

Roots and coefficients of polynomials

Remark on the textbook

M211 review

Revision sessions, 3704 and M211

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