Langlands: Global Foundations¶

Job: state the global Langlands correspondence for $\mathrm{GL}_n$ as a matching of Galois (or motivic) data with automorphic representations, mediated by equal $L$-functions and compatible with local Langlands.

Core¶

Galois side¶

Let $F$ be a number field (or global function field). An $n$-dimensional $\ell$-adic Galois representation is a continuous homomorphism $$ \rho:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}n(\overline{\mathbb{Q}}\ell) $$ unramified outside a finite set $S$ of places (after conjugation), typically pure of some weight when arising from geometry. Frobenius conjugacy classes $\mathrm{Frob}_v$ for $v\notin S$ determine $$ L(s,\rho)=\prod_{v\notin S}\det\bigl(1-\rho(\mathrm{Frob}_v)\,q_v^{-s}\bigr)^{-1}\times\text{(ramified factors)}. $$ Artin representations (finite image, $\ell$-independent after comparison) are the classical special case.

Geometric source. Étale cohomology $H^i_{\mathrm{\acute{e}t}}(X_{\overline{F}},\mathbb{Q}_\ell)$ of a smooth proper $X/F$ yields Galois representations; purity (Deligne) controls eigenvalues of Frobenius. See arithmetic geometry and AG cohomology.

Automorphic side¶

An automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ is an irreducible admissible constituent of $L^2\bigl(\mathrm{GL}_n(F)\backslash\mathrm{GL}_n(\mathbb{A}_F), \omega\bigr)$ (or its smooth/ $K$-finite avatar), with central character $\omega$. Restricted tensor product $$ \pi\cong\bigotimes_v{}'\pi_v, $$ $\pi_v$ smooth irreducible of $\mathrm{GL}_n(F_v)$ (archimedean: $(\mathfrak{g},K)$-module / Casselman–Wallach).

Standard $L$-function. Godement–Jacquet / Rankin–Selberg / Langlands–Shahidi packages produce $$ L(s,\pi)=\prod_v L(s,\pi_v), $$ entire (or meromorphic of controlled poles for $n=1$) of order $1$ after completing with archimedean $\Gamma$-factors, with functional equation $L(s,\pi)=\varepsilon(s,\pi)L(1-s,\widetilde{\pi})$.

Object	Galois / motivic	Automorphic
Local data at $v$	$\mathrm{WD}(\rho	{W)$}
Unramified invariant	char. poly of $\mathrm{Frob}_v$	Satake parameters of $\pi_v$
Global invariant	$L(s,\rho)$, $\varepsilon(s,\rho)$	$L(s,\pi)$, $\varepsilon(s,\pi)$
Functoriality	$\rho\mapsto r\circ\rho$	transfer $\pi\mapsto\Pi$ along $r:{}^LG\to{}^LH$

The correspondence (conjectural package)¶

Conjecture (Langlands for $\mathrm{GL}_n/F$, schematic). There is a bijection (up to stated equivalence) between:

irreducible continuous $\rho:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}_n(\overline{\mathbb{Q}}_\ell)$ that are “geometric / automorphic type” (de Rham at $\ell$-adic places, pure, etc.), and
cuspidal automorphic $\pi$ on $\mathrm{GL}_n(\mathbb{A}_F)$ with matching algebraic infinity type,

such that $L(s,\rho)=L(s,\pi)$ (including $\varepsilon$-factors) and local–global compatibility holds at every place. For general reductive $G$, replace $\mathrm{GL}_n$ by $G$ and $\mathrm{GL}_n$ dual by ${}^LG$; packets and endoscopy appear.

Known pillars.

Case	Status
$n=1$	class field theory — theorem
$n=2$, $F=\mathbb{Q}$, modular elliptic curves	modularity theorem
Many $2$-dim odd irreps over $\mathbb{Q}$	Serre modularity (Khare–Wintenberger)
Cyclic base change, solvable base change	Arthur–Clozel et al.
Function fields (Drinfeld, L. Lafforgue, V. Lafforgue)	much stronger theorems
General $n$, number fields	largely open

$L$-functions matching (what “equal” means)¶

Matching requires: same Euler factors at unramified $v$; same Artin conductors / local $\varepsilon$-factors at ramified and archimedean places; same functional equation. Analytic continuation and boundedness in vertical strips on the automorphic side are theorems for $\mathrm{GL}_n$; on the Galois side they are conjectural in general (Brauer induction reduces Artin $L$ to Hecke, but general geometric $\rho$ need motivic / automorphy).

