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#1 2007-09-30 00:45:20

ganesh
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Registered: 2005-06-28
Posts: 21,319

Elliptic curves

An elliptic curve is of the form

y²=x³+ax²+bx+c where a,b, and c  are integers.

When a=0, b=0 and c=-2, the equation becomes

y²=x³-2

Leonard Euler was the first to prove that this equation has only one solution, y=5 and x=3. Thus, he proved that 26 is the only number among the infinity of numbers which is jammed between a perfect square and a perfect cube.

The proof is lengthy and sophisticated even for the present day mathematicians.

The Prince of Mathematicians, Euler, who is belived to have contributed more to mathematics than any other individual was also the first to make amajore breakthrough in proving Fermat's Last theorem. He had proved that

a³+b³ = c³ has no solution where a,b, and c are whole numbers, the only exception being a=b=c=0.

Generations of mathematicians later proved Fermat's Last theorem for certain prime number values of n. It was only during the fag end of the twentieth century that Andrew Wiles finally succeeded in proving Fermat's Last Theorem.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#2 2007-09-30 01:46:02

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Elliptic curves

Elliptic curves over the field of rationals are related to modular forms – a result which was crucial to the proving of Fermat’s last theorem by Andrew Wiles. cool

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#3 2007-09-30 02:01:29

ganesh
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Registered: 2005-06-28
Posts: 21,319

Re: Elliptic curves

You're correct, JaneFairfax!

The relation between elliptic curves and modular forms was studied by Yukata Taniyama and Goro Shimura about 10 years after the second world war in Japan leading to the famous Shimura-Taniyama Conjecture on which Andrew Wiles' proof is based.

Go to this link.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#4 2007-12-06 08:10:45

100'
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Registered: 2007-12-06
Posts: 8

Re: Elliptic curves

ganesh wrote:

Leonard Euler was the first to prove that this equation has only one solution, y=5 and x=3. Thus, he proved that 26 is the only number among the infinity of numbers which is jammed between a perfect square and a perfect cube.

The proof is lengthy and sophisticated even for the present day mathematicians.

The Prince of Mathematicians, Euler, who is belived to have contributed more to mathematics than any other individual was also the first to make amajore breakthrough in proving Fermat's Last theorem. He had proved that

a³+b³ = c³ has no solution where a,b, and c are whole numbers, the only exception being a=b=c=0.

Generations of mathematicians later proved Fermat's Last theorem for certain prime number values of n. It was only during the fag end of the twentieth century that Andrew Wiles finally succeeded in proving Fermat's Last Theorem.

It was Fermat who proved that 26 is the only number between a perfect square and cube.

Also, the proof that a³ + b³ = c³ has no solutions was only a tiny part of the solution. Find a piece of evidence in an infinity to support a hypothesis (regardless of how hard it was to find) is useless (this has been demonstrated by the Riemann Hypothesis in which millions of zeroes have been found upon Riemanns critical line, not furthering research particularly).

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#5 2016-05-23 19:27:51

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

One can define a "group law" on a cubic (and therefore an elliptic curve, by reducing the cubic to Weierstrass form).

Suppose we have an elliptic curve
, and denote the set of rational points on
by
. Poincaré saw that given three points
on a line, one may form a group law by taking
. That is, to add two points
, we draw a line through
and
and take the third point of intersection to be
. We then reflect
in the x-axis to obtain a new point
. We can then obtain a group law for the set of rational points
and the group operation
. The operation is closed, and it suffices to check that
, that is,
. We take the point at infinity
to be the identity of the group. One can verify associativity in various ways -- one of which is by drawing lots of lines and verifying it geometrically -- and hence we have a group.

Some authors write
instead of
. A nice photo from Wikipedia illustrates this:

ECClines.svg

Last edited by zetafunc (2016-05-23 19:42:48)

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#6 2016-05-23 19:41:24

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

With a little algebra, we can write a "doubling formula" for a point
. We can obtain a new point
whose co-ordinates are given by:

And hence we can obtain explicit formulae for
,
and so on. (We'll see the importance of this later when we compute torsion subgroups of elliptic curves.)

A natural question to ask is: given the algorithm for finding new rational points in the previous post, how many can we generate? Will we eventually return to our original point, or will this go on forever? It turns out that the answer to both of these questions is yes. If we do return to our original point, then we say that P has finite order, and write
, to mean that P has order n. Such a point is called a torsion point.

If we do not return to our original point, then P is a point of infinite order.

