On the structure of Selmer groups
Let \(E\) be a modular elliptic curve over \(\mathbb{Q}\) and \(K\) be an imaginary quadratic field of discriminant \(\equiv 0\pmod{4\times\text{conductor }E)}\). For a rational prime \(\ell\) a set \(T\subset\bigcup_ n H^ 1(K,E_{\ell^ n})\) of explicit cohomology classes was constructed earlier by the author [Euler systems, The Grothendieck Festschr., Vol. II, Prog. Math. 87, 435-483 (1990; Zbl 0742.14017)]. Let \(\ell\) be such that the image of the \(\ell\)-adic representation of \(\text{Gal}(\overline {\mathbb{Q}}/\mathbb{Q})\) corresponding to \(E\), is maximal and \(\ell\) does not divide the discriminant of \(\text{End}(E)\). All sufficiently large \(\ell\) satisfy this condition. Then, conjecturally, \(T\neq 0\). If \(T\neq 0\) we obtain some expressions for invariants of Selmer groups \(S(K,E_{\ell^ n})\) in terms of \(T\). Analogous results for the remaining primes are formulated.
scientific article; zbMATH DE number 59957