The Langlands programme: an orientation

MotivationH

Three polynomials modulo p

Fix a polynomial with integer coefficients and ask, for each prime $p$ : how many roots does it have modulo $p$ ?

$x^2 + 1$ . For odd $p$ , it has two roots modulo $p$ when $p \equiv 1 \pmod 4$ and none when $p \equiv 3 \pmod 4$ . The answer depends only on $p \bmod 4$ .
$x^3 - x - 1$ . It has no roots modulo $13$ and three modulo $59$ , although $13 \equiv 59 \pmod{23}$ — and one can show that no congruence condition, to any modulus, decides the number of roots. Yet there is a rule. Expand
$\eta(z)\eta(23z) = q\prod_{n \ge 1}(1 - q^n)(1 - q^{23n}) = q - q^2 - q^3 + q^6 + q^8 - q^{13} - q^{16} + q^{23} - \cdots$
Then for every prime $p \neq 23$ , the number of roots of $x^3 - x - 1$ modulo $p$ is the coefficient of $q^p$ plus one. (The coefficient of $q^{13}$ is $-1$ : no roots. The coefficient of $q^{59}$ is $2$ : three roots.)
$y^2 + y = x^3 - x^2 - 10x - 20$ , an elliptic curve. Let $a_p = p - \#\{(x, y) \bmod p\}$ . Then $a_p$ is the coefficient of $q^p$ in $q\prod_{n\ge1}(1 - q^n)^2(1 - q^{11n})^2$ , for every prime $p$ .

The infinite products are modular forms: holomorphic functions on the upper half-plane with an enormous group of symmetries. That arithmetic data — how polynomials factor mod $p$ , how many points a curve has mod $p$ — should be encoded in the Fourier coefficients of such symmetric functions is the heart of the Langlands programme. In a 1967 letter to André Weil, Robert Langlands proposed that this is a universal phenomenon: arithmetic objects (Galois representations) correspond to analytic objects (automorphic representations), with matching $L$ -functions, and that the analytic objects for different groups are related by a principle of functoriality.

This lesson is a map. It builds on algebraic number theory, analytic number theory and representation theory, and on the companion lesson on elliptic curves.

IntuitionH

Two sides of one dictionary

On the Galois side live the symmetries of algebraic numbers. The absolute Galois group $G_\Q = \operatorname{Gal}(\overline{\Q}/\Q)$ is a huge profinite group; we study it through its representations $\rho: G_\Q \to GL_n$ . For almost every prime $p$ there is a conjugacy class $\operatorname{Frob}_p$ in $G_\Q$ , and its characteristic polynomial under $\rho$ records how $p$ behaves — how a polynomial factors mod $p$ , or how many points a variety has mod $p$ .

On the automorphic side live functions with symmetry: modular forms, and more generally automorphic forms on $GL_n$ over the adèles. They are eigenfunctions of commuting Hecke operators $T_p$ , one for each prime, and the eigenvalues are numbers attached to primes.

Reciprocity says that every (suitable) $\rho$ matches an automorphic $\pi$ with "Frobenius eigenvalues = Hecke eigenvalues", and conversely. Both sides make $L$ -functions, and matching means equal $L$ -functions:

The two sides of the Langlands correspondence meet in their L-functions:

\operatorname{tr}\rho(\operatorname{Frob}_p)

matches the Hecke eigenvalue of

\pi

at

p

.

For $n = 1$ this dictionary is class field theory: one-dimensional Galois representations correspond to Dirichlet characters (and Hecke characters), and the first example above is its simplest case. For $n = 2$ over $\Q$ , example 2 is an instance of Hecke's theory of theta series and example 3 of the work of Eichler and Shimura, now a special case of the modularity theorem. For $n \ge 3$ most of the dictionary is conjectural.

Analogy, not a formal construction

Think of a prime $p$ as a "frequency". The Galois side tells you how an arithmetic object "vibrates" at frequency $p$ (through $\operatorname{Frob}_p$ ); the automorphic side is a single function whose Fourier coefficients list all these vibrations at once. Reciprocity says such a function always exists. The analogy is informal: the precise statements are about traces of Frobenius and Hecke eigenvalues.

