The most concrete example of a Selmer group is that of an elliptic curve. Fix an integer n > 0. Then the mod n Selmer group Sel(Q,E[n]) of an elliptic curve E over Q sits in an exact sequence
0 → E(Q)/nE(Q) → Sel(Q,E[n]) → Ш(Q,E)[n] → 0.That is, Sel(Q,E[n]) has a subgroup isomorphic to E(Q)/n, with quotient isomorphic to the n-torsion in the Shafarevich-Tate group Ш(Q,E) of E.
We are being slightly sloppy in our notation here, as the Selmer group Sel(Q,E[n]) depends on E and not merely on the n-torsion E[n] of E. (This phenomenon is why congruent elliptic curves need not have the same rank.) Nevertheless, if &rho is a p-adic Galois representation, it is possible to define a Selmer group Sel(Q,&rho) of an integral model of &rho via Galois cohomology and p-adic Hodge theory. If &rho is the p-power torsion representation of an elliptic curve E, then Sel(Q,&rho) agrees with the inverse limit of the p-power Selmer groups Sel(Q,E[pn]) of E.
Finally, then, the adjoint Selmer group Sel(Q,ad &Delta) of the modular form &Delta is simply the Selmer group of the three dimensional p-adic Galois representation obtained by composing the two-dimensional Galois representation &rho&Delta associated to the modular form &Delta with the map GL2 → GL3 given by the conjugation action of 2 × 2 matrices on the three dimensional vector space of trace zero 2 × 2 matrices. Viewing this latter action as the adjoint action of SL2 on its lie algebra, one finds that Sel(Q,ad &Delta) is closely related to the deformation theory of the reduction of &rho&Delta. It is for this reason that Selmer groups similar to this one were crucial to the proof of Fermat's last theorem by Andrew Wiles.Back