Elliptic Curves

Finally, Witt Vectors

ellipticcurves — Mon, 01 Jun 2009 22:51:46 +0000

Now, let us try to do arithmetic operations on -adic numbers written in terms of Teichmüller representatives. Suppose

and

In other words,

and ditto for Then

So, if we define polynomials in and

then

One can also show that there exist polynomials and for all integer such that

and

for all , or, more curtly,

and

In other words, the addition and multiplication of -adic integers can be written in polynomial form when those numbers are written in terms of Teichmüller representatives.

More on Teichmüller Representatives

ellipticcurves — Mon, 01 Jun 2009 22:49:55 +0000

As I mentioned before, Teichmüller’s great insight was that one might have better luck if using representatives

for , instead of integers themselves. One of the advantages, for example is that this set of representatives is closed under multiplication. In other words, it is an monomorphism of multiplicative monoids from to

But is this really a set of representative? In other words, can every -adic integer be written as a -series in Since for each and integers form a set of representatives, the answer to this question is yes.

Let’s offer a demonstration. Say we would like to write -adic integer 7 in terms of Teichmüller representatives. Say,

Then

so let
i.e., , so we take .
i.e., , so we take .
i.e., , so we take .
etc.

So,

Therein lies a seed of the inductive proof of the fact that

is a set of representatives. If for a fixed -adic integer one can choose so that

or, equivalently,

then

and there exists such that

Then

or, equivalently,

The short version is that any overshot modulo can be corrected modulo using only multiples of

Motivation for Witt Vectors

ellipticcurves — Mon, 01 Jun 2009 21:55:28 +0000

If we write two -adic integers and as -series in integers and then multiply them together to get it goes something like this:

Therefore, So far so good. But as we move to the next congruence class, we get

Good luck figuring out the formula for the carry part.

Teichmüller Representatives

ellipticcurves — Wed, 21 Jan 2009 07:00:49 +0000

Let us solve a particular equation in -adic numbers:

Notice that since this equation has precisely distinct solutions in (actually, all of its elements), by -adic mojo, it has solutions in .

It is easy to figure out the sequence representations of these solutions:

for . By Euler’s Theorem,

which takes care of both the condition on sequences defining a -adic number and the fact that every element of the sequence is a solution of our equation in its respective modular ring.

Teichmüller’s insight was that it might be easier to write down the arithmetic laws on -adic numbers expressed as -series in these numbers (zero and st roots of unity) rather than just the integers themselves. But more on that later.

Topological construction of p-adic integers

ellipticcurves — Fri, 26 Dec 2008 09:49:01 +0000

One of the interesting things to observe is what multiplication by does to -adic integers:

But what matters is the congruence class of the components modulo and so

Iterating gives us

This is, of course, a lot easier when using the series notation.

So, we can tell that divides precisely when the first components of are zero. As a corollary, we obtain that if , then there exist a largest such that divides (i.e., does not divide ). This is precisely the number of the initial components of that are congruent to zero in their respective

Definition. The number as above is called the -adic valuation of and is denoted by :

We set

The -adic valuation function satisfies the following properties:

if and only if
and

The last property is referred to as the ultrametric property for reasons to be explained right now.

Given any number (customarily, ), we can define

Then the properties of become

for all , with if and only if

and

In other words, is an ultrametric norm. We can also define a metric

which makes into an utrametric space. The topology (convergence of sequences) induced by this norm does not change if we choose a different , so from now on, . Then

In other words, the higher power of divides you, the smaller you are.

Note that this norm can be restricted to the rational integers contained in . The ring of integers is not complete under this norm, and therefore we can consider its completion, i.e., the smallest complete normed space containing . Then the standard construction of the completion coincides for the -adic norm with the construction of -adic integers as sequences. Specifically, the condition that translates into the Cauchy criterion as for sequences.

In light of this norm, the series notation also starts to make sense. Indeed, the truncated series form a Cauchy sequence, since has norm no greater than , which goes to zero as goes off to infinity. Thus the infinite series we work with converge with respect to the ultrametric -adic norm.