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#7 2016-05-23 19:59:18

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

We denote the set of torsion points -- that is, the set of points of finite order on
-- by
. It turns out that this torsion subgroup is not only a subgroup of E(Q), but it is also finite. The point at infinity
has order 1, by default. We have the following theorem:

The Lutz-Nagell Theorem

Let
have order
. Then:

(a)

(b) If
then
. If
, then
. Thus,
is finite.

We can explicitly write the possibilities Tors(E(Q)) by Mazur's theorem.

Mazur's Theorem

Either
for some
or
, or
for some
.

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#8 2016-05-23 20:13:44

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

Examples of Computing Tors(E(Q))

Let
be an elliptic curve. We compute the discriminant to be
, and note that if we have a point of order k > 2, then by the Lutz-Nagell theorem we must have
. But the only possible values for y in that case are
. But substituting these values of y into E, we see that there are no possible solutions for x over the rationals. Note that if y = 0, then we again have no solutions, so there are no points of order k = 2. Therefore, we only have the trivial point of order 1, the point at infinity,
.

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#9 2016-05-23 20:30:21

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 105,288

Re: Elliptic curves

Hi;

Do you use the elliptic discriminant http://mathworld.wolfram.com/EllipticDiscriminant.html to get that


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
No great discovery was ever made without a bold guess.

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#10 2016-05-23 20:34:08

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

Hi,

I am using:
for an elliptic curve
. (in other words, without the multiplication by -16)

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#11 2016-05-23 23:36:13

thickhead
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Registered: 2016-04-16
Posts: 817

Re: Elliptic curves

I am surprised to know about elliptic curves of rational numbers and corresponding modular form.

Last edited by thickhead (2016-05-24 00:19:48)


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{Gods rejoice at those places where ladies are respected.}

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#12 2016-05-24 09:05:27

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

Another Example of Computing Tors(E(Q))

Let 
  be an elliptic curve. We compute the discriminant to be
, so that if we have a point of order k > 2 on E, then by the Lutz-Nagell theorem we must have
, so the possibilities are
. Substituting these values into E we obtain the following points:

. But using the duplication formula, these cannot be torsion points. Thus, there are no points of order k > 2.

Since a point has order 2 if and only if its y co-ordinate is 0, then we see that
is the only point of order 2. Moreover, as usual, the point at infinity
is also a torsion point (of order 1).

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#13 2016-05-24 18:41:59

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

Height and Elliptic Curves

For a given rational number
where m,n are coprime, define its height to be
. If
denotes the x co-ordinate of a point
, then write
. We take
.

If
is an elliptic curve, define the height of an elliptic curve to be
. From this we can derive the inequality:

A useful consequence: if
satisfies
, then
has infinite order!

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#14 2016-05-24 19:09:11

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

Néron-Tate Height

Suppose we define the height in a slightly different way. Let
, so that the inequality in post #13 implies that
. Replacing
with
and dividing by
, we get:

Summing over k from m+1 to n, we get:

so that we have a Cauchy sequence (and therefore it converges). We call this limit the Néron-Tate height:

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#15 2016-05-24 20:22:39

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

The Weierstrass Elliptic Function

We say a function
is doubly periodic if for some
we have
.

Define the lattice
, and the parallelogram
. Let
.

Define the Eisenstein series of weight k to be:

.

We define the Weierstrass elliptic function as follows:

.

It can be shown that it converges absolutely, is periodic on the lattice and is even. Moreover, any meromorphic,

-periodic function
is a rational function of
and
.

We have an isomorphism of groups:
. Hence, every lattice may be associated with an elliptic curve! (And the converse is true as well.)

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#16 2016-05-24 20:28:03

zetafunc
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Registered: 2014-05-21
Posts: 1,351
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Re: Elliptic curves

Mordell's Theorem

The group of rational points
is finitely generated, and so can be written as
, where T is a finite torsion subgroup, and r is a non-negative integer. r is called the rank of the elliptic curve E.

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#17 2016-05-27 23:05:13

zetafunc
Member
Registered: 2014-05-21
Posts: 1,351
Website

Re: Elliptic curves

Embedding Torsion in

Theorem: If for a prime
we have
, then
contains a subgroup which is isomorphic to
.

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#18 2016-05-31 20:19:49

zetafunc
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Registered: 2014-05-21
Posts: 1,351
Website

Re: Elliptic curves

The j-invariant

Let
be an elliptic curve over
. We define the j-invariant of
to be:

Two elliptic curves are isomorphic over
if they have the same j-invariant.

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