MILLENNIUM PRIZE SERIES: The Millennium Prize Problems are seven mathematics problems laid out by the Clay Mathematics Institute in 2000. They’re not easy – a correct solution to any one results in a US$1,000,000 prize being awarded by the institute.
Russian mathematician Grigori Perelman was awarded the Prize on March 18 last year for solving one of the problems, the Poincaré conjecture – as yet the only problem that’s been solved. Famously, he turned down the $1,000,000 Millennium Prize.
Over the coming weeks, each of these problems will be illuminated by experts from the Australian Mathematical Sciences Institute (AMSI) member institutions.
Here, Daniel Delbourgo explains the Birch and Swinnerton-Dyer Conjecture. Enjoy.
In simplest terms, one can describe these curves by using a cubic equation of the form
where A and B are fixed rational numbers (to ensure the curve E is nice and smooth everywhere, one also needs to assume that its discriminant 4A3 + 27B2 is non-zero).
To illustrate, let’s consider an example: choosing A=-1 and B=0, we obtain the following picture:
At this point it becomes clear that, despite their name, elliptic curves have nothing whatsoever to do with ellipses! The reason for this historical confusion is that these curves have a strong connection to elliptic integrals, which arise when describing the motion of planetary bodies in space.
The ancient Greek mathematician Diophantus is considered by many to be the father of algebra. His major mathematical work was written up in the tome Arithmetica which was essentially a school textbook for geniuses. Within it, he outlined many tools for studying solutions to polynomial equations with several variables, termed Diophantine Equations in his honour.
One of the main problems Diophantus considered was to find all solutions to a particular polynomial equation that lie in the field of rational numbers Q. For equations of “degree two” (circles, ellipses, parabolas, hyperbolas) we now have a complete answer to this problem. This answer is thanks to the late German mathematician Helmut Hasse, and allows one to find all such points, should they exist at all.
Returning to our elliptic curve E, the analogous problem is to find all the rational solutions (x,y) which satisfy the equation defining E. If we call this set of points E(Q), then we are asking if there exists an algorithm that allows us to obtain all points (x,y) belonging to E(Q).
At this juncture we need to introduce a group law on E, which gives an eccentric way of fusing together two points (p₁ and p₂) on the curve, to obtain a brand new point (p₄). This mimics the addition law for numbers we learn from childhood (i.e. the sum or difference of any two numbers is still a number). There’s an illustration of this rule below:
Under this geometric model, the point p₄ is defined to be the sum of p₁ and p₂ (it’s easy to see that the addition law does not depend on the order of the points p₁, p₂). Moreover the set of rational points is preserved by this notion of addition; in other words, the sum of two rational points is again a rational point.
where the number of copies of the integers Z above is called the “rank r(E) of the elliptic curve E”. The finite group ΤE(Q) on the end is uninteresting, as it never has more than 16 elements.
Thanks to Mordell’s efforts, in order to completely understand the group of rational points, we need to be able to find independent points p₁,p₂,…,pr(E) generating all of E(Q), by taking combinations of their sums and differences.
To use a biological analogy, these points could be thought of as the DNA sequence of the elliptic curve. Easy, eh?
At this point the harsher realities of life intervene. To date there is no known algorithm to determine the rank of an elliptic curve, although in practice there are tools that work well for specific examples.
Therefore the elephant in the room becomes:
Question. How do we determine the rank r(E) in general, for each elliptic curve E?
At that time computing was yet to make a major impact in mathematics, and there was no such thing as a laptop computer (unless your lap belonged to the Statue of Liberty).
They conducted experiments with the so-called L-function of an elliptic curve E, denoted as L(E,s). This mysterious object is defined analogously to the Riemann zeta-function, but instead counts points on the curve satisfying various congruence relations.
The precise definition of the L-function itself is highly non-trivial. As an illustration, tabulated below are the first few terms calculated for the elliptic curve E : y² + y = x³ - x² -10x - 20:
Hopefully you now get the picture! Many computer programs have been devised with the sole purpose of producing these types of expansions, for arbitrary elliptic curves. The above expression converges nicely if the real part of s is at least three halves, but it’s a labour of love to show that it extends to the whole complex plane as a well behaved function.
Birch and Swinnerton-Dyer made the quite astonishing discovery that the order of vanishing at s=1 of the function L(E,s) appears to be governed by the dimension of the lattice Z+Z+ … +Z sitting inside the group of rational points.
More succinctly, we have their famous prediction
The Birch and Swinnerton-Dyer Conjecture:
r(E) = orders=1 L(E,s).
Let’s have a closer look at another specimen, in this case the elliptic curve E : y² = 4x³ - 28x + 25. The size of the lattice inside E is three-dimensional:
Correspondingly, the L-function for E must then vanish to exact order r(E)=3 at the point s=1, that is:
L(E,s) = c(E) × (s-1)^r(E) + higher order terms in (s-1)
for some non-zero constant c(E). In fact Birch and Swinnerton-Dyer’s predictions give you a conjectural formula, describing how to compute each c(E) in terms of the arithmetic invariants of E.
One cannot emphasise enough how revolutionary this conjecture has subsequently turned out to be. It’s no exaggeration to say it has driven a significant portion of the last 50 years research in number theory.
Perhaps its most beautiful facet is that the Birch and Swinnerton-Dyer conjecture links together two radically different realms of mathematics: the algebraic world of polynomial equations (where elliptic curves reside), and the gritty world of complex functions (where the L-function naturally lives).
Aside from when the order of vanishing is tiny, there is no proof for the BSD conjecture! The largest known rank for an elliptic curve is 28, but it’s expected that the rank can be arbitrarily large in general.
This is the fifth part of the Millennium Prize Series. To read the other instalments, follow the links below.