Selmer groups are intersection of two direct summands of the adelic cohomology

Abstract

We give a positive answer to a conjecture by Bhargava, Kane, Lenstra Jr., Poonen and Rains, concerning the cohomology of torsion subgroups of elliptic curves over global fields. This implies that, given a global field k and an integer n, for 100% of elliptic curves E defined over k, the nth Selmer group of E is the intersection of two direct summands of the adelic cohomology group H1false(boldA,E[n]false). We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.