Saito and Langer in Duke J 1996 prove a strange finiteness result for certain Selmer groups.
arXiv:1205.4456 Generalized explicit descent and its application to curves of genus 3 from arXiv Front: math.AG by Nils Bruin, Bjorn Poonen, Michael Stoll We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically-defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over Q of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus-3 examples defined by polynomials with small coefficients.
arXiv:1108.3364 Unfaking the fake Selmer group from arXiv Front: math.AG by Michael Stoll, Ronald van Luijk For any abelian variety J over a global field k and an isogeny phi: J -> J, the Selmer group Sel^phi(J,k) is a subgroup of the Galois cohomology group H^1(Gal(k^s/k),J[phi]), defined in terms of local data. When J is the Jacobian of a cyclic cover of P^1, the Selmer group has a quotient by a subgroup of order at most 2 that is isomorphic to the “fake Selmer group”, whose definition is more amenable to explicit computations. In this paper we define in the same setting the `unfake Selmer group', which is isomorphic to the Selmer group itself and just as amenable to explicit computations as the fake Selmer group.
arXiv:1301.4724 Selmer groups as flat cohomology groups fra arXiv Front: math.AG av Kestutis Cesnavicius Given a prime number p, Bloch and Kato showed how the p^\infty-Selmer group of an abelian variety A defined over a number field K is determined by the p-adic Tate module T_p(A). In general one cannot hope that the p^m-Selmer group of A would be determined by the mod p^m Galois representation A[p^m]. We show, however, that this is the case if p is large enough. More precisely, we determine a finite explicit set of primes S depending on K and A, such that the p^m-Selmer group of A is determined by A[p^m] for all p not in S. To do this we observe that for such p, the p^m-Selmer group agrees with the flat cohomology group of the ring of integers of K with coefficients in the p^m-torsion of the Neron model of A. We give a Selmer-type description for this flat cohomology group in terms of local conditions, which, for p not in S, agree with the local conditions defining the p^m-Selmer group and depend only on A[p^m].
nLab page on Selmer group