21.10.01 · number-theory / langlands

Langlands Philosophy Survey

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Anchor (Master): Langlands 1967 *Problems in the theory of automorphic forms* (Letter to A. Weil, Yale, January 1967; published *Lectures in Modern Analysis and Applications III*, LNM 170, Springer 1970, 18-86) — originator paper containing functoriality, the $L$-group, and the reciprocity conjecture; Borel 1979 *Automorphic forms, representations and L-functions* (Proc. Symp. Pure Math. 33, Corvallis, two volumes) — formal packaging at Corvallis; Bump 1997 (Cambridge Studies 55) — modern $\mathrm{GL}_n$ treatment; Drinfeld 1974 *Funct. Anal. Appl.* 8 + 1980 *Soviet Math. Dokl.* 21 (function-field $\mathrm{GL}_2$, shtukas; Fields Medal 1990); Deligne 1973 *Formes modulaires et représentations de $\mathrm{GL}_2$* (in Antwerp II, LNM 349); L. Lafforgue 2002 *Invent. Math.* 147 (function-field $\mathrm{GL}_n$, Fields Medal 2002); Harris-Taylor 2001 *The Geometry and Cohomology of Some Simple Shimura Varieties* (Ann. Math. Studies 151) — local Langlands $\mathrm{GL}_n(\mathbb{Q}_p)$; Henniart 2000 *Invent. Math.* 139 (independent proof, numerical Langlands); V. Lafforgue 2018 *J. Amer. Math. Soc.* 31 (automorphic-to-Galois for reductive groups over function fields via excursion operators); Fargues-Scholze 2024 *Geometrization of the local Langlands correspondence* (Astérisque 449; arXiv:2102.13459); Beilinson-Drinfeld c. 1991-2004 *Quantization of Hitchin's integrable system and Hecke eigensheaves* (preprint); Gaitsgory-Raskin et al. 2024 (arXiv:2405.03648 and companion papers — geometric Langlands for $\mathrm{GL}_2$ over $\mathbb{C}$); Frenkel 2007 *Lectures on the Langlands Program and Conformal Field Theory* (in *Frontiers in Number Theory, Physics, and Geometry II*, Springer, 387-533); Manin-Panchishkin 2005 Ch. 6 of Part II

Intuition Beginner

The Langlands programme is the most ambitious unifying scheme in modern number theory. Its core claim, stated in one sentence, is that two completely different kinds of objects — the symmetries of algebraic numbers, and the harmonic-analytic patterns on a certain class of infinite-dimensional spaces — describe the same arithmetic information. The symmetries live on one side of the bridge; the harmonic patterns live on the other; and the bridge is built out of matching invariants called $L$ -functions.

On the symmetry side, one starts with the absolute Galois group of a number field, the group of all field-theoretic symmetries of the algebraic numbers fixing the rationals. This group is enormous and inaccessible directly. The way it becomes visible is through its actions on finite-dimensional vector spaces: a Galois representation is a continuous homomorphism from the Galois group into a matrix group, and each such representation encodes some piece of arithmetic information — most famously, the way the group acts on the torsion points of an elliptic curve, or on the étale cohomology of a variety.

On the harmonic side, one starts with a reductive group like the general linear group $GL_{n}$ and considers its automorphic representations: irreducible representations of the group $GL_{n}$ acting on an adelic version of itself, occurring as eigenspaces of certain integral operators. The simplest example, when $n = 2$ and the field is the rationals, is a modular form: a holomorphic function on the upper half-plane satisfying a transformation rule under a discrete group. The Langlands reciprocity conjecture is the claim that these two pictures match, with matching $L$ -functions: every Galois representation comes from an automorphic representation, and the bridge is named.

Visual Beginner

A three-panel diagram. Left panel: a cloud labelled "Galois side" with an arrow from the absolute Galois group $G_{F} = Gal (\overline{F} / F)$ to a matrix group $GL_{n}$ , representing a Galois representation $ρ$ . Middle panel: a cloud labelled "automorphic side" with the group $GL_{n} (A_{F})$ at the top and an irreducible representation $π$ at the bottom, with arrows from each local factor $GL_{n} (F_{v})$ feeding into the global $π$ . Right panel: a double-headed arrow labelled $L (s, ρ) = L (s, π)$ joining the two clouds, with a small box underneath labelled "matching $L$ -functions" containing an Euler product over places $v$ , multiplying local factors $L_{v} (s, ρ)$ on the Galois side against local factors $L_{v} (s, π)$ on the automorphic side, with equality place by place.

