---
tag: JQEYEQJY2G
kind: concept
name: Galois representation
msc: [11F80, 11R34, 11S37]
specializes: [WYGRXCG2RN]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_representation
same_as:
  wikidata: Q3427418
  wikipedia: Galois representation
---

## Summary

A Galois representation is a continuous [[representation|WYGRXCG2RN]] of the absolute [[Galois group|0XTY86BZKY]] 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|427N6S6QQW]], from the cohomology of varieties, and from [[modular forms|GWKN5VG78A]].
Matching them with automorphic objects is the content of the [[Langlands program|J9T2M084W5]].

## Explanations

### L1 human

The absolute Galois group of $\Q$ records all the symmetries of all algebraic numbers at once; it is far too large to study directly.
A Galois representation looks at this group through matrices: each symmetry becomes an invertible matrix, compatibly with composition.
For example, the symmetries act on the roots of unity of order $\ell^k$ for all $k$, and this gives the cyclotomic character, a $1 \times 1$ representation.
The torsion points of an elliptic curve give a $2 \times 2$ representation, which encodes how many points the curve has modulo each prime.