Poster for the number theory year at the Centre de Recherches Mathématiques in Montreal
The four chapters entitled How to Explain Number Theory at a Dinner Party progress from a review of elementary facts about prime factorization and quadratic equations to an impressionistic account of the Birch-Swinnerton-Dyer (BSD) conjecture, introduced by name as a Guiding Problem for more than a generation of number theorists. At no point is the BSD conjecture stated; it is rather explained that it gives a way of determining the number of rational solutions of a cubic equation in two variables — an elliptic curve — with rational coefficients, in terms of the number of solutions, for every prime number p, of the congruence modulo p given by the same elliptic curve.
This is not incorrect, although a number theorist who reads this description knows that the literal “number” of rational solutions can be infinite and that the BSD conjecture measures the collection of solutions in a different way. At the same time, this description is incorrect insofar as the BSD conjecture is understood not only a way of understanding rational solutions to cubic equations but also as the first of a long line of conjectures about many different kinds of equations in any number of variables. Elliptic curves are the simplest case on which to test conjectures, and any general result of any kind about elliptic curves is likely to be treated as an avatar of a conjecture of immense scope.
Moreover, most of what is known in support of the BSD conjecture derives at least in part from the Gross-Zagier formula, mentioned in several places in the book. When this theorem was announced in 1983, it was seen as an isolated case, an identity of two infinite sums that had nothing obvious to do with one another but that miraculously gave the same result. Benedict Gross himself at one point wrote that he thought it would be the last theorem of its kind; but a few years later he realized that this identity was an avatar of another infinite class of conjectural identities, each of which would prove a case of the conjectures mentioned in the previous paragraph. (One of my own papers with Steve Kudla provided early indirect evidence for this viewpoint.) Partial generalizations of the Gross-Zagier formula and related formulas have grown into a full-fledged research program that was the subject of a two-week summer school in Paris last June.
These are the sorts of matters being discussed at the conference I am attending in Montreal. Unfortunately, there is absolutely no hope of doing justice to these topics at a dinner party. To tell our dinner companions that we have become number theorists in order to make online shopping more secure is, for most of us, one kind of lie. To claim that our goal is to solve cubic equations in two variables is a different kind of lie, for those of us who see cubic equations as the first step on an endless avatar ladder whose first rungs have already been visible for a long time.
Of course, some people really are working on cubic equations. Manjul Bhargava was awarded the Fields Medal in 2014 in part because of his work (with collaborators) in establishing that the BSD conjecture is true for most elliptic curves. The point I’m trying to make is not that cubic equations are anything other than fascinating but rather that, even when a number theorist has convinced a dinner party to sit still long enough to listen to an explanation of what number theory says about elliptic curves, the guests will still go home with only the vaguest idea of what it is that number theorists do.