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 →That is, Sel(E(Q)/nE(Q) → Sel(Q,E[n]) → Ш(Q,E)[n] → 0.

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*[*p*^{n}]) 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
GL_{2} → GL_{3}
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
SL_{2} 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.