One of the seven Millennium Prize Problems—the Birch and Swinnerton-Dyer conjecture—is being reformulated to make it simpler to state and potentially easier to prove. This conjecture sits at the heart of number theory, asking whether certain equations called elliptic curves have finitely or infinitely many rational solutions. Elliptic curves are not abstract curiosities: they underpin much of modern cryptography, securing online transactions and communications. Currently, the conjecture is expressed through difficult analytic functions and an obscure object called the Tate-Shafarevich group. The researcher proposes a new version that compares the number of rational solutions to the number of solutions modulo prime numbers—a more geometric, less technically demanding formulation. If the reformulation succeeds, it could shift how mathematicians attack the conjecture, potentially leading to a full proof. That proof would unlock a deeper understanding of solving equations in integers, with ripple effects across cryptography and other fields that rely on elliptic curves. This is fundamental science: no immediate practical application is promised, but the Clay Institute’s prize problems have historically driven breakthroughs—Andrew Wiles’s proof of Fermat’s Last Theorem, for instance—that later reshaped entire areas of mathematics and its applications.
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One of the seven Millennium Prize Problems listed by the Clay Mathematics Institute is the Birch and Swinnerton-Dyer conjecture. This is one of the biggest open problems in number theory. By now, we know some results but a complete proof still seems far away -- and there are generalisations that put this at the very heart of a vast programme to understand how to solve equations in integers. The aim of this proposal is to reformulate the Birch and Swinnerton-Dyer conjecture and to put it into a new perspective. Let A and B be two integers and consider the equation y2 = x3+A x+B. Such equations are called elliptic curves and they have important applications in cryptography among many other areas of mathematics. We are interested in solving this equation in rational numbers x and y. For some choices of A and B there might be no solution at all, like for A=B=2; for some only finitely many, like when A=1 and B=2; but quite often there are infinitely many, like in the case A=B=1. It is hard to predict or calculate for a given A and B in which case we are. In the infinite case, we can refine the question and count the number N(T) of solutions (x,y) such that both numerators and denominators are between -T and T for a given number T. The new formulation of the conjecture now compares this counting function to the number of solutions of this equation "modulo p" for prime numbers p below T. Essentially, this means that we are looking for x and y between 0 and p-1 such that y2 and x3+A x+B have the same remainder when dividing by p. The usual formulation of the conjecture involves difficult analytic functions and intricate arithmetic terms like the mysterious Tate-Shafarevich group. The new version is simpler to state and has a more geometric flavour. But beyond the fact that the reformulation drops the level of difficulty of the mathematical objects involved, there is also hope that one can use it to change the way mathematicians look at it. One hope is to find somewhere a patch of finite area containing the points counted by the function N(T). It will not be a patch in the usual plane, but instead in the plane of adèles, a mathematical construction invented to bring tools from mathematical analysis into number theory.
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