Average Ranks of Elliptic Curves

Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins

Relevant Computer Code
The Stein-Watkins Database


Suppose you are given an algebraic curve C defined, let us say, as the locus of zeroes of a polynomial f(x,y) in two variables with rational coefficients. Suppose you are told that C has at least one rational point, i.e., there is a pair of rational numbers (a,b) such that f(a,b)=0. How likely is it that C will have infinitely many rational points?

Rational points on elliptic curves are the gems of the arithmetic theory: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit, and control---thanks to ideas of Kolyvagin---the existence of rational points and other mathematical structures elsewhere. Despite all that we presently know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today. The subject of this conflict is average ranks of elliptic curves.

We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data supports our conjectures; and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery.