Introduction and Motivation

In mathematics, it is common to reframe problems in different contexts to find solutions. For instance, certain geometric challenges can be transformed into algebraic ones, and vice versa.

A classic problem, dating back thousands of years, asks whether a positive integer n can represent the area of a right triangle with rational side lengths, meaning that all three sides can be expressed as fractions. When this is true, n is referred to as a congruent number. For example, 6 is a congruent number because it corresponds to the area of a right triangle with sides measuring 3, 4, and 5.

In 1640, Fermat famously demonstrated that 1 is not a congruent number using his renowned method of proof by infinite descent.

As the brilliant mathematician Keith Conrad points out regarding this result:

"This leads to a curious proof that ?2 is irrational. If ?2 were rational, then ?2, ?2, and 2 would form the sides of a rational right triangle with an area of 1. This contradicts the fact that 1 is not a congruent number!"

Since Fermat's time, the quest to prove or disprove the congruence of numbers has continued.

Astonishingly, one can show using elementary techniques that for every triple of rational numbers (a, b, c) satisfying a² + b² = c² and 1/2 ab = n, there exist two rational numbers x and y such that y² = x³ - n²x with y ? 0. Conversely, for each rational pair (x, y) such that y² = x³ - n²x and y ? 0, we can find three rational numbers a, b, c that satisfy a² + b² = c² and 1/2 ab = n.

In essence, right triangles with an area of n correspond directly to rational solutions to the equation y² = x³ - n²x where y ? 0, and vice versa. Mathematicians refer to this as a bijection between the two sets.

Thus, a rational number n is congruent if and only if the equation y² = x³ - n²x has a rational solution (x, y) with y ? 0. For instance, since 1 is not congruent, the only rational solutions to y² = x² - x yield y = 0.

For those interested in the exact correspondence:

When we apply this correspondence to the triangle with sides 3, 4, and 5, which has an area of 6, the corresponding solution is (x, y) = (12, 36).

> "To me, this is absolutely amazing. One starts with a problem in number theory and geometry and transforms it through algebra into a problem about rational points on plane curves!"

The equation y² = x³ - n²x represents an example of an elliptic curve.

Elliptic Curves

In general, if f(x) signifies a third-degree polynomial with a non-zero discriminant (meaning all roots are distinct), then y² = f(x) describes an elliptic curve, with the addition of a unique feature known as a "point at infinity."

> "Essentially, a point at infinity is a theoretical point where parallel lines can intersect. While explaining this concept fully is beyond the scope of this article, it is a fascinating topic that I encourage you to explore further."

By employing a minor algebraic transformation, we can make a rational change of coordinates, resulting in a new curve of the form y² = x³ + ax + b, where rational points on both curves correspond one-to-one. The transformed curve is typically simpler to analyze.

Consequently, it is common to assume that an elliptic curve is expressed in this form. Thus, when we refer to an "elliptic curve," we will consider it as a curve of the form y² = x³ + ax + b alongside a point at infinity.

Throughout this article, unless stated otherwise, we will assume that the coefficients a and b are rational numbers.

Elliptic curves exhibit two typical shapes, as illustrated below.

However, when we consider x and y as complex variables, the curves take on an entirely different appearance. They then manifest as a complex torus, resembling a doughnut!

> "So why do we study elliptic curves, and what can we accomplish with them?"

First and foremost, many problems in number theory can be translated into challenges involving Diophantine equations. Furthermore, elliptic curves are interconnected with discrete geometric entities known as lattices and are deeply related to important constructs called modular forms—highly symmetric complex functions rich in number-theoretic information.

In fact, the relationship between elliptic curves and modular forms was pivotal in proving Fermat’s Last Theorem, a feat accomplished by Andrew Wiles in the 1990s after years of intensive work on this connection.

The tale of this endeavor and the proof of the theorem is, in my view, one of the most beautiful pursuits in all of science. Unfortunately, as my friend Kenneth Nielsen pointed out, the margins in this Medium post are too narrow to encompass it!

I guess I’ll have to write another article.

Elliptic curves also play a crucial role in cryptography, facilitating the encryption of messages and online transactions.

What is most astounding about them, however, is the fact that they are more than mere curves and geometry. They possess an algebraic structure known as an Abelian group structure, complete with a fascinating geometric operation—a geometric addition rule for combining points on the curve.

If you are unfamiliar with Abelian groups, think of them as a set of objects equipped with an operation that mirrors the structure of integers concerning addition (though they can be finite).

In more specific terms, for a group with the operation , it must remain stable under this operation (i.e., if *a and b belong to the group, then a * b is also in the group). There exists an identity element e (0 for integers) such that a * e = a for all elements a in the group, and each element a has an inverse element c, satisfying a * c = e. Additionally, the group operation must be associative, meaning a * (b * c) = (a * b) * c. If the commutative property holds (i.e., a * b = b * a), then the group is termed an Abelian group.

Examples of Abelian groups include:

1. The integers ? with respect to addition (+).
2. The action of rotating a square clockwise by 90 degrees.
3. Vector spaces, where vectors are the elements and vector addition serves as the operation.

