Representations

A representation of a group G is a homomorphism of G to the automorphisms of a vector space V. In studying the representations of a group we can use linear algebra and try to obtain some information about a perhaps not well understood group. Representatons often arise naturally from any symmetries in the mathematical situation that we study.

Number theory is in may ways the study of the Galois groups of separable normal extensions of the field of rational numbers. Since we do not understand all such extensions (for example, Hilbert's question whether every finite group is the Galois group of an extension of the rationals still remains open) we can try instead to study the representations of these Galois groups and thereby linearize the groups.

Both elliptic curves and modular forms give rise to representations of Galois groups.