The picture captures the central organising claim of the Langlands programme: two completely different kinds of mathematical objects — Galois representations of arithmetic origin and automorphic representations of analytic origin — pair up perfectly when sorted by their $L$ -functions. The matching is conjectural in general but proved in many important special cases.

Worked example Beginner

Trace through the smallest non-abelian case of Langlands reciprocity: the modularity theorem identifying an elliptic curve over the rationals with a weight- $2$ modular form.

Step 1. Pick the smallest-conductor elliptic curve over the rationals: $E : y^{2} + y = x^{3} - x^{2} - 10 x - 20$ , of conductor $N = 11$ . Count its points modulo small primes. At $p = 2$ , one gets $5$ points on the projective curve, so $a_{2} = 2 + 1 - 5 = - 2$ . At $p = 3$ , $a_{3} = - 1$ (after a careful count). At $p = 5$ , $a_{5} = 1$ . At $p = 7$ , $a_{7} = - 2$ .

Step 2. The Galois side. The $ℓ$ -adic Tate module $T_{ℓ} E$ for any prime $ℓ$ is a free $Z_{ℓ}$ -module of rank $2$ , and the absolute Galois group of the rationals acts on it continuously. This gives a Galois representation $ρ_{E, ℓ} : G_{Q} \to GL_{2} (Z_{ℓ})$ . The trace of Frobenius at a prime $p$ of good reduction equals $a_{p} = p + 1 - N_{p}$ , and the determinant equals $p$ (the cyclotomic character). So the Galois-side data is the sequence $(a_{p})_{p} = (- 2, - 1, 1, - 2, \dots)$ .

Step 3. The automorphic side. The space $S_{2} (Γ_{0} (11))$ of weight- $2$ cusp forms on $Γ_{0} (11)$ is one-dimensional, generated by a unique normalised newform $f_{11}$ . Its $q$ -expansion is $f_{11} (z) = q - 2 q^{2} - q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{7} + \dots$ . The Hecke eigenvalues $a_{p} (f_{11}) = - 2, - 1, 1, - 2$ at $p = 2, 3, 5, 7$ are exactly the integers from the Galois side.

Step 4. The bridge. The modularity theorem says these two sequences of integers are the same sequence by design: the Galois representation $ρ_{E, ℓ}$ and the automorphic representation generated by $f_{11}$ correspond under Langlands reciprocity for $GL_{2} / Q$ , with matching $L$ -functions $L (s, ρ_{E, ℓ}) = L (s, f_{11})$ given by the same Euler product.

What this tells us: Langlands reciprocity is not abstract. In the smallest non-abelian case, it identifies a geometric object (an elliptic curve) with an analytic object (a modular form), and the matching is a sequence of integers computable in either picture. The conjecture in higher rank predicts the same kind of identification for every Galois representation that comes from arithmetic geometry.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Langlands reciprocity conjecture for $GL_{n}$ over a global field $F$ asserts a correspondence between:

A. Continuous representations of the absolute Galois group $G_{F}$ into $GL_{n} (C)$ and characters of the multiplicative group $F^{\times}$ .
B. Irreducible $n$ -dimensional $ℓ$ -adic Galois representations of $G_{F}$ and cuspidal automorphic representations of $GL_{n} (A_{F})$ , with matching $L$ -functions.
C. Subgroups of $GL_{n} (Q)$ and finite-index subgroups of the modular group.
D. Elliptic curves over $F$ and weight- $2$ holomorphic modular forms.

Hint

The bridge has a Galois side (representations) and an automorphic side (representations of $GL_{n} (A_{F})$ ), with $L$ -functions matching across the bridge.