DefinitionsH

Definitions

Definition— Frobenius element

Let $L/\Q$ be a finite Galois extension and $p$ a prime unramified in $L$ . For a prime $\mathfrak{P}$ of $L$ above $p$ , the Frobenius element $\operatorname{Frob}_{\mathfrak{P}} \in \operatorname{Gal}(L/\Q)$ is the unique $\sigma$ with $\sigma(x) \equiv x^p \pmod{\mathfrak{P}}$ for all algebraic integers $x$ of $L$ ; it generates the decomposition group of $\mathfrak{P}$ . Changing $\mathfrak{P}$ conjugates it, so $p$ determines a conjugacy class $\operatorname{Frob}_p$ . If $L$ is the splitting field of $f$ and $p \nmid \operatorname{disc} f$ , the cycle type of $\operatorname{Frob}_p$ acting on the roots of $f$ is the list of degrees of the irreducible factors of $f \bmod p$ .

Definition— Galois representation

A Galois representation of a number field $K$ is a continuous homomorphism $\rho: G_K = \operatorname{Gal}(\overline{K}/K) \to GL_n(E)$ , where $E$ is a topological field — $\C$ (Artin representations, which have finite image), or a finite extension of $\Q_\ell$ ( $\ell$ -adic representations, such as the Tate module of an elliptic curve). Its $L$ -function is $L(\rho, s) = \prod_{\mathfrak{p}} \det\big(1 - \rho(\operatorname{Frob}_{\mathfrak{p}}) N\mathfrak{p}^{-s} \mid V^{I_\mathfrak{p}}\big)^{-1}$ .

Definition— class field theory

Class field theory describes the abelian extensions of a number field $K$ in terms of data intrinsic to $K$ . In idelic form: there is a continuous surjective homomorphism, the Artin map, $K^\times \backslash \mathbb{A}_K^\times \to \operatorname{Gal}(K^{\mathrm{ab}}/K)$ , sending a uniformiser at an unramified prime $\mathfrak{p}$ to $\operatorname{Frob}_\mathfrak{p}$ , whose kernel is the connected component of the identity. Equivalently, finite abelian extensions of $K$ correspond to open subgroups of finite index of the idèle class group (classically: to subgroups of ray class groups).

Definition— modular forms and cusp forms

For $\Gamma_0(N) = \left\{\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in SL_2(\Z) : N \mid c\right\}$ , a modular form of weight $k$ and level $N$ is a holomorphic $f$ on the upper half-plane $\mathbb{H}$ with

f\left(\frac{az + b}{cz + d}\right) = (cz + d)^k f(z) \quad \text{for all } \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N),

that is holomorphic at the cusps. Since $z \mapsto z + 1$ lies in $\Gamma_0(N)$ , $f$ has a $q$ -expansion $f = \sum_{n \ge 0} a_n q^n$ , $q = e^{2\pi i z}$ ; $f$ is a cusp form if it vanishes at every cusp ( $a_0 = 0$ at $\infty$ , and similarly at the other cusps). The spaces $M_k(\Gamma_0(N)) \supset S_k(\Gamma_0(N))$ are finite-dimensional. (Weight-one forms with a character, such as $\eta(z)\eta(23z)$ , are defined in the same way with a factor $\chi(d)$ .)

Definition— Hecke operators and eigenforms

For $p \nmid N$ , the Hecke operator $T_p$ acts on $S_k(\Gamma_0(N))$ by

a_m(T_p f) = a_{mp}(f) + p^{k-1} a_{m/p}(f) \qquad (a_{m/p} = 0 \text{ unless } p \mid m),

and in general $a_m(T_n f) = \sum_{d \mid \gcd(m, n),\ \gcd(d, N) = 1} d^{k-1} a_{mn/d^2}(f)$ . The $T_n$ commute, $T_mT_n = T_{mn}$ when $\gcd(m, n) = 1$ , and $T_{p^{r+1}} = T_pT_{p^r} - p^{k-1}T_{p^{r-1}}$ for $p \nmid N$ . A normalised eigenform is a simultaneous eigenvector with $a_1 = 1$ ; a newform is a normalised eigenform not coming from a lower level.