The ring of p-adic integers

ellipticcurves — Fri, 26 Dec 2008 08:20:43 +0000

In what way can we impose operations of addition, subtraction, and multiplication on the set of -adic integers? One thing on our wish list is that these operation be compatible with the corresponding operations on the ring of integers embedded in . Another, is that they be compatible with the operation on the component considered as elements of It seems that the obvious way to define these operations is componentwise:

Another way to define these operations is using the “series” notation. We perform these operations just like base calculations, only digits extend infinitely to the left (some people write instead of the series notation ). For example, We carry 1 toward ‘s and 2 towards ‘s. It’s a bit of an exercise to show that these definitions agree. It is not that difficult to check that these operations make into a commutative ring.

Using Abstract Nonsense, the construction of the -adic numbers as sequences can be reformulated as follows. For each we have a map

of reduction modulo , so rings form an inverse system. Then

the inverse limit of this system. Recall that the inverse limit in the category of rings is constructed by taking all elements of the direct product of all rings in the system (a.k.a. sequences) that “agree” with the reduction maps.

Introduction to p-adic integers

ellipticcurves — Fri, 26 Dec 2008 05:24:01 +0000

I feel like taking a break from elliptic curves and talking about something completely different. For motivation, let’s take a look at solutions of the equation for different values of For we have two solutions: For still two solutions: For we get And so on, and so forth.

Notice something interesting. Reduce 91 modulo 25, and you get 16. Reduce 16 modulo 5, and you get 1. Similarly, reduce 33 modulo 25 to get 8, which reduces to 3 modulo 5. It seems like a solution modulo 5 begets a solution modulo 25, which then yields a solution modulo 125, etc. No surprise there, since all we are doing is reducing the equation modulo

The question is this. Can we go in the opposite direction and keep this string of solutions going ad infimum? Given a solution modulo 125, can we find one modulo 625?

To make things more general, let us denote polynomial by and by Futher, we will always assume that is prime. We know that there exists a solution to equation Let us for a second suppose that there exists a number such that and Then for some Plug this equality into the given equation to get

Since is a polynomial, we can multiply things out, reduce modulo where possible, to obtain

(Do this for to see that this is precisely what you get when the dust settles.) Move things around, to get

If is not zero modulo then it is also invertible modulo and we can divide both sides of this congruence by it to get

Finally, add to both sides, and remember the definition of to see that

This formula looks precisely like Newton’s Method in Calculus.

One can also quickly check that if is another integer satisfying the defining conditions of then Note that the only requirement for us to be able to build this chain of solutions is that But Hence the sole condition for using Newton’s Method to build a chain of solutions modulo for starting with is that be a simple root of modulo

So, we are now dealing with sequences of integer numbers

such that

So far as we only care about the congruence class of each modulo we identify two sequences and if for all Check for yourself that this identification is an equivalence relation. The equivalence classes are called the -adic integers.

Alternative and much more suggestive notation is also available. If we let then we can write a -adic integer as a formal infinite sum

The from the previous paragraph are equal to truncations (reductions modulo ) of this sum. To make our lives easier, we choose from the set of representatives of congruence classes modulo

The set of -adic integers contains the set of rational integers via the map

In other words, the “usual” integers are represented by constant sequences/finite series.

Going back to our original equation, we find that it has two solutions in

and

Definition of Elliptic Curves

ellipticcurves — Sat, 20 Sep 2008 03:41:18 +0000

I think it is about time we defined what an elliptic curve is. It is a nonsingular cubic curve with a distinguished point, usually denoted by . We say that the elliptic curve is defined over a field if the curve is defined over and .

It turns out that for every field , the set can be endowed with a structure of an abelian group as follows.

The line through any two points and of intersects in a third point, which we denote by . If , we use the line tangent to at this point. As we saw above, . Notice that this operation is commutative, though not associative. Also notice the following “cancellation rule”:

Now we define

and

Observe that this addition operation is commutative.
Also

so is the identity for the addition operation.
In addition,

This means that is the additive inverse of .
The only property we need to check is the associativity of the addition operation. This is not trivial, so we postpone this until later (unspecified) date.

If is a flex point, then , and the negation operation simplifies to .

