Heuristics for the arithmetic of elliptic curves

Abstract

© Proceedings of the International Congress of Mathematicians, ICM 2018. All rights reserved. This is an introduction to a probabilistic model for the arithmetic of elliptic curves,a model developed in a series of articles of the author with Bhargava, Kane, Lenstra,Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, andwe make predictions about elliptic curves based on corresponding theorems provedabout the model. In particular, the model suggests that all but finitely many ellipticcurves over Q have rank - 21, which would imply that the rank is uniformly bounded.