Wei Ho

Abstract: In 2007, Bektemirov, Mazur, Stein, and Watkins published a beautiful paper ("Average ranks of elliptic curves: tension between data and conjecture") that described the state of the art in the study of the distribution of ranks of elliptic curves over Q. The last decade has seen progress on these questions in many directions, perhaps the most surprising being genuine theorems giving upper bounds on the average ranks of elliptic curves over Q (led by work of Bhargava-Shankar). In these theorems, the set of elliptic curves over Q are ordered by height, which is a measure of the size of the coefficients defining each curve, as opposed to by conductor, a more arithmetic invariant used to order curves in most previous work.

In this talk, we will discuss the updated heuristics and conjectures in this field as well as some of these theorems and the main ideas behind their proofs. We will compare the expected behavior with data from a new database of elliptic curves over Q ordered by height (joint work with Balakrishnan, Kaplan, Spicer, Stein, and Weigandt); perhaps the new data resolves some of the old tension?