Category Archives: algebraic geometry



Madrid Lecture

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.
Hang on, what about


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.

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.

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

D: G=0


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


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


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




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,


But u+p^{m-n}v\equiv u \mod p, so u+p^{m-n}v is also a unit. Hence,

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?

Part C Elliptic Curves, Week 1

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.

Non-examinable material

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.

Roots and coefficients of polynomials

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.

Remark on the textbook

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).