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These curves defined by very simple equations are shrouded in mystery and elegance. In fact, the equations that describe them are so simple that even high school students can understand them.

However, despite the unremitting efforts of some of the world's greatest mathematicians, there are still a large number of simple questions about them that remain unsolved. But that's not all. As you will soon see, this theory connects all important fields of mathematics, because elliptic curves are not just planar curves.

An old problem in mathematics, some geometric problems can be transformed into algebraic problems, and vice versa. For example, look at a classic question thousands of years ago, whether a positive integer n equals the area of a right triangle whose side length is a rational number. In this case, n is called congruence. For example, 6 is a congruence because it is the area of a right triangle with sides 3 and 5. In 1640, Fermat proved that 1 is not congruent. Since Fermat's proof, research has been going on to prove that a number is (or is not) a congruence.

Amazingly, we can use elementary methods to prove that for each set of rational numbers, if there is any

We can find two rational numbers x and y, so that

Conversely, for each pair of rational numbers (x, y) such that y ^ 2 = x ^ 3-(n ^ 2) x and y ≠ 0, we can find three rational numbers a, b, c such that a ^ 2 + b ^ 2 = c ^ 2 and 1max 2 ab = n.

In other words, when y ≠ 0, a right triangle with an area of n corresponds to the understanding of the equation y ^ 2 = x ^ 3-(n ^ 2) x, and vice versa. Mathematicians would say that there is a bijection between the two sets.

Therefore, if and only if the equation y ^ 2 = x ^ 3-(n ^ 2) x has an understanding (x, y) and y ≠ 0, n > 0 is a congruence. For example, because 1 is not a congruence, the only understanding of y ^ 2 = x ^ 2-x is y = 0.

The specific correspondence is as follows

If we try this correspondence on a triangle with an area of 6 and a side length of 3 and 4, then the corresponding solution is (x ~ 4) = (12 ~ 36). This is incredible. One starts with the problem of number theory and geometry, and through algebra, transforms it into a problem about rational points on a plane curve!

Generally speaking, if f (x) denotes a cubic polynomial with non-zero discriminant (that is, all roots are different), then y ^ 2 = f (x) describes an elliptic curve, except for the "infinity" (that is, the unit elements in the group formed by addition of points on the elliptic curve).

Now, with a little algebraic technique, we can make appropriate (reasonable) changes to the coordinates and get a form called

So that the rational points on the two curves correspond one by one. From now on, when we say "elliptic curve", we mean a curve in the form of y ^ 2 = x ^ 3 + ax + b and a point 𝒪 at infinity. In addition, we assume that the coefficients an and b are rational.

The elliptic curve has two typical shapes, as shown in the following illustration.

Wikipedia

However, if we think of x and y as complex variables, the curve looks completely different. They look like doughnuts.

So why are we studying elliptic curves and what can we do with them?

First of all, many number theory problems can be transformed into Diophantine equations. Secondly, elliptic curves are related to discrete geometric objects called lattices (lattices) and are closely related to some very important objects called modular forms, which are extremely symmetric complex functions containing a lot of number theory information.

In fact, the connection between elliptic curve and modular form is the key to proving Fermat's Great Theorem. Andrew Wiles established this connection through several years of efforts in the 1990s, thus proving Fermat's Great Theorem. In cryptography, elliptic curves are also used to encrypt information and trade online.

Their most important feature, however, is the exciting fact that they are not just curves and geometry. In fact, they have an algebraic structure called the Abelian group structure, which is a geometric operation (rule) used to add points on a curve. For an Abelian group, you can think of it as a group of objects and operate on them so that they have the same structure as integers in addition (except that they can be finite).

Examples of Abelian groups are:

The integer ℤabout addition operations.

The operation of rotating a square 90 degrees clockwise.

Take the vector as the element and the vector addition as the vector space of operation.

The magic of the elliptic curve is that we can define an operation (called "⊕") between the rational points on the elliptic curve (that is, the x and y coordinates are both rational), so that the set of these points on the curve becomes an abelian group about the operation "⊕" and the unit element 𝒪 (points at infinity).