Definition— automorphic representation (rough)

Let $\mathbb{A}$ be the adèles of $\Q$ and $G = GL_n$ . A cuspidal automorphic representation $\pi$ of $GL_n(\mathbb{A})$ is, roughly, an irreducible representation of $GL_n(\mathbb{A})$ occurring in the space of cusp forms on $GL_n(\Q) \backslash GL_n(\mathbb{A})$ (with a fixed central character). It factors as a restricted tensor product $\pi = \bigotimes'_v \pi_v$ of local representations; for almost all $p$ , $\pi_p$ is unramified and determined by a conjugacy class (its Satake parameter) in $GL_n(\C)$ , which defines $L(\pi, s) = \prod_p \det(1 - A_p p^{-s})^{-1}$ up to finitely many factors. A newform $f$ of weight $k$ gives a cuspidal automorphic representation of $GL_2(\mathbb{A})$ ; a Dirichlet character gives one of $GL_1(\mathbb{A}) = \mathbb{A}^\times$ . For a general reductive group $G$ the Satake parameters live in the Langlands dual group $\widehat{G}(\C)$ , whose root datum is dual to that of $G$ (roots and coroots exchanged): $\widehat{GL_n} = GL_n$ , $\widehat{SO_{2n+1}} = Sp_{2n}$ , $\widehat{SL_n} = PGL_n$ .

Worked examplesH

Examples

Example— Quadratic fields: reciprocity with a conductor

An odd prime $p \nmid d$ splits in $\Q(\sqrt d)$ exactly when $d$ is a square mod $p$ . For $d = -1$ : when $p \equiv 1 \pmod 4$ . For $d = 5$ : when $p \equiv \pm 1 \pmod 5$ (formalised below). For $d = -23$ : when $p$ is a square mod $23$ . In each case the condition is a congruence modulo $|D|$ , the discriminant of the field: this is the conductor, and it is quadratic reciprocity in action. The splitting is described by the Dirichlet character $\chi_D(p) = \left(\frac{D}{p}\right)$ , a one-dimensional Galois representation that is also a $GL_1$ -automorphic form.

Example— Cyclotomic fields: Frobenius is p mod n

In $\Q(\zeta_n)$ , the automorphism $\sigma_a: \zeta_n \mapsto \zeta_n^a$ gives $\operatorname{Gal}(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^\times$ . For $p \nmid n$ , $\sigma_p(x) \equiv x^p$ modulo any prime above $p$ (expand $(\sum c_i\zeta^i)^p$ in characteristic $p$ ), so $\operatorname{Frob}_p = \sigma_p$ : the Frobenius at $p$ is the residue class $p \bmod n$ . Consequently $p$ splits into $\varphi(n)/f$ primes of degree $f = \operatorname{ord}_n(p)$ , and splits completely iff $p \equiv 1 \pmod n$ . All of this is in Mathlib (see the formal section).

Example— A weight-one form and a non-abelian cubic

The cubic $x^3 - x - 1$ has discriminant $-23$ and Galois group $S_3$ ; its splitting field $H$ is the Hilbert class field of $K = \Q(\sqrt{-23})$ , whose class group has order $3$ . The two-dimensional irreducible representation $\rho$ of $S_3$ is induced from a character of order $3$ of the class group of $K$ , and Hecke's theory of theta series makes it modular: $L(\rho, s) = L(f, s)$ with

f = \tfrac12\big(\theta_{x^2 + xy + 6y^2} - \theta_{2x^2 + xy + 3y^2}\big) = \eta(z)\eta(23z),

a weight-one cusp form of level $23$ with character $\left(\frac{-23}{\cdot}\right)$ . Since $\operatorname{tr}\rho(\operatorname{Frob}_p) = (\text{number of roots of } x^3 - x - 1 \bmod p) - 1$ , this gives the rule of the motivation. Concretely, for $p \neq 23$ : three roots if $p = x^2 + xy + 6y^2$ (e.g. $59 = 5^2 + 5\cdot 2 + 6 \cdot 2^2$ ), none if $p = 2x^2 \pm xy + 3y^2$ (e.g. $13 = 2 \cdot 2^2 + 2 + 3$ ), and one root if $\left(\frac{-23}{p}\right) = -1$ . Because $H/\Q$ is not abelian, no congruence condition can separate the first two cases.