Answer

B. Langlands reciprocity matches irreducible continuous $n$ -dimensional $ℓ$ -adic Galois representations $ρ : G_{F} \to GL_{n} (\overline{Q_{ℓ}})$ with cuspidal automorphic representations $π$ of $GL_{n} (A_{F})$ , identified by $L (s, ρ) = L (s, π)$ . The other choices are wrong: A is too special (characters of $F^{\times}$ correspond to $GL_{1}$ only and the codomain is wrong); C is a subgroup question, not a representation question; D is a special case of B at $n = 2$ over the rationals, not the general statement.

Exercise (easy, true-false).

True or false: the case $n = 1$ of Langlands reciprocity for $GL_{n}$ is class field theory.

Hint

A one-dimensional Galois representation is a continuous character of the abelianisation of $G_{F}$ . Class field theory describes the abelianisation in terms of the idele class group, which is the $GL_{1}$ automorphic side.

Answer

True. A continuous one-dimensional Galois representation $ρ : G_{F} \to GL_{1} (\overline{Q_{ℓ}}) = \overline{Q_{ℓ}}^{\times}$ factors through the abelianisation $G_{F}^{ab}$ . Class field theory (Takagi 1920, Artin 1927) identifies $G_{F}^{ab}$ with the idele class group $F^{\times} \ A_{F}^{\times}$ via the Artin reciprocity map. Hecke characters of $F$ are by definition continuous characters of the idele class group, so they correspond to $GL_{1}$ -automorphic representations. The matching $L$ -functions are the Hecke $L$ -function and the Artin $L$ -function of the corresponding one-dimensional Galois representation, equal by Artin reciprocity. This is exactly Langlands reciprocity at $n = 1$ .

Formal definition Intermediate+

Fix a global field $F$ — either a number field (a finite extension of $Q$ ) or a function field (the field of rational functions on a smooth projective curve $X$ over a finite field $F_{q}$ ). The places of $F$ are the equivalence classes of absolute values on $F$ other than the constant value $1$ on $F^{\times}$ ; at each place $v$ the completion $F_{v}$ is a local field, and the integers of $F_{v}$ form a maximal compact subring $O_{F_{v}}$ . The adele ring is the restricted topological product $$ \mathbb{A}F = {\prod_v}' F_v = {(x_v) \in \prod_v F_v : x_v \in \mathcal{O}{F_v} \text{ for almost every } v}, $$ a locally compact topological ring containing $F$ diagonally as a discrete cocompact subring (in the function-field case) or with compact quotient up to the volume of the connected component (in the number-field case, via the strong approximation theorem of Cassels-Fröhlich 1967 ^{[Bump 1997]}).

Definition (reductive group over $F$ ). A reductive group $G$ over $F$ is a connected smooth affine algebraic group over $F$ whose unipotent radical is the identity. Standard examples are $G = GL_{n}$ , $SL_{n}$ , $SO_{n}$ , $Sp_{2 n}$ , and $GSp_{2 n}$ . The group of $A_{F}$ -points is the locally compact group $G (A_{F})$ , and $G (F) ↪ G (A_{F})$ embeds diagonally as a discrete subgroup with $G (F) \ G (A_{F})$ of finite volume against an invariant measure (for $G$ semisimple or $G$ with anisotropic centre modulo a centre-fixing).

Definition (cuspidal automorphic representation). An automorphic form for $G$ is a function $f : G (F) \ G (A_{F}) \to C$ with prescribed analytic / smoothness conditions: $K$ -finite under a maximal compact $K \subset G (A_{F})$ , $Z$ -finite under the centre $Z$ of the universal enveloping algebra at the archimedean places, and of moderate growth. A cuspidal automorphic form vanishes against every unipotent integral along every proper parabolic subgroup of $G$ . The space $L_{cusp}^{2} (G (F) \ G (A_{F}))$ (with fixed central character if applicable) decomposes as a Hilbert direct sum of irreducible $G (A_{F})$ -representations $π$ ; each such $π$ is a cuspidal automorphic representation, and the Flath-Casselman tensor product theorem ^{[Bump 1997]} factors $π = \otimes_{v}^{'} π_{v}$ as a restricted tensor product of irreducible smooth admissible representations $π_{v}$ of $G (F_{v})$ , with $π_{v}$ unramified (containing a $G (O_{F_{v}})$ -fixed vector) for almost every $v$ .

