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

TL;DR

This paper proves a conjecture that for most elliptic curves over global fields, the Selmer group is the intersection of two direct summands in adelic cohomology, with some exceptions provided.

Contribution

It confirms a conjecture relating Selmer groups to adelic cohomology for most elliptic curves over global fields, advancing understanding of their structure.

Findings

Proves the conjecture for 100% of elliptic curves over global fields.

Provides counterexamples where the conjecture does not hold.

Establishes a new perspective on Selmer groups as intersections of summands.

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.