In this section, we will describe very roughly the Birch and Swinnerton-Dyer conjecture [1] which is related to the number of points with rational coordinates on “elliptic curves”. This amounts to solving a special class of Diophantine equations, that is, polynomial equations whose coefficients are integers or rational numbers, and looking for their integer or rational solutions. The conjecture was developed at the beginning of the 1960s by the mathematicians Bryan Birch (born in 1931) and Peter Swinnerton-Dyer (1927-2018) with the help of EDSAC (an ENIAC descendant), at University of Cambridge Computer Laboratory. An elliptic curve E over the field of rational numbers is the set of solutions to an equation of the form $y^2=x^3+ax+b$, where $a$ and $b$ are rational numbers such that $4a^3+27b^2\neq 0$. [2]
On the adjacent figure, we show an example with $a=-1$ and $b=0$. The point $(2,\sqrt{6})$ lies on E but it is not an integer-valued point. By setting $y = 0$, we obtain the integer-valued points (0,0), (1,0) and (−1,0). Fermat proved by his method of infinite descent that these three points are the only rational points on E .
A striking feature of rational solutions to elliptic curve equations is that these solutions form an abelian group which is fintely generated the Mordell-Weil theorem. We refer to e.g. [3] for details. Of course, number theorists want to calculate this group. That involves finding a system of generators : rational solutions from which all others can be deduced by repeatedly using the group operation. At least, we would like to know how big this group is. By Mordell’s theorem, as any finitely generated abelian group, it can be decomposed into the direct sum of a finite number of disjoint copies of the group of integers $\mathbb{Z}$, and a finite abelian group, called torsion (which, again, cane be decomposed into the direct sum of finite cyclic factors). The finite-part piece is rather well understood. The difficult part, which is the subject of the Birch and Swinnerton-Dyer conjecture, is to find the number of copies of $\mathbb{Z}$ that appear in the infinite-part piece. This number is called the “rank” of the elliptic curve, and it is denoted by r. To be more precise, to get a group one has to add a point “at infinity” to the rational points of the given elliptic curve E, because the “natural” way to to view an elliptic curve is to view it as a curve in projective space P2. Then this group can be written as the direct sum of $\mathbb{Z}^r$ and a “torsion subgroup” which is characterized by a theorem of Mazur ; see [1].
Let us come back to Birch and Swinnerton-Dyer. Their idea was to numerically look for the points on an elliptic curve modulo a given prime p, as there are only a finite number of possibilities to check. Of course, for large primes it is computationally intensive. They computed the number of points modulo $p$ (denoted by $N_p$ ) on the elliptic curves $y^2=x^3-dx$, for five values of d for which the rank was known. Birch noticed that Swinnerton-Dyer’s computer experiments produce an interesting pattern if you divide $N_p$ by the prime concerned. Then multiply all of these fractions together, for all primes less than or equal to a given one, and plot the results against successive primes on logarithmic graph paper. The data all seem to lie close to a straight line, whose slope is the rank of the elliptic curve. They obtained the graphical plots shown in the figure below. They conjectured that
$$\pi_{E_d}(X) :=\prod_{p\leq X}\frac{N_p}{p} \sim C\, (\log X)^r,\quad X\to\infty$$
where $C$ is a constant. In fact, the function $\pi_E$ is difficult to work with, so they stated a related conjecture involving the so-called $L$-function of $E$ in place of $\pi_E$. We will not give any detail here and again refer to [3].
Let us finish this section by quoting Ian Stewart [4] :
“In the 1960s, when computers were just coming into being, the University of Cambridge had one of the earlier ones, called EDSAC. Which stands for electronic delay storage automatic calculator, and shows how proud its inventors were of its memory system, which sent sound waves along tubes of mercury and redirected them back to the beginning again. It was the size of a large truck, and I vividly remember being shown round it in 1963. Its circuits were based on thousands of valves-vacuum tubes. There were vast racks of the things along the walls, replacements to be inserted when a tube in the machine itself blew up. Which was fairly often.”
