# Langlands Program

## A Vast Web of Correspondences

The Langlands program is a broad collection of conjectures connecting number theory, representation theory, harmonic analysis, and algebraic geometry. Its central idea is that two very different kinds of objects should secretly encode the same arithmetic information.

On one side are Galois representations, which come from field extensions, algebraic varieties, and étale cohomology. On the other side are automorphic representations, which come from harmonic analysis on adelic groups.

The expected bridge has the rough form

$$
\text{Galois representations}
\quad \longleftrightarrow \quad
\text{automorphic representations}.
$$

This correspondence generalizes many older results in number theory, including class field theory and the modularity of elliptic curves.

## Class Field Theory as the First Case

Class field theory describes abelian extensions of number fields. For a number field $K$, it relates the abelianized Galois group

$$
\operatorname{Gal}(K^{\mathrm{ab}}/K)
$$

to the idele class group

$$
K^\times\backslash \mathbb{A}_K^\times.
$$

This may be viewed as the $1$-dimensional case of the Langlands program.

Indeed, characters of the idele class group correspond to one-dimensional Galois representations. In this sense, Langlands generalizes class field theory from one-dimensional representations to higher-dimensional ones.

## Galois Representations

Let $K$ be a number field. A Galois representation is a homomorphism

$$
\rho:
\operatorname{Gal}(\overline{K}/K)
\to
\operatorname{GL}_n(E),
$$

where $E$ is usually a field such as $\mathbb{C}$, $\mathbb{Q}_\ell$, or a finite extension of $\mathbb{Q}_\ell$.

Such representations arise naturally from arithmetic geometry. For example, if $X$ is an algebraic variety over $K$, then étale cohomology produces representations of the absolute Galois group on vector spaces such as

$$
H^i_{\mathrm{\acute{e}t}}
(X_{\overline{K}},\mathbb{Q}_\ell).
$$

The eigenvalues of Frobenius elements at primes encode arithmetic data such as point counts modulo primes.

## Automorphic Representations

Automorphic representations arise from harmonic analysis on adelic quotients. For an algebraic group $G$, one studies representations occurring in spaces of functions on

$$
G(K)\backslash G(\mathbb{A}_K).
$$

When

$$
G=\operatorname{GL}_n,
$$

these representations are expected to match $n$-dimensional Galois representations.

The local factors of an automorphic representation

$$
\pi=\bigotimes_v \pi_v
$$

carry information at every place $v$ of $K$, just as a Galois representation has local behavior at every prime.

## Matching by $L$-Functions

One of the main ways to compare the two sides is through $L$-functions.

A Galois representation $\rho$ has an associated $L$-function

$$
L(s,\rho).
$$

An automorphic representation $\pi$ also has an associated $L$-function

$$
L(s,\pi).
$$

The Langlands philosophy predicts that corresponding objects satisfy

$$
L(s,\rho)=L(s,\pi),
$$

up to the expected local factors at bad primes.

This equality means that the same arithmetic information is being expressed in two different languages.

## Frobenius and Hecke Eigenvalues

At an unramified prime $p$, a Galois representation has a Frobenius element

$$
\mathrm{Frob}_p.
$$

The characteristic polynomial of

$$
\rho(\mathrm{Frob}_p)
$$

encodes local arithmetic information.

On the automorphic side, Hecke operators act on automorphic forms. Their eigenvalues encode local automorphic information.

The Langlands correspondence predicts that Frobenius eigenvalues match Hecke eigenvalues.

This principle is already visible in the theory of modular forms and elliptic curves.

## Example: Elliptic Curves and Modular Forms

Let

$$
E/\mathbb{Q}
$$

be an elliptic curve. For each prime $p$ of good reduction, define

$$
a_p=p+1-\#E(\mathbb{F}_p).
$$

The modularity theorem states that $E$ corresponds to a modular form

$$
f(q)=\sum_{n\geq1}a_nq^n
$$

such that the coefficient $a_p$ of the modular form matches the point-counting coefficient of the elliptic curve.

This is a concrete Langlands-type correspondence:

$$
\text{elliptic curve}
\quad \longleftrightarrow \quad
\text{modular form}.
$$

The proof of Fermat’s Last Theorem depended crucially on this connection.

## Local Langlands Correspondence

The Langlands program has both local and global forms.

The local Langlands correspondence concerns representations over local fields such as

$$
\mathbb{Q}_p
$$

or finite extensions of it.

For $\operatorname{GL}_n$, it relates irreducible admissible representations of

$$
\operatorname{GL}_n(K_v)
$$

to $n$-dimensional representations of the Weil-Deligne group of $K_v$.

This local theory describes what happens prime by prime.

The global theory assembles these local correspondences into statements about number fields and automorphic representations.

## Functoriality

Functoriality is one of the deepest principles in the Langlands program.

Suppose there is a homomorphism between dual groups

$$
{}^LG\to{}^LH.
$$

Langlands functoriality predicts a corresponding transfer of automorphic representations from $G$ to $H$.

This principle explains many mysterious identities between $L$-functions and automorphic forms.

Functoriality includes important cases such as:

$$
\operatorname{GL}_2
\to
\operatorname{GL}_3
$$

through symmetric square lifts, and many more general transfers.

## Geometric Langlands

There is also a geometric version of the Langlands program.

Instead of number fields, geometric Langlands studies curves over algebraically closed fields. It replaces automorphic forms with sheaves on moduli spaces of bundles and replaces Galois representations with local systems.

The rough correspondence becomes

$$
\text{local systems}
\quad \longleftrightarrow \quad
\text{sheaves on moduli spaces}.
$$

Geometric Langlands has deep connections with algebraic geometry, representation theory, mathematical physics, and category theory.

## Why the Program Matters

The Langlands program is not a single theorem. It is a framework that organizes a vast part of modern mathematics.

It connects:

| Galois Side | Automorphic Side |
|---|---|
| Field extensions | Harmonic analysis |
| Frobenius elements | Hecke operators |
| Galois representations | Automorphic representations |
| Étale cohomology | Representation theory |
| Arithmetic varieties | Automorphic forms |
| Motivic $L$-functions | Automorphic $L$-functions |

This dictionary allows problems in one domain to be translated into another, where different tools become available.

## Conceptual Importance

The Langlands program suggests that arithmetic has a hidden spectral structure.

Galois groups describe algebraic symmetries of fields. Automorphic representations describe analytic symmetries of adelic spaces. The Langlands correspondence predicts that these two symmetry theories are manifestations of the same underlying arithmetic order.

For modern number theory, this program provides one of the central organizing principles.