I would like to thank Mr H P F Swinnerton-Dyer for his very generous help in the preparation of this paper.

A game with perfect information has an equilibrium-point of pure strategies; this was first proved for two-person games by Zermelo and extended to n-person games by the reviewer. More recently, Dalkey and Otter and Dunne have proved the stronger result: If in the complete inflation of a game $\Gamma$ every player has complete information about every other player, then $\Gamma$ has an equilibrium-point of pure strategies.

The present paper extends this result by defining a game $\Gamma$ to have almost complete information for a given player if he has complete information about every player, and every other player has complete information about him. Theorem. If $\Gamma$ is any game and $\overline{\Gamma}$ is its complete inflation, there is an equilibrium-point of $\overline{\Gamma}$ at which those players for whom $\overline{\Gamma}$ has almost complete information have pure strategies. (A slightly restricted converse is also shown.) The proof is based on a theory of the decomposition of games initiated by the reviewer. A key result obtained in this paper asserts that every game structure can be completely decomposed into indecomposable components; this decomposition is unique and leads to a tree of substructures. Another result, too complicated to reproduce here, characterises the equilibrium-points of a game structure in terms of those of a subgame and a difference game. A sketch of a decomposition theory relevant to behaviour strategies is also presented.

I would like to thank Dr Cassels for his helpful advice on the preparation of this note.

... as a teaching fellow I got to know him well; he taught me to love opera (I have happy memories of sitting on the floor listening to his recording of Callas singing Casta Diva) ...

... followed Harold Davenport in applying analytic methods to prove results about the zeros of rational polynomials in many variables. For instance, if such a polynomial of odd degree has enough variables, it will certainly have a rational zero.

... while I was there [at Princeton] I both wrote joint papers with Davenport by transatlantic mail, and also learnt a great deal of new mathematics. In particular I learnt of the beautiful reformulation of Siegel's work on quadratic forms in terms of a natural "Tamagawa measure" for linear algebraic groups. I seem to remember that Tamagawa gave a lecture, and Weil's comments made it exciting.

Siegel has shown that the density of rational points on a quadric surface can be expressed in terms of the densities of p-adic points; which for almost all primes p depends directly on the number of solutions of the corresponding equation in the finite field with p elements. More recently, Siegel's work has been fruitfully extended and simplified by Tamagawa and independently by Kneser.

It is natural to hope that something similar will happen for the elliptic curve

          $\Gamma : y^{2} = x^{3} - Ax - B$

where A, B are rational. In particular, one hopes that if for most p the curve has unusually many points in the finite field with p elements, then it will have a lot of rational points.

Several years ago we carried out by hand some calculations which tended to support these hopes. Since then, we have used the Cambridge University electronic computer EDSAC 2 for extensive calculations on elliptic curves. As a result, we have been able to put our conjectures into an exact form and support them with a good deal of numerical evidence. However, it has become clear to us that we are unlikely to be able to prove these conjectures.

We therefore feel it best to publish the material we have obtained, in a series of short notes. Much of the information in these notes is the result of machine computation; however the theoretical basis of these computations is not always trivial.

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve $\mod p$ to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorisation of numbers into primes, and cryptography, to name three.
...
When the solutions [of an elliptic equation] are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behaviour of an associated zeta function $\zeta (s)$ near the point $s = 1$. In particular this amazing conjecture asserts that if $\zeta (1)$ is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if $\zeta (1)$ is not equal to 0, then there is only a finite number of such points.

The conjecture discovered jointly by Peter Swinnerton-Dyer and Bryan Birch in the early 1960s both surprised the mathematical world, and also forcefully reminded mathematicians that computations remained as important as ever in uncovering new mysteries in the ancient discipline of number theory. Although there has been some progress on their conjecture, it remains today largely unproven, and is unquestionably one of the central open problems of number theory. It also has a different flavour from most other number-theoretic conjectures in that it involves exact formulae, rather than inequalities or asymptotic questions.