Definition (Langlands dual group). For $G$ a split connected reductive group over $F$ with root datum $(X^{*}, R, X_{*}, R^{\lor})$ , the Langlands dual group $G$ is the complex reductive group with dual root datum $(X_{*}, R^{\lor}, X^{*}, R)$ ; its character lattice is the cocharacter lattice of $G$ and vice versa. Examples: $GL_{n} = GL_{n} (C)$ , $SL_{n} = PGL_{n} (C)$ , $SO_{2 n + 1} = Sp_{2 n} (C)$ , $Sp_{2 n} = SO_{2 n + 1} (C)$ . The full $L$ -group is the semidirect product $$ {}^L G = \widehat{G}(\mathbb{C}) \rtimes \mathrm{Gal}(\overline{F}/F), $$ which reduces to $G (C)$ for split $G$ (identity Galois action) and packages the form of $G$ at non-split cases.

Definition (Langlands $L$ -function). Fix a finite-dimensional representation $r :^{L} G \to GL_{N} (C)$ of the $L$ -group. For a cuspidal automorphic representation $π = \otimes_{v}^{'} π_{v}$ of $G (A_{F})$ and a place $v$ where $π_{v}$ is unramified, the Satake isomorphism ^{[Borel 1979]} identifies the spherical Hecke algebra at $v$ with the representation ring of $G$ and produces a semisimple conjugacy class $c (π_{v}) \in G (C)$ , called the Satake parameter. The local $L$ -factor at $v$ is $$ L_v(s, \pi, r) = \det(1 - r(c(\pi_v)) \cdot q_v^{-s})^{-1} $$ with $q_{v}$ the cardinality of the residue field at $v$ , and the global $L$ -function is the Euler product $L (s, π, r) = \prod_{v} L_{v} (s, π, r)$ , convergent in a right half-plane. For $G = GL_{n}$ and $r$ the standard representation, $L (s, π, r)$ is the standard automorphic $L$ -function $L (s, π)$ .

Langlands reciprocity for $GL_{n}$ (statement). Let $F$ be a global field and fix a prime $ℓ$ (a prime number when $F$ is a number field; a prime different from the characteristic when $F$ is a function field). There is a conjectural bijection $$ \left{ \begin{array}{c} \text{irreducible continuous semisimple} \ n\text{-dimensional } \ell\text{-adic Galois} \ \text{representations } \rho : G_F \to \mathrm{GL}n(\overline{\mathbb{Q}\ell}) \ \text{unramified outside a finite set } S \end{array} \right} ;\longleftrightarrow; \left{ \begin{array}{c} \text{cuspidal automorphic} \ \text{representations } \pi \text{ of } \mathrm{GL}_n(\mathbb{A}_F) \ \text{unramified outside } S \end{array} \right} $$ matching local parameters at every unramified place and matching $L$ -functions $L (s, ρ) = L (s, π)$ .

In the function-field case ( $F$ the function field of a smooth projective curve over $F_{q}$ ), this bijection is a theorem: Drinfeld 1974 / 1980 for $n = 2$ and L. Lafforgue 2002 for general $n$ ^{[L. Lafforgue 2002]}. In the number-field case it is conjectural except for $n = 1$ (class field theory) and specific families at $n = 2$ (modular elliptic curves, weight- $\geq 2$ regular cusp forms, certain mod- $ℓ$ cases via Khare-Wintenberger 2009).

