Monthly Archives: January 2013

A quiz about p-adic numbers

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?

Advertisements

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

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<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?

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.