Galois representation

Summary

A Galois representation is a continuous representation of the absolute Galois group of a field on a finite-dimensional vector space or free module. The field is usually a number field or a local field. Typical coefficients are the complex numbers, finite fields and $\ell$-adic numbers. Galois representations arise from the torsion points of elliptic curves, from the cohomology of varieties, and from modular forms. Matching them with automorphic objects is the content of the Langlands program.