Example— Ramanujan's Δ

$\Delta(z) = q\prod_{n \ge 1}(1 - q^n)^{24} = \sum_{n \ge 1} \tau(n) q^n = q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 - \cdots$ is a cusp form of weight $12$ for $SL_2(\Z)$ . Ramanujan (1916) conjectured that $\tau$ is multiplicative, that $\tau(p^{r+1}) = \tau(p)\tau(p^r) - p^{11}\tau(p^{r-1})$ , and that $|\tau(p)| \le 2p^{11/2}$ . The first two were proved by Mordell (1917; Theorem 5 below), the third by Deligne (1974) as a consequence of his proof of the Weil conjectures. Deligne (1969) attached to $\Delta$ an $\ell$ -adic Galois representation $\rho_{\Delta, \ell}: G_\Q \to GL_2(\Q_\ell)$ with $\operatorname{tr}\rho(\operatorname{Frob}_p) = \tau(p)$ , which explains congruences such as Ramanujan's $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$ : modulo $691$ the representation becomes a sum of two characters.

Example— An elliptic curve of conductor 11

$E = 11a1$ and $f = q\prod(1 - q^n)^2(1 - q^{11n})^2 \in S_2(\Gamma_0(11))$ , the unique newform of that level: $a_p(E) = a_p(f)$ for all $p$ (Eichler 1954, Shimura). In representation-theoretic terms, the Galois representation on the Tate module of $E$ and the automorphic representation generated by $f$ correspond. The modularity theorem says this happens for every elliptic curve over $\Q$ .

ProofH

Central results

Theorem— Lemma 1 (the quadratic Gauss sum)

Let $q$ be an odd prime, $\zeta = e^{2\pi i/q}$ , and $g = \sum_{a=1}^{q-1} \left(\frac{a}{q}\right)\zeta^a$ . Then $g^2 = q^* := (-1)^{(q-1)/2} q$ . In particular $\sqrt{q^*} \in \Q(\zeta_q)$ .

Proof

Write $\chi(a) = \left(\frac{a}{q}\right)$ , a multiplicative character of $(\Z/q)^\times$ with $\sum_{a \ne 0}\chi(a) = 0$ . Then $g^2 = \sum_{a, b \ne 0} \chi(ab)\zeta^{a+b}$ . Substitute $b = ac$ (for fixed $a$ , $c$ runs over the non-zero residues as $b$ does) and use $\chi(a^2c) = \chi(c)$ :

g^2 = \sum_{c \ne 0} \chi(c) \sum_{a \ne 0} \zeta^{a(1 + c)}.

The inner sum is $q - 1$ if $c = -1$ and $-1$ otherwise (a sum of all non-trivial $q$ -th roots of unity). Hence $g^2 = \chi(-1)(q - 1) - \sum_{c \ne 0, -1}\chi(c) = \chi(-1)(q-1) + \chi(-1) = \chi(-1)\,q$ , and $\chi(-1) = (-1)^{(q-1)/2}$ by Euler's criterion.

∎

Theorem— Theorem 2 (quadratic reciprocity, Gauss 1796)

For distinct odd primes $p, q$ : $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$ .

Proof

We give the proof that explains why: via Frobenius elements in $\Q(\zeta_q)$ . Let $G = \operatorname{Gal}(\Q(\zeta_q)/\Q) \cong (\Z/q)^\times$ , a cyclic group of even order $q - 1$ . It has a unique subgroup $H$ of index $2$ , the squares, and so $\Q(\zeta_q)$ has a unique quadratic subfield; by Lemma 1 it is $\Q(\sqrt{q^*})$ . Now compute, for the prime $p \ne q$ (unramified in $\Q(\zeta_q)$ ):

$p$ splits in $\Q(\sqrt{q^*})$ $\iff$ $q^*$ is a square mod $p$ $\iff$ $\left(\frac{q^*}{p}\right) = 1$ (Dedekind–Kummer: $x^2 - q^*$ factors mod $p$ ; $p$ is odd and does not divide $q^*$ ).
$p$ splits in $\Q(\sqrt{q^*})$ $\iff$ $\operatorname{Frob}_p$ acts trivially on $\Q(\sqrt{q^*})$ $\iff$ $\operatorname{Frob}_p \in H$ $\iff$ $p \bmod q$ is a square $\iff$ $\left(\frac{p}{q}\right) = 1$ , because $\operatorname{Frob}_p = \sigma_p$ corresponds to $p \bmod q$ (the cyclotomic example).