Let's define this operation. If you take two rational points on a curve (such as P and Q) and consider a straight line that passes through them, then the line intersects the curve at another rational point (possibly at infinity). We call this point-R.

Now, because the curve is symmetrical about the x axis, we get another rational point R.

This reflection point (R in the image above) is the addition of the two points mentioned earlier (P and Q). We can write it as

It can be proved that this operation satisfies the law of association, which is really surprising. In addition, the point at infinity is the (only) identity of this operation, and each point has an inverse point.

The great mystery has proved that two different elliptic curves can have very different groups. An important invariant, in a sense, is the most defining feature, which is the rank of the so-called curve (or group).

There can be a finite or infinite number of rational points on a curve. We are interested in how many points are needed to generate all the other points according to the addition rules mentioned earlier. These generators are called base points.

Rank is a measure of dimension, just like the dimension of a vector space, indicating how many independent base points (on a curve) have infinite degree. If the curve contains only a finite number of rational points, then the rank is zero. There is still a group, but it is limited.

Calculating the rank of an elliptic curve is notoriously difficult, but Model tells us that the rank of an elliptic curve is always finite. In other words, we only need a limited number of base points to generate all the rational points on the curve.

One of the most important and interesting problems in number theory is called Birch and Swinnaton-Dayard conjecture (the Birch and Swinnerton-Dyer Conjecture), which is all about the rank of elliptic curves. In fact, it is so difficult and important that it has become one of the millennium problems.

It is difficult to find rational points on elliptic curves with rational coefficients. One method is to simplify the curve p by modulus, where p is a prime number. This means that we do not consider the understanding set of the equation y ^ 2 = x ^ 3 + ax + b, but consider congruence.

We may have to eliminate the denominator by multiplying integers on both sides in order to make it meaningful.

So we're thinking about two numbers, and the remainder is the same when divided by p, which is equal in this new space. The advantage of this is that there are only a limited number of things to check. Let us denote the number of simplified curves that have an understanding of p modulo in terms of Numberp.

In the early 1960s, Dell used EDSAC-2 computers in the Cambridge University computer Lab to calculate the number of points taking p-modules on elliptic curves with known ranks. He worked with mathematician Brian John Birch to study elliptic curves, and after the computer processed a bunch of elliptic curves in the following form

For the growth of x, they get the following output from the data related to curve E: y ^ 2 = x ^ 3-5x (as an example). I should notice that the x-axis is log log xrem and the y-axis is log y.

On this graph, the slope of the regression line seems to be 1. The rank of curve E is 1, and when they try curves with different ranks, they find the same pattern every time. The slope of the fitted regression line always seems to be equal to the rank of the curve.

More precisely, they put forward a bold conjecture.

Where C is a constant. This kind of computer operation, coupled with great foresight, makes them make a general guess about the behavior of the Hasse-Weil L-function L of the curve when s = 1. The L function is defined as follows. Let

Let the discriminant of the curve be marked as Δ. Then we can define the L function related to E as the following Euler product

We think of it as a function of the complex variable s. Porgy and Swinnadon-Daya guessed that it now looks like this:

Let E be an arbitrary elliptic curve on ℚ. The rank of the Abelian group E (ℚ) of the rational point of the curve E is equal to the order of the zero point of L (E, s) when s = 1.

It is visionary because, at the time, they didn't even know whether all such L functions had so-called analytic extensions. The problem is that the L (E, s) defined above is only if Re (s) > 3 Unix 2.

They can all be evaluated at s = 1 by analytic extension, which was first proved in 2001, through the close connection with the modular form proved by Andrew Wiles. Sometimes this conjecture is expressed by the Taylor expansion of the L function, but it expresses the same thing in different ways. The field of rational numbers can be replaced by more general fields.

The elliptic curve is a beautiful dance between number theory, abstract algebra and geometry. There is a lot to say about them besides what I have described here, and I hope you can feel or see something shocking.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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