1. Introduction

L. J. Mordell began his famous paper with the words “Mathematicians have been familiar with very few questions for so long a period with so little accomplished in the way of general results, as that of finding the rational points on elliptic curves" [1]. Elliptic curves have a lot of applications, this because it possible to take two Points on the curve and generate a third point. In fact, we will show that by defining an addition operation and introducing an extra point called the point of infinity, the points on an elliptic curve form an additive abelian group [12]. There are still a number of significant open questions specific to the theory of elliptic curves themselves, such as the conjecture of Birch and Swinnerton-Dyer which would give a much more precise description of the beautiful arithmetic that exists for points on elliptic curves [14].

2. The Algebra and Geometry of Elliptic Curves

Elliptic curves are functions defined by equation of the form

(1)

has no multiple roots. The cubic equation

does not define an elliptic curve, because

has 1 as a multiple root. Similarly

is not an elliptic curve, but

is an elliptic curve.

The general form of an elliptic curve is given below

(2)

Where belong to a specified field such as

General we use to represent an elliptic curve.

If we wish to consider points in a field we write which is defined as below.

(3)

2.1. We can Use Geometry to Make the Points of an Elliptic Curve into a Group

Figure 1. P + Q

2.2. Properties of "Addition" on Elliptic Curve

The addition law on an Elliptic curve has the following properties:

In other words, the addition law + makes the points of E into a commutative group. [11]

2.3. The Addition Operation is Summarized below

Let E be an elliptic curve defined by Let and be points on with We the define as follows

1. If then where .

2. If but then

3. If and then where .

4. If and then

Also we define for all points [12].

2.3.1. Example 1

Let E be the curve and suppose we know the point and lies on the curve. To find another point on. In the notation of elliptic curve addition we have:

Figure 2. y2 - x + 1 and its addition law

2.3.2. Example 2

The set is the set of integers that satisfy

We can see that is the set as

To find all the points in we find all the possible values and then see what values of will match. There are 11 choices of the integers Subbing these values in turn into the cubic and reducing modulo 11 will give us the possible values of

So we can see that the possible values of cannot be 0, 1, 2 or 10.

Next examine the 10 possible values of and identify which values of they could be paired with to give a point on the curve.

So in we find

3. Elliptic Curves over Finite Fields

Let be a finite field and an elliptic curve defined over Since there are only a finite number of pairs with the group must itself be finite. In this section. We discuss the basic theory of elliptic curves over finite fields and also we will state Hasse’s theorem which gives a bound of the size of the group defined by We also look at methods to find the order of a point in [12].

3.1. Example 4

Let be thr curve over

has order 9. is cyclic and generated by (0,1).

3.2. Methods to determine the order of

Hasse’s theorem gave bounds for the group of points on an elliptic curve over a finite field. In this section we discuss methods for determining the group order exactly. Suppose we have an elliptic curve defined over a finite field we can determine the order of by listing the points. We can then determine the order of for all [12]

3.2.1. Theorem 1

Let write then for all

(4)

3.3. Example 3

Let be an elliptic curve satisfies

Therefore and we obtain the polynomial

We could compute the last expression directly, but better use the recurrence relation

So, (as we calculated when listing points).

3.4. Legendre Symbol

To make a list of all the points on over a finite field, we listed every possible value of and then found the square roots, of if they existed. This procedure will be the basis for a simple point counting algorithm.

The Legendre symbol we can generalize this to a finite field odd, by defining for

We can now give a more accurate solution to the number of points on

3.4.1. Theorem 2

Let E be an elliptic curve over Then

(5)

Proof.

Consider a point There are points on with coodinate if is a non-zero square in There is one such point if it is zero, and no such points if it is square It follows that the number of points in with coordinate is

So to find the order of we must sum over all and add 1 for the point at infinity:

3.5. Example 5

Let be the curve over

So the non-zero squares modulo 5 are 1 and 4.

3.6. Theorem 3 (Hasse’s Theorem)

Let E be an elliptic curve over the finite field Then the order of satisfies the following inequality.

3.7. The Frobenius Endomorphism

The Frobenius Map is the function

(4)

The quality is called the Trace of Frobenius. [16]

What are rational solutions? This question is even more difficult in general. If the degree of the equation is higher than three, little is known. If the degree is exactly three, we have essentially an elliptic curve. [17]

3.8. Theorem 4 (Henri Poincare’s Theorem, 1901)

Let be an elliptic curve defined over a field Then is an abelian group under +.

4. Conjecture 1 (Henri Poincare’s Conjecture, 1901)

Let be an elliptic curve. Then is finitely generated Mordell gave a good partial answer in 1923 (based on a conjecture of Henri Poincare in 1901), known as Mordell’s Theorem. This result states that the group of rational points on an elliptic curve is "finitely generated". This means that, if there are any rational solutions, then they can all be determined from a certain finite subset of them.

