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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   n  th Selmer group of   E   is the intersection of two direct summands of the adelic cohomology groupH  1    (  A  ,  E   [  n  ]   )    . We also give examples of elliptic curves for which the conclusion of this conjecture does not hold.

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