So $\left(\frac{p}{q}\right) = \left(\frac{q^*}{p}\right) = \left(\frac{-1}{p}\right)^{(q-1)/2}\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}\left(\frac{q}{p}\right)$ , using Euler's criterion $\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2}$ . Multiply by $\left(\frac{q}{p}\right) = \pm 1$ .

∎

The structure of this proof is the structure of the subject: an arithmetic question about splitting of primes (Galois side) is answered by identifying the Frobenius with an "automorphic" invariant, the residue class $p \bmod q$ , which is a character of the idèle class group of $\Q$ . The proof also shows that $\Q(\sqrt{q^*})$ lies in a cyclotomic field — the quadratic case of the next theorem.

Theorem— Theorem 3 (Kronecker–Weber; Kronecker 1853, Weber 1886, Hilbert 1896)

Every finite abelian extension of $\Q$ is contained in a cyclotomic field $\Q(\zeta_n)$ .

Kronecker stated it, Weber gave a proof with a gap, and Hilbert gave the first complete proof. Standard modern proofs reduce to local fields and analyse ramification (Washington, Introduction to Cyclotomic Fields, Ch. 14), or deduce it from class field theory. With the cyclotomic example it gives class field theory over $\Q$ : abelian extensions of $\Q$ correspond to subgroups of the groups $(\Z/n)^\times$ , and Frobenius elements to residue classes. Over a general number field, Takagi (1920) and Artin (1927) proved the corresponding theorem with ray class groups replacing $(\Z/n)^\times$ ; generating the abelian extensions explicitly (Hilbert's twelfth problem) is understood for $\Q$ and imaginary quadratic fields (complex multiplication), and Dasgupta and Kakde (2021–23) gave a $p$ -adic answer for totally real fields.

Theorem— Proposition 4 (eigenforms have multiplicative coefficients)

Let $f = \sum a_n q^n \in S_k(SL_2(\Z))$ be non-zero with $T_n f = \lambda_n f$ for every $n \ge 1$ . Then $a_n = \lambda_n a_1$ for all $n$ , so $a_1 \neq 0$ . Normalising $a_1 = 1$ : $a_{mn} = a_m a_n$ when $\gcd(m, n) = 1$ , and $a_{p^{r+1}} = a_pa_{p^r} - p^{k-1}a_{p^{r-1}}$ for every prime $p$ .

Proof

By the formula for $T_n$ with $m = 1$ , the only divisor of $\gcd(1, n)$ is $d = 1$ , so $a_1(T_nf) = a_n(f)$ . But $a_1(T_nf) = a_1(\lambda_nf) = \lambda_na_1$ . Hence $a_n = \lambda_na_1$ for every $n$ ; if $a_1$ were $0$ , all coefficients would vanish and $f = 0$ . With $a_1 = 1$ we have $a_n = \lambda_n$ , and the relations $T_mT_n = T_{mn}$ and $T_{p^{r+1}} = T_pT_{p^r} - p^{k-1}T_{p^{r-1}}$ between the operators become the stated relations between their eigenvalues.

∎

The same argument works at level $N$ for $n$ prime to $N$ ; to conclude $a_1 \neq 0$ one also needs the operators $U_p$ for $p \mid N$ , or one restricts to newforms.

Theorem— Theorem 5 (Mordell 1917)

$\tau(mn) = \tau(m)\tau(n)$ for coprime $m, n$ , and $\tau(p^{r+1}) = \tau(p)\tau(p^r) - p^{11}\tau(p^{r-1})$ .

Proof

The space $S_{12}(SL_2(\Z))$ is one-dimensional, spanned by $\Delta$ (by the dimension formula for level one: $\dim M_{12} = 2$ , with $M_{12} = \C E_{12} \oplus \C\Delta$ ). Each $T_n$ preserves $S_{12}$ , so $T_n\Delta = \lambda_n\Delta$ for every $n$ . Since $\tau(1) = 1$ , Proposition 4 applies.

∎

The proof is a template: one-dimensionality forces an eigenform, and eigenforms have Euler products, $L(\Delta, s) = \prod_p (1 - \tau(p)p^{-s} + p^{11 - 2s})^{-1}$ . In general $S_k(\Gamma_0(N))$ has a basis of newforms and oldforms, and multiplicity one (Atkin–Lehner) identifies newforms with their Hecke eigenvalues.

