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

Abstract: We give a positive answer to a Conjecture by Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra Jr., Bjorn Poonen and Eric 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 $n$-th Selmer group of $E$ is the intersection of two direct summands of the adelic cohomology group $H^1(\mathbf{A},E[n])$. We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.