Local Langlands pointer¶

At each place $v$, local Langlands for $\mathrm{GL}_n(F_v)$ identifies irreducible smooth $\pi_v$ with $n$-dimensional Frobenius-semisimple Weil–Deligne representations. Global reciprocity is constrained by this local dictionary. Details: local.

Modularity theorem as special case¶

Theorem (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor, …). Every semistable (then all) elliptic curve $E/\mathbb{Q}$ is modular: there is a weight-$2$ cusp form $f$ with $L(E,s)=L(f,s)$. Via Eichler–Shimura / Deligne, $f$ produces a compatible system of $2$-dimensional Galois representations isomorphic to those of $T_\ell(E)$.

Thus modularity $=$ global Langlands for the $2$-dimensional representations attached to elliptic curves over $\mathbb{Q}$. Potential modularity and automorphy lifting (Kisin, Emerton, Barnet-Lamb–Gee–Geraghty–Taylor, …) extend the method far beyond elliptic curves.

Functoriality (one paragraph)¶

Langlands functoriality: a map of $L$-groups $r:{}^LG\to{}^LH$ should induce a transfer of automorphic packets $G\rightsquigarrow H$. Instances: cyclic base change, symmetric powers (partial), endoscopic transfer (Ngo’s fundamental lemma $\Rightarrow$ Arthur for classical groups). Functoriality implies analytic properties of many $L$-functions by reducing them to $\mathrm{GL}_n$.

See it¶

Artin conjecture via Langlands. For irreducible continuous $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_n(\mathbb{C})$, Artin conjectured $L(s,\rho)$ entire if $\rho$ is nontrivial. If $\rho$ is automorphic (Langlands), entirety follows from known properties of cuspidal $L(s,\pi)$. Proven for $n=1$ (Hecke) and many $n=2$ cases (modular forms of weight $1$, Deligne–Serre; Langlands–Tunnell for icosahedral pieces used by Wiles). The general Artin conjecture remains open and is a poster child for needing global Langlands.

Check¶

Sketch why the modularity theorem implies FLT for semistable freeness (Frey–Serre–Ribet strategy).

A hypothetical nontrivial solution of $a^p+b^p=c^p$ ($p>2$) yields a semistable elliptic curve $E_{a,b,c}$ (Frey) whose mod-$p$ residual representation is irreducible and modular of level $2$ after Ribet’s level lowering — contradicting that the only weight-$2$ level-$2$ cusp forms space is $0$. Wiles’ modularity of semistable $E/\mathbb{Q}$ closes the contradiction.

What data does an unramified local component $\pi_v$ of $\mathrm{GL}_n$ contribute to $L(s,\pi)$?

Satake isomorphism: unramified $\pi_v\leftrightarrow$ semisimple conjugacy class $t_{\pi_v}\in\mathrm{GL}_n(\mathbb{C})$ (up to permutation of eigenvalues $\alpha_{1,v},\ldots,\alpha_{n,v}$); local factor $L(s,\pi_v)=\prod_{i=1}^n(1-\alpha_{i,v}q_v^{-s})^{-1}$.

Pitfalls¶

Cuspidal vs isobaric: general automorphic $L$-functions on $\mathrm{GL}_n$ factor through isobaric sums of cuspidal ones (Jacquet–Shalika); Galois side should match semisimple $\rho$.
$\ell$-adic vs complex: comparison isomorphisms and coefficient fields need care; “the” associated $\pi$ is independent of $\ell$ in expected systems.
Infinity type: algebraic automorphic representations must match Hodge–Tate weights / infinity components; not every unitary $\pi$ is expected from geometry.
Beyond $\mathrm{GL}_n$: multiplicity, $A$-packets, and endoscopic classification replace naive bijections.

Refs¶

Arthur, The Endoscopic Classification of Representations (classical groups)
Taylor, Galois representations (ICM / Bourbaki surveys)
Gelbart, An elementary introduction to the Langlands program (Bull.\ AMS)
Bump–Cogdell–de Shalit–Gaitsgory–Kowalski–Kudla, An Introduction to the Langlands Program (Jerusalem lectures)

Next¶

Langlands index · Local Langlands · Class field · L-functions · Analytic NT · Arithmetic geometry · AG / Galois reps & Fontaine–Mazur — the geometric Galois reps / the Fontaine–Mazur / the modularity · Representation theory · Geometric Langlands (derived) — the categorical upgrade of the global Langlands conjecture · SAG / Morava / Lubin–Tate — the Lubin–Tate tower realizes the local Langlands; the Morava $E$-theory connects