The geometric distribution of Selmer groups of elliptic curves over function fields.

Fix a positive integer n and a finite field F q . We study the joint distribution of the rank rk (E) , the n-Selmer group Sel n (E) , and the n-torsion in the Tate–Shafarevich group as E varies over elliptic curves of fixed height d ≥ 2 over F q (t) . We compute this joint distribution in the large q limit. We also show that the "large q, then large height" limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains. [ABSTRACT FROM AUTHOR]