Let C be an elliptic curve defined over the rationals with at least one rational point; so C is an abelian variety of dimension 1. The set of rational points of C form a subgroup, call it $C (\mathbb{Q})$, of the group of points of C; Mordell has shown that $C (\mathbb{Q})$ is finitely generated. There are effective methods of determining the points of $C (\mathbb{Q})$ of finite order; the problem is, how may one determine g, the number of independent generators of infinite order? Apparently there is a considerable theory waiting to be discovered; at any rate, there are many interrelated conjectures as to what such a theory should be. At the Stockholm congress, Cassels described certain conjectures due to Swinnerton-Dyer and the speaker, by which g was related to the zeta function of the curve C. ...

Modular curves and their close relatives, Shimura curves attached to multiplicative subgroups of quaternion algebras, are equipped with a distinguished collection of points defined over class fields of imaginary quadratic fields and arising from the theory of complex multiplication: the so-called Heegner points. It is customary to use the same term to describe the images of degree zero divisors supported on these points in quotients of the Jacobian of the underlying curve (a class of abelian varieties which we now know is rich enough to encompass all elliptic curves over the rationals). It was Birch who first undertook, in the late 70's and early 80's, a systematic study of Heegner points on elliptic curve quotients of Jacobians of modular curves. Based on the numerical evidence that he gathered, he observed that the heights of these points seemed to be related to first derivatives at the central critical point of the Hasse-Weil L-series of the elliptic curve (twisted eventually by an appropriate Dirichlet character). The study initiated by Birch was to play an important role in the number theory of the next two decades, shedding light on such fundamental questions as the Gauss class number problem and the Birch and Swinnerton-Dyer conjecture.

Heegner was a fine mathematician, with a rather low-grade post in a gymnasium in East Berlin; he clearly knew Heinrich Weber's book well. He was interested in the congruence number problem: recollect that m is a congruence number if it is the area of a right-angled triangle with rational sides (most people call this a Pythagorean triangle; Heegner called it a Harpedonapten triangle). In his famous, very eccentrically written, paper he begins with a historical introduction concerning the congruence number problem, then he quotes various things from Weber and proves some highly surprising theorems showing that the congruence number problem is soluble for certain families of m; and then he suddenly (correctly but over succinctly) solves the classical class number one problem. Unhappily, in 1952 [when Heegner published his paper] there was no one left who was sufficiently expert in Weber's 'Algebra' to appreciate Heegner's achievement.
...
Heegner's paper was written in an amateurish and rather mystical style, so perhaps it was not surprising that at the time no one tried very hard to understand it. It was thought that his solution of the class number problem contained a gap, and though his work on the congruence number problem was clearly correct, no one realised that it contained the germs of a valuable new method. Sadly, he died in obscurity.

Looking back at old diaries and suchlike, I find that I first saw Heegner's paper in 1966 (a little later than Stark, he tells me); I had been told it was wrong, but so far as I could see, it followed from results in Weber's 'Algebra'; and his results on points on elliptic curves were exciting. It took a while to decide he was right (one had to read Weber first, and I hadn't even got good German) but this was achieved by the end of 1967. It took very much longer to understand it properly, maybe until 1973; it was necessary to both simplify and generalise. One needed to replace Heegner's rather miraculous construction of rational points on certain elliptic curves by a theorem that modular elliptic curves, indeed modular curves, are born with natural points on them, defined over certain classfields.

The Senior Whitehead Prize is awarded to B J Birch for his work in number theory, and in particular for his outstanding contributions to the arithmetic of elliptic curves.

... in recognition of his influential contributions to modern number theory. His joint work with Peter Swinnerton-Dyer on elliptic curves created an exciting new area of arithmetic algebraic geometry: the Birch-Swinnerton-Dyer conjecture remains after 40 years one of the most tantalising problems in modern mathematics. Birch's work on Heegner points has led to huge advances in the arithmetic of elliptic curves.

... for work that has played a major role in driving the theory of elliptic curves, through the Birch-Swinnerton-Dyer conjecture and the theory of Heegner points.

I was fortunate enough to be working on the arithmetic of elliptic curves when comparatively little was known, but when new tools were just becoming available, and when forgotten theories such as the theory of automorphic function were being rediscovered. At that time, one could still obtain exciting new results without too much sophisticated apparatus: one was learning exciting new mathematics all the time, but it seemed to be less difficult!