The sophisticated term for a curve endowed with an Abelian group structure is an Abelian variety.

What is remarkable about elliptic curves is that we can define an operation (denote it as ?) between rational points on them (where both x and y coordinates are rational numbers) such that the set of these points forms an Abelian group regarding the operation ? and the identity element being the point at infinity.

Let us define this operation.

When you take two rational points on the curve (let’s say P and Q) and consider a line passing through them, the line intersects the curve at another rational point (possibly the point at infinity). We’ll denote this intersection point as -R.

Since the curve is symmetric about the x-axis, reflecting -R across it yields another rational point R. The illustration below depicts this.

This reflected point (R in the image above) represents the sum of the two points mentioned earlier (P and Q). We can express this as P ? Q = R.

It can be proven (and this is actually quite complex) that this operation is associative, which surprises me. Furthermore, the point at infinity serves as a (unique) identity for this operation, and each point possesses an inverse point (obtained by reflecting across the x-axis). This structure is also Abelian (i.e., P ? Q = Q ? P).

The Mystery

It turns out that distinct elliptic curves can possess significantly different associated groups. A crucial invariant—arguably the most defining feature—is known as the rank of the curve (or group).

A curve may have either a finite or an infinite number of rational points. This can be challenging to navigate, so we focus on how many points are necessary to generate all others using the aforementioned addition rule. These generating points are referred to as basis points.

The rank serves as a measure of dimensionality, akin to the dimension of a vector space, indicating how many independent basis points (on the curve) have infinite order (meaning we can continually add them without returning to the starting point). If the curve contains only a finite number of rational points, then the rank is zero. While there still exists a group, it is finite.

Calculating the rank of an elliptic curve is notoriously difficult, but a valuable result from Mordell assures us that the rank is always finite. Thus, we need only a finite number of basis points to generate all rational points on the curve.

Among the most significant and intriguing challenges in number theory is the Birch and Swinnerton-Dyer Conjecture, which revolves around the rank of elliptic curves. It is so pivotal and challenging that it constitutes one of the Millennium Problems.

> "You actually receive a million dollars if you solve it!"

Finding rational points on elliptic curves with rational coefficients is a complex task. One method to tackle this is by reducing the curve modulo p, where p is a prime number. This approach means that instead of examining the rational solution set of the equation y² = x³ + ax + b, we investigate the rational solution set of the congruence y² ? x³ + ax + b (mod p). For this to be meaningful, we may need to clear denominators by multiplying both sides by an integer.

In this new context, two numbers with the same remainder when divided by p are considered equal. The advantage of this technique is that it limits the number of possible solutions to check. We denote the number of rational solutions to this reduced curve modulo p as Np.

In the early 1960s, Peter Swinnerton-Dyer employed the EDSAC-2 computer (not exactly a modern laptop!) at the University of Cambridge Computer Laboratory to determine the number of points modulo p on elliptic curves of known rank. Collaborating with mathematician Bryan John Birch, they aimed to deepen their understanding of elliptic curves. After the computer processed numerous products of the form:

for increasing values of x, they obtained the following output associated with the curve E: y² = x³ - 5x (as an example). It should be noted that the x-axis reflects log log x while the y-axis shows log y.

As a mathematician, I am not a statistician, but even I can discern a clear trend suggesting that the regression line has a slope of 1 in this plot.

The curve E has a rank of 1, and when they tested various curves of differing ranks, they consistently observed the same pattern. The slope of the fitted regression line seemed to correspond directly to the rank of the examined curve.

More precisely, they boldly conjectured that:

Here, C is some constant.

This computational exploration, combined with considerable foresight, led them to formulate a general conjecture regarding the behavior of a curve’s Hasse-Weil L-function L(E, s) at s = 1.

This L-function is defined as follows. Let

and let the curve's discriminant be denoted as ?. We can then define the L-function associated with E as the following Euler product:

We regard this as a function of the complex variable s.

Their conjecture can be stated as follows:

> Conjecture (Birch and Swinnerton-Dyer): Let E be any elliptic curve over ?. The rank of the abelian group E(?) of rational points of the curve E is equal to the order of the zero of L(E, s) at s = 1.

This conjecture was particularly prescient because, at the time, they were unaware if an analytic continuation existed for all such L-functions. The challenge was that L(E, s), as defined, converges only when Re(s) > 3/2.

The fact that all of them can be evaluated at s = 1 through analytic continuation was first established in 2001, leveraging the close relationship to modular forms that Andrew Wiles demonstrated.

Occasionally, the conjecture is expressed using the Taylor expansion of the L-function, but it conveys the same idea in an alternative format. The field of rational numbers can be substituted with a broader field, although I aimed to avoid excessive abstraction.

The study of elliptic curves represents a beautiful intertwining of number theory, abstract algebra, and geometry. Much more could be said about them than what I have outlined here, but I hope this article has provided you with a sense of the remarkable aspects of this subject.

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