Theorem— Theorem 6 (Galois representations from modular forms)

Let $f$ be a newform of weight $k \ge 1$ , level $N$ and character $\varepsilon$ , with coefficients in a number field $E_f$ . For each prime $\lambda$ of $E_f$ above $\ell$ there is a continuous irreducible representation $\rho_{f,\lambda}: G_\Q \to GL_2(E_{f,\lambda})$ , unramified outside $N\ell$ , with $\operatorname{tr}\rho_{f,\lambda}(\operatorname{Frob}_p) = a_p(f)$ and $\det\rho_{f,\lambda}(\operatorname{Frob}_p) = \varepsilon(p)p^{k-1}$ for $p \nmid N\ell$ . For $k = 2$ (Eichler–Shimura) it is cut out of the Tate module of the Jacobian of $X_0(N)$ ; for $k \ge 2$ (Deligne, 1969–71) from the étale cohomology of Kuga–Sato varieties; for $k = 1$ (Deligne–Serre, 1974) it comes from a representation into $GL_2(\C)$ with finite image.

The converse direction is the hard one.

Theorem— Theorem 7 (modularity; Wiles, Taylor–Wiles 1995; Breuil–Conrad–Diamond–Taylor 2001)

For every elliptic curve $E/\Q$ there is a newform $f$ of weight $2$ and level $N_E$ with $\rho_{E,\ell} \cong \rho_{f,\ell}$ for all $\ell$ , equivalently $a_p(E) = a_p(f)$ for all $p$ .

Related reciprocity theorems for $GL_2$ over $\Q$ : Serre's modularity conjecture — every odd irreducible $\bar\rho: G_\Q \to GL_2(\overline{\mathbb{F}}_\ell)$ comes from a newform — was proved by Khare and Wintenberger (2009, with Kisin), and implies Artin's conjecture for odd two-dimensional complex representations of $G_\Q$ (earlier, Langlands (1980) and Tunnell (1981) had done the tetrahedral and octahedral cases using base change). See the elliptic-curves lesson for the consequence for Fermat's Last Theorem.

Theorem— Conjecture 8 (Langlands reciprocity, one form)

Irreducible $\ell$ -adic Galois representations $\rho: G_\Q \to GL_n(\overline{\Q}_\ell)$ that are unramified almost everywhere and de Rham at $\ell$ (the Fontaine–Mazur "geometric" ones) correspond bijectively to cuspidal automorphic representations $\pi$ of $GL_n(\mathbb{A}_\Q)$ that are algebraic at infinity, with $L(\rho, s) = L(\pi, s)$ (suitably normalised) — that is, the characteristic polynomial of $\rho(\operatorname{Frob}_p)$ matches the Satake parameter of $\pi_p$ for almost all $p$ .

Known: $n = 1$ (class field theory, over any number field); much of $n = 2$ over $\Q$ and over totally real fields; the direction "automorphic $\Rightarrow$ Galois" for regular algebraic self-dual representations of $GL_n$ over CM and totally real fields (Clozel, Harris–Taylor, Shin, Chenevier–Harris and others) and, since Harris–Lan–Taylor–Thorne (2016) and Scholze (2015), without self-duality; and many "potential" cases (Taylor and collaborators), which give the Sato–Tate theorem. Open: the general case, notably the direction "Galois $\Rightarrow$ automorphic" in dimension $n \ge 3$ .

Theorem— Conjecture 9 (functoriality)

Let $G$ and $H$ be connected reductive groups over $\Q$ (say quasi-split) and $r: {}^LH \to {}^LG$ an $L$ -homomorphism between their $L$ -groups. For every automorphic representation $\pi$ of $H(\mathbb{A})$ there is an automorphic representation $\Pi$ of $G(\mathbb{A})$ whose local Satake parameters are the images under $r$ of those of $\pi$ at almost all places.

Functoriality for $H = GL_2$ , $G = GL_{n+1}$ and $r = \operatorname{Sym}^n$ would give the Ramanujan conjecture and, via analytic properties of the symmetric-power $L$ -functions, the Sato–Tate law. Known cases include: Jacquet–Langlands (1970, inner forms of $GL_2$ ); $\operatorname{Sym}^2$ (Gelbart–Jacquet, 1978); cyclic base change for $GL_n$ (Arthur–Clozel, 1989); $\operatorname{Sym}^3$ (Kim–Shahidi, 2002) and $\operatorname{Sym}^4$ (Kim, 2003); all symmetric powers of holomorphic newforms (Newton–Thorne, 2021); and endoscopic transfer from classical groups to $GL_n$ (Arthur, 2013), which needed Ngô Bảo Châu's proof of the Fundamental Lemma (2008–10). Functoriality in general is an open conjecture.

