Heuristics for the arithmetic of elliptic curves

Bjorn Poonen

Published 2017-11-28Version 1

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, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over $\mathbb{Q}$ have rank $\le 21$, which would imply that the rank is uniformly bounded.