Stanley Yao Xiao, Department of Pure Mathematics, University of Waterloo
“Binary quartic forms and average ranks of elliptic curves”
In this talk I aim to describe the approach taken by Bhargava and Shankar to show that the average rank of elliptic curves is bounded. It turns out that the key observation is controlling the average size of various Selmer groups. Various improvements on the bound for the rank depends on knowledge of 2,3,5-Selmer groups, respectively. In the 2-Selmer case, the key is to count 2-Selmer elements via a bijection with binary quartic forms, thus turning the problem of bounding the rank of elliptic curves into counting equivalence classes of integral binary quartic forms.