ExplanationH

The landscape: known, open, and parallel programmes

Local Langlands. Over a $p$ -adic field $F$ , irreducible smooth representations of $GL_n(F)$ correspond to $n$ -dimensional Weil–Deligne representations of the Weil group of $F$ : proved by Harris–Taylor and by Henniart (1998–2000), with a third proof by Scholze (2013). For general groups the correspondence is expected to be finite-to-one (" $L$ -packets"); Fargues and Scholze (2021) constructed the semisimplified parameters for every reductive group by "geometrising" the local correspondence on the Fargues–Fontaine curve.

Function fields. Replace $\Q$ by the field of rational functions of a curve over $\mathbb{F}_q$ . Drinfeld proved global Langlands for $GL_2$ (1974–), Laurent Lafforgue for $GL_n$ (2002), and Vincent Lafforgue constructed the automorphic-to-Galois direction for every reductive group (2018), using moduli of shtukas.

Geometric Langlands. Replace the curve over $\mathbb{F}_q$ by a complex algebraic curve $X$ . Galois representations become local systems (flat $\widehat G$ -bundles) on $X$ , and automorphic forms become sheaves (D-modules) on the moduli stack $\operatorname{Bun}_G$ of $G$ -bundles. The conjecture, in the form of Beilinson–Drinfeld and Arinkin–Gaitsgory, is an equivalence of categories. A proof of the unramified global geometric Langlands conjecture in characteristic $0$ was announced in 2024 by Gaitsgory, Raskin and collaborators; it spans several long papers. This is analogous to, but logically independent of, the arithmetic conjectures: progress in one does not settle the other.

Open problems (well documented; consult recent surveys for the current state): functoriality in general; reciprocity for $GL_n$ , $n \ge 3$ , over $\Q$ in general; Artin's conjecture that $L(\rho, s)$ is entire for every non-trivial irreducible Artin representation (open, for instance, for even icosahedral representations in dimension $2$ , and in general in dimension $\ge 3$ ); the Ramanujan conjecture for Maass forms, whose archimedean part is Selberg's eigenvalue conjecture $\lambda_1 \ge \tfrac14$ for congruence subgroups (the best bound, $\lambda_1 \ge \tfrac{975}{4096}$ , is due to Kim and Sarnak); the Birch and Swinnerton-Dyer conjecture and its generalisations (Beilinson–Bloch–Kato), which concern special values of the $L$ -functions this programme produces.

Formal mathematicsF

Reciprocity in Lean and Mathlib

Mathlib has Legendre and Jacobi symbols with quadratic reciprocity, cyclotomic fields with their Galois groups and the splitting of primes in them, Dirichlet characters and $L$ -series with Dirichlet's theorem, and modular forms for congruence subgroups including Ramanujan's $\Delta$ . It does not have class field theory, the Kronecker–Weber theorem, automorphic representations, or the Galois representations attached to modular forms.

Quadratic reciprocity and the splitting laws for $x^2 + 1$ — including a proof from Fermat's little theorem that $-1$ is not a square modulo $p \equiv 3 \pmod 4$ , and the parity argument for sums of two squares:

quadratic_reciprocityLean 4Applies a library theoremFormally verified

/-- **Quadratic reciprocity** (Gauss, 1796): for distinct odd primes `p` and `q`,
`(q/p)(p/q) = (−1)^((p−1)/2 · (q−1)/2)`. (In Lean, `p / 2 = (p − 1)/2` for odd `p`, and
`legendreSym p a` is the Legendre symbol `(a/p)`.) This applies Mathlib's
`legendreSym.quadratic_reciprocity`. -/
theorem quadratic_reciprocity (p q : ℕ) [Fact p.Prime] [Fact q.Prime] (hp : p ≠ 2)
    (hq : q ≠ 2) (hpq : p ≠ q) :
    legendreSym q p * legendreSym p q = (-1) ^ (p / 2 * (q / 2)) :=
  legendreSym.quadratic_reciprocity hp hq hpq