Bezout’s Theorem

ellipticcurves — Wed, 03 Sep 2008 23:20:19 +0000

We are coming across this theorem so often, that I think it deserves a post of its own. We keep wishing that the number of intersection of two curves defined by equation of degrees and respectively, be but we constantly hit snags. In this post, we try to iron out things out all the way.

For simplicity, we will refer to the degree of the equation defining the curve as the degree of the curve, and denote it by

One of the problems is that some curves might have infinitely many points of intersection, but that happens precisely when they have a component in common. We rule that possibility out in out conditions.

The other problem is that there is some inhomogeneity to the affine plane. We fix this by working in the projective plane.

Finally, just like with solving polynomial equations in one variable, we should be looking for points with coordinates in an algebraically closed field and counting them with appropriate multiplicity. The latter actually requires a definition.

First let us look at the one variable case. Let be a polynomial with a root Let denote the ring of formal power series in How can you figure out the multiplicity of (the number of times divides ) using these two. Sounds like a stupid question, so I will just give the answer. It equals the dimension over of the vector space where denotes the ideal of generated by Notice that if we write so that does not have a root at then is invertible in and therefore Finally, the basis is formed by while any higher power of is congruent to zero modulo Therefore, as expected.

Through an appropriate change of variables, we can move any point to the origin of the affine plane. We defined the intersection multiplicity of two projective curves and at a point by analogy:

Here, and

Bezout’s Theorem. Let and be two curves in that do not have a common component. Then

Notice that we do not specify the field over which we are searching for points of intersection of and This means we are looking for them over all fields in the universe. One doesn’t have to look that far however. All point of intersection are defined over the algebraic closure of the common field of definition of these curves.

Affine Maps

ellipticcurves — Wed, 27 Aug 2008 01:00:44 +0000

What are acceptable maps between irreducible curves in ? We start by defining rational maps, which are precisely what they sound like.

Definition. A rational map between two irreducible curves is a map given on points of by for some rational functions , such that

the denominators of do not identically vanish on
for all points where is defined.

For now, we say that is defined over a field if we can choose

The first requirement says that should be defined in at least one point of . We can say more.

Theorem. A rational map is defined on all but finitely many points of the curve .

Proof. Consider the (not necessarily) irreducible curves and defined by the denominators of and The points of intersection of these curves with are precisely the points where is undefined. The first condition of the definition means that the defining polynomial of does not divide the denominators of and which implies that is not a component of and By Bezout’s Theorem, each intersection is finite. So is their union, which is precisely the set at which is undefined.

Remark. The second condition can be restated as follows. If is given by , then

for all points where is defined.

Definition. Similarly, a rational map is given by a pair of rational functions in one variable whose image lies in . A rational map is called a rational function on , because that is what it is. The only restriction is that the denominator of this function not identically vanish on .

We say that two rational maps of irreducible curves are equivalent (write ) if they agree on all but finitely many points of From now on, we consider rational maps modulo this equivalence relation. In this way, domains of rational maps can sometimes be extended by switching to another pair of rational functions that define an equivalent function and are defined in the chosen point If this is possible, we say that is regular at

Example. For every curve the identity map is a rational map.

Definition. Let be a rational map. If there exists a rational map such that and we call a birational map, and say that curves and are birationally isomorphic, or just birational.

Definition. A rational map on a curve is said to be regular if it is regular at every point of If a map of curves is regular, with a regular inverse, we call it an isomorphism and say that curves and are isomorphic.

Example. Go back to the parameterization of points on the unit circle. Let be the unit circle defined over , and consider maps

given by

and

given by

Both maps are rational. The map is defined on the whole of and therefore is regular. The map is undefined at and so it is not rational. These maps are inverses of each other. Therefore, in the affine plane, the circle is birationally isomorphic to an affine line, but is not isomorphic.

One of the important points about the previous example is that a map being a birational isomorphism and regular does not guarantee that the inverse is regular. In other words, a regular birational isomorphism is not necessarily an isomorphism.

Example. Generalizing the previous example, we showed using projections that any irreducible conic is birationally isomorphic to

Example. Any line in is isomorphic to Indeed, if is the line in question with , then rational maps and are regular and inverse to each other.