nLab Galois representation

Idea

A Galois representation is a linear representation of a Galois group (often the absolute Galois group of some field). In other words, given a field extension $E$ of $F$, and a vector space $V$ over some field $k$, a Galois representation $\rho$ is a group homomorphism

$$\rho:\Gal(E/F)\to \GL(V)$$

where $\GL(V)$ is the group of linear transformations of $V$. If $V$ is an $n$-dimensional vector space, $\GL(V)$ is the same as the general linear group $\GL_{n}(k)$. A Galois module is a generalization of a Galois representation to modules instead of vector spaces.

Continuous Galois representations

When the Galois group is the absolute Galois group of a number field or a finite extension of the p-adic numbers, and the field of scalars of the underlying vector space of the representation is a topological field, one often considers continuous Galois representations (noting that the absolute Galois group is a topological group).

Examples

An example of a continuous Galois representation can be obtained by taking the $\ell$-adic Tate module as discussed in the article on Galois modules and taking its tensor product with $\mathbb{Q}_{\ell}$.

p-adic Galois representations

Continuous Galois representations of $\mathrm{Gal}(\overline{F}/F)$, where $F$ is a finite extension of the p-adic numbers $\mathbb{Q}_{p}$, over an underlying vector space over $E$, where $E$ is another extension of $\mathbb{Q}_{p}$ (often with some condition that $E$ is big enough) are called p-adic Galois representations (in contrast to $\ell$-adic Galois representations, where the underlying vector space is over some extension of $\mathbb{Q}_{\ell}$, where $\ell$ is another prime not equal to $p$). p-adic Galois representations have a very rich theory (significantly richer and more complicated than $\ell$-adic Galois representations), and their study is part of p-adic Hodge theory.

Conjectures on Galois Representations

Fontaine-Mazur Conjecture

The Fontaine-Mazur conjecture gives a criterion for when an $\ell$-adic Galois representation “comes from geometry”, i.e. is obtained from the $\ell$-adic etale cohomology of a variety. Namely, a Galois representation comes from geometry if and only if it is unramified at almost all places, and de Rham (see p-adic Hodge theory) at the places over $\ell$ (LiLFunctions).

Langlands Correspondence

A restricted version of the global Langlands correspondence (BuzzardMSRI) for $\GL_{n}$ states that algebraic automorphic representations of $\GL_{n}$ over $\mathbb{Q}$ are in bijection with compatible systems of $\ell$-adic Galois representations. A more general form is conjectured for number fields instead of $\mathbb{Q}$.

The local Langlands correspondence for $\GL_{n}$ states that irreducible admissible representations of $\GL_{n}(\mathbb{Q}_{p})$ are in correspondence with F-semisimple Weil-Deligne representations of $\Gal(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$. This has been proved. A more general form is also true for more general local fields instead of $\mathbb{Q}_{p}$.

For reductive groups other than $\GL_{n}$, Galois representations need to be replaced by the more general concept of L-parameters (and the statement becomes a classification, in terms of a partition into L-packets, rather than a bijection).

Serre’s Modularity Conjecture

Serre’s modularity conjecture states that an odd (this means the image of complex conjugation has determinant $-1$) irreducible two-dimensional Galois representation over a finite field comes from a modular form, and furthermore (in its strong form) gives a recipe for the level and weight of the modular form as well. This conjecture was proved by Khare and Wintenberger in KhareWintenberger09a and KhareWintenberger09b.

Modern formulations of the weight part of Serre’s conjecture are stated differently, inspired by the observation by Ash and Stevens (AshStevens86) that a Galois representation over a finite field is modular of prime-to-$p$ level $N$ and weight $k$ if and only if the corresponding system of Hecke eigenvalues appears in the group cohomology $H^1(\Gamma(N),\Sym^{k-2}\overline{\mathbb{F}}_{p}^{2})$. This is the same as requiring that the system of Hecke eigenvalues appears in $H^{1}(\Gamma(N),V)$, where $V$ is a Jordan-Holder factor of $\Sym^{k-2}\overline{\mathbb{F}}_{p}^{2}$.

Therefore, modern formulations of the weight part of Serre’s conjecture consists of associating to a Galois representation over a finite field $\mathbb{F}_{p}$ a set of Serre weights, which are irreducible $\overline{\mathbb{F}}_{p}$ representations of $\GL_{2}(\mathbb{F}_{p})$ (there are also generalizations to other groups). See also GHS18 for more discussion of this point of view.

Related concepts

• class field theory

• etale cohomology

• Galois cohomology

• p-adic Hodge theory

• Langlands correspondence

References

• Richard Taylor, Galois representations (pdf)

In the context of the Langlands program:

• Kevin Buzzard, MSRI Summer School on automorphic forms (web)

The statement of the Fontaine-Mazur conjecture as stated in this article comes from

• Chao Li, Arithmetic of L-Functions (notes taken by Pak-Hin Lee) (pdf)

The proof of Serre’s modularity conjecture is given in

• Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), Serre’s modularity conjecture (I), Inventiones Mathematicae, 178 (3): 485–504

• Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), “Serre’s modularity conjecture (II)”, Inventiones Mathematicae, 178 (3): 505–586

The modern formulation of the weight part of Serre’s conjecture is discussed in

• Toby Gee, Florian Herzig, David Savitt, General Serre Weight Conjectures, J. Eur. Math. Soc. 20 (2018), no. 12, 2859-2949 (arXiv:1509.02527)

The historical inspiration for the previous entry can be found in

• Avner Ash and Glenn Stevens, Modular forms in characteristic l and special values of their L-functions, Duke Math. J. 53 (1986), no. 3, 849–868.

Review of the fact that Galois representations encode local systems are are hence analogs in arithmetic geometry of flat connections in differential geometry includes

• Tom Lovering, Étale cohomology and Galois Representations, 2012 (pdf)

See also at function field analogy.

• Peter Scholze, Locally symmetric spaces, and Galois representations, Harvard CMD conference 2015, yt