## Monthly Archives: January 2013

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.

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. 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.