An *n*-dimensional *Galois representation* is a continuous
homomorphism

&rho : Gal(whereF/Q) → GL_{n}(K)

If *K* is a
*p*-adic field, then there exist &rho with infinite
image. Important sources of two-dimensional *p*-adic
Galois representations are
elliptic curves
(via their *p*-power
torsion points) and
modular forms.
More generally, the
étale
cohomology
of algebraic varieties is a plentiful source of *p*-adic
Galois representations.
The
Langlands' conjectures predict that automorphic representations
should also give rise to Galois representations; see
Richard Taylor's excellent expository account
*Galois representations* for more details.

If ρ is a *p*-adic Galois representation and *k* is the
residue field of the ring of integers of *K*, then there is
a well-defined semisimple reduction

&rho : Gal(TwoF/Q) → GL_{n}(k).