How many rational points does a random curve have?
A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over
, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank 0 and rank
, with higher ranks being negligible. We will describe these conjectures and discuss some results on bounds for average rank, highlighting recent work of Bhargava and Shankar.