分野別セミナー

Elliptic curves defined over the rational numbers are of great interest in modern number theory. The rank of an elliptic curve is a crucial invariant; indeed, there is a million-dollar prize problem about the rank!

There is great interest in the average rank of an elliptic curve. The minimalist conjecture is that the average rank should be 1/2. In 2007, Bektemirov-Mazur-Stein-Watkins, using well-known databases of elliptic curves, set out to numerically compute the average rank of elliptic curves, ordered by conductor. They found that "there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either."

The exciting recent work of Bhargava-Shankar has produced new theoretical upper bounds on average rank: when elliptic curves are ordered by height, the average rank is bounded above by 0.885. It was of interest to revisit the question of numerically computing average rank, under the ordering by height.

In joint work with Ho, Kaplan, Spicer, Stein, and Weigandt, we have assembled a new database of elliptic curves ordered by height. I will describe the database and some computations we have carried out.