Unfortunately, there are two things that Mordell’s result does not do. First, it provides no way to tell whether any rational points exist (other than the "point at infinity"). Second, it does not provide an "effective" means (i.e. an algorithm) for finding a set of generators for the group of rational points. In some cases Mordell’s methods are able to do this. And it has been conjectured, but not yet proven, that the methods will work in all cases. There is a general theorem about finitely generated abelian groups such as It states that any finitely generated abelian group is the "direct sum" of the subgroup consisting of elements of finite order and zero or more copies of the additive group of integers. [14]

4.1. Theorem 5 (Mordell Theorem)

If E is an elliptic curve over then the commutative group is finitely generated.

By Mordell’s theorem we can write

Where is a nonnegative integer and is the Torsion subgroup of elements of finite order in This subgroup is called the torsion subgroup of The integer is called the rank of and is written

Determining theoretically and in practice is currently the main problem of arithmetic elliptic curve theory. As it happens, much more is known about the torsion part of the group denoted by

A theorem due to Elisabeth Lutz and Trygve Nagell in the 1930’s showed how to compute in any particular case. [1]

4.2. Torsion Subgroups

The torsion subgroup is "well-understood". First, there is an effective algorithm to determine given

4.2.1. Theorem 6 (Nagell-Lutz)

Let be the elliptic curve . If and Then then

1.

2. either or divides

4.2.2. Corollary 1

Let bean elliptic curve defined over Then the torsion subgroup is finite.

Proof: Suppose By Lutz-Nagell, divides so there are only finitely many possibilities for Fixing there are at most 3 solutions to in thus is finite group.

4.2.3. Example 6

Let then Torsion subgroup are

1) (-1, 0) has order 2

2) (0,±1) has order 3

3) (2,±3) has order 6

4.2.4. Theorem 7 (Mazur Theorem)

If E is an elliptic curve, then T is one of the following 15 groups:

Figure 3. Examples of torsion subgroups of elliptic curves [9]

Each of the groups in Theorem 6 occurs infinitely often as the torsion subgroup of an elliptic curve over

4.3. Ranks

The rank of an elliptic curve is a measure of the size of the set of rational points. There is no analogue of Theorems 12 or 14 for ranks:

Ÿ there is no known algorithm guaranteed to determine rank of

Ÿ it is not known exactly which integers can occur as the rank of an elliptic curve. [1]

4.4. Reduction of an Elliptic Curve Modulo p

Let be an elliptic curve given by an equation

with We can reduce the coefficients of modulo a prime to get an elliptic curve with coefficients in

With However, remember we must check that is not singular, which means that we need the discriminant

We say that E has Good Reduction at p if p does not divide the discriminant and we say that has Bad Reduction at if p does divide the discriminant.

When we talk about reduction modulo we will generally assume that we have good reduction at [13]

4.5. The Reduction Modulo p Homomorphism

It is hard to overstate the importance of reduction modulo p. A first indication is:

4.6. Theorem 8

If has good reduction, then the reduction modulo map

is a group homomorphism.

4.7. Example 6

Let be the elliptic curve

Some points in are

The reduction modulo 11 map

5. The Birch and Swinnerton-Dyer Conjecture

Fix an elliptic curve over For every prime number not dividing the discriminant.

of we can reduce modulo and view as an elliptic curve over the finite field

Reduction modulo induces a group homomorphism.

The idea of Birch and Swinnerton-Dyer was that the large is, the larger should be "on average" as varies. The size of can be measured by rank of but how can one measure the average size of the [1]

5.1. What does look like?

The group is obviously a finite group. Indeed, it clearly has no more than points. For each there is a "50% chance" that the value of is a square in And if is a square, then we (usually) get two points in Plus there’s the point at infinity Thus we might expect to contain approximately [13]

A famous theorem of Hasse makes this precise.

5.2. Theorem 9 (Hasse’s Theorem)

Let be an elliptic curve

with Then

For every prime number p not dividing

Let

5.2.1. Numerical experiments of the Birch and Swinnerton-Dyer

To test their idea, in the 1950’s Birch and Swinnerton-Dyer computed

as grows, for certain elliptic curves

Figure 2 shows the behaviour of for up to about for five different curves

The horizontal axis is and the vertical axis is

From their data Birch and Swinnerton-Dyer were led to conjecture that

(5)

As for some constant depending only on (Note that this relation is consistent with the data in Figure 2. if the axes were to scale, then the slopes of the lines would be the ranks of the curves.) The function does not behave very nicely and therefore is difficult to work with. Birch and Swinnerton-Dyer stated a related conjecture, using the L-function of E in place of [1]

Figure 4. Birch and swinnerton-Dyer data for

5.3. The L-Series of an Elliptic Curve

Let Analogous to the Euler factors of the Riemann zeta function, we define the local L-factor of E to be

(6)

The variable s is a complex variable

When evaluating its value at we retrieve the arithmetic information at

(7)

Notice that each point in reduces to a point in So when then tends to be small. Birch and Swinnerton-Dyer observed that if is infinite, then the reduction of the points in tend to make large than usual. So they conjectured Birch and Swinnerton-Dyer did numerical experiments and suggested

The L-function of is defined to be the product of all local

(8)

So intuitively the rank of will correspond to the value of L(E, s) at s=1: the larger r is, the "smaller" L(E, 1) is. However, the value of L (E, s) at s = 1 does not make sense since the product of L (E, s) only converges when R(s) > 3/2.

Nevertheless, if L(E, s) can be continued to an analytic function on the whole of it may be reasonable to believe that the behavior of L(E, s) at s = 1 contains the arithmetic information of the rank of

A deep theorem of Wiles et al., which many consider the crowning achievement of 1990s number theory, implies that L(E, s) can be analytically continued to an analytic function on all This implies that L(E, s) has a Taylor series expansion about s = 1. [12]

5.4. Theorem 10 (Wiles’ Theorem)

The function L(E, s) extends to an analytic function on all of and satisfies a function equation

(9)

where and

(10)

for some positive

Where

5.5. Taylor expansion of L(E,s) about s = 1

with the analytic rank

Define the analytic rank of E to be the order of vanishing of L (E, s) as s = 1. [5]

The famous Birch and Swinnerton-Dyer conjecture asserts that Birch and Swinnerton-Dyer conjecture.

5.6. Conjecture 1

The Birch and Swinnerton-Dyer Conjecture can then be stated simply as: for any elliptic curve Then the algebraic and analytic ranks of E are the same. Goldfeld also proved the following surprising result, which says in particular that, the connection between and is off by a factor of [17]

5.7. Theorem 11. (Goldfled Theorem)

Suppose where with constants and . Then and

Where Euler’s constant. In particular, if r = 0 then

The lines figure 2 were calculated using equation 5, Theorem 11, and the full Birch and Swinnerton-Dyer Conjecture to determine

5.8. Theorem 12

Suppose is an elliptic curve over and that Then the algebraic and analytic ranks of E are the same.

A quote from William A. Stein:

In 2000, Conjecture 1 was declared a million dollar millennium prize problem by the Clay Mathematics Institute, which motivated even more work, conferences, etc., on the conjecture. Since then, to the best of my knowledge, not a single new result directly about Conjecture 1 has been proved. The class of curves for which we know the conjecture is still the set of curves over with along with a finite set of individual curves on which further computer calculations have been performed (by Cremona, Watkins, myself, and others).

"A new idea is needed".

Nick Katz on BSD, at a 2001 Arizona Winter School.

The following theorem, a combination of work of Kolyvagin, Gross and Zagier, and others, is the best result to date in the direction of the Birch and Swinnerton-Dyer Conjecture. [17]

5.9. Theorem 13 (Gross-Zagier, Kolyvagin Theorem)

Assertion (i) can be rephrase as is finite”Assertion (ii) can be rephrase as and then r = 1, and there is an efficient method for calculating

The case remains completely open problem.

5.9.1. Example 23

If is the curve then

Thus theorem 22 (i) shows that is finite.

The sign in the functional equation for determines the parity of

The Birch and Swinnerton-Dyer Conjecture predicts in particular that rank(E) and rankan(E) have the same parity, so the following is a consequence of the Birch and Swinnerton-Dyer Conjecture.

5.10. Conjecture 2. (Parity Conjecture)

To explain the recent progress concerning the Parity Conjecture, we need to introduce the Tate-Shafarevich group and the Selmer group. The Tate-Shafarevich group is a torsion group that measures the failure of the Hasse’s Principle for curves that is principal homogeneous spaces for

5.11. Conjecture 3 (Birch and Swinnerton-Dyer)

Let be an elliptic curve over of rank Then and

And another quote from Bertolini-Darmon (2001):

"The following question stands as the ultimate challenge concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves over Provide evidence for the Birch and Swinnerton-Dyer conjecture in cases where [17]

6. Conclusions

In conclusion, although there has been little success in the last fifty years in finding the number of rational points on an elliptic curve, there are still almost no methods for finding such points. It is to be hoped that a proof of the Birch and Swinnerton-Dyer conjecture will give some insight on the number of rational points on an elliptic curve.

ACKNOWLEDGEMENTS

The authors thank the staff at the Mathematics Department of Kwame Nkrumah University of Science and Technology for their support and kindness during the period this paper was written.