An n-dimensional Galois representation is a continuous homomorphism

&rho : Gal(F/Q) → GLn(K)
where F is a Galois extension of Q and K is a field. If K = C, then &rho is called an Artin representation. In this case, or if K is a finite field, then the continuity of ρ forces it to have finite image.

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(F/Q) → GLn(k).
Two p-adic Galois representations are said to be congruent if they have isomorphic reductions.