Counterexamples to common slips

"Every Galois representation is automorphic." The reciprocity conjecture restricts to continuous semisimple representations that are unramified outside a finite set. A continuous representation with infinite image and ramification at every prime would not satisfy the unramification hypothesis. Moreover the Fontaine-Mazur 1995 refinement ^{[Manin-Panchishkin 2005]} restricts further to geometric representations: those unramified outside a finite set and de Rham at every place above $ℓ$ . The geometric hypothesis is conjecturally equivalent to coming from a sub-quotient of étale cohomology of a smooth proper variety.
"The automorphic side is just modular forms." Modular forms cover only $GL_{2}$ over $Q$ (and via Hilbert / Bianchi forms, $GL_{2}$ over totally real / imaginary quadratic fields). The full automorphic side is the cuspidal spectrum of $GL_{n} (A_{F})$ for arbitrary $n$ and $F$ , an infinite-dimensional unitary representation theory built from the regular representation $L^{2} (G (F) \ G (A_{F}))$ . Even for $n = 2$ , the automorphic spectrum contains Maass forms (non-holomorphic) and Eisenstein series alongside the holomorphic modular forms.
"Langlands reciprocity is a theorem." The full reciprocity conjecture for $GL_{n}$ over number fields is open. What is proved: $n = 1$ via class field theory; $n = 2$ for cuspidal Hecke eigenforms over totally real fields via potential modularity and Wiles-Taylor (modularity for $Q$ , Freitas-Le Hung-Siksek for real quadratic, Skinner-Wiles and the Skinner-Diamond machinery for higher totally real); function-field $GL_{n}$ via Drinfeld + L. Lafforgue; certain symmetric-power $L$ -functions $Sym^{k} ρ_{E}$ via Newton-Thorne 2021. The full reciprocity for $GL_{3} / Q$ remains open in general.

Key theorem with proof Intermediate+

The signature theorem of this survey unit is the local Langlands correspondence for $GL_{n}$ over a $p$ -adic field, proved independently by Harris-Taylor 2001 and Henniart 2000. It is the local pillar on which global reciprocity rests: a global automorphic representation $π = \otimes_{v}^{'} π_{v}$ has local components $π_{v}$ at each place, and the local Langlands correspondence pairs each $π_{v}$ with a Langlands parameter, which then assembles into a global Galois representation.

Theorem (local Langlands for $GL_{n} (Q_{p})$ ; Harris-Taylor 2001, Henniart 2000). Fix a prime $p$ and a positive integer $n$ . There is a bijection $$ \mathrm{LL}_n : \left{ \begin{array}{c} \text{irreducible smooth admissible} \ \text{representations of } \mathrm{GL}_n(\mathbb{Q}p) \end{array} \right} ;\xrightarrow{\sim}; \left{ \begin{array}{c} n\text{-dimensional Frobenius-semisimple} \ \text{Weil-Deligne representations of } W{\mathbb{Q}_p} \end{array} \right} $$ characterised by (i) matching local $L$ -factors $L (s, π) = L (s, LL_{n} (π))$ , (ii) matching local $ε$ -factors $ε (s, π, ψ) = ε (s, LL_{n} (π), ψ)$ for every additive character $ψ$ of $Q_{p}$ that is not identically $1$ , (iii) matching central characters via the determinant character, (iv) compatibility with twists by $GL_{1}$ -characters, and (v) compatibility with parabolic induction (the irreducible discrete-series parameters are the irreducible Weil-Deligne parameters, and the rest are parametrised by Langlands data).

Proof. The two proofs (Harris-Taylor, Henniart) take different routes; each establishes the bijection via existence and uniqueness arguments, the bijectivity being the deep content.

The Harris-Taylor route. Harris and Taylor work with the étale cohomology of a tower of compact Shimura varieties of certain unitary type. Specifically, they consider a totally imaginary CM extension $E / E^{+}$ with $E^{+}$ a totally real field, and a unitary similitude group $G$ over $E^{+}$ of signature $(1, n - 1)$ at one infinite place and definite at the others. The Shimura variety $Sh_{K} (G, X) / E$ at a level $K$ has dimension $n - 1$ , and its étale cohomology in the middle degree $$ H^{n-1}{\mathrm{et}}(\mathrm{Sh}K(G, X){\overline{E}}, \overline{\mathbb{Q}\ell}) $$ carries commuting actions of $G (A_{E^{+}, f})$ and $Gal (\overline{E} / E)$ . Decomposing along the cuspidal automorphic representations $π$ of $G (A_{E^{+}})$ , each $π$ -isotypic component is a tensor $π_{f} \otimes R_{ℓ} (π)$ with $R_{ℓ} (π)$ an $n$ -dimensional $ℓ$ -adic Galois representation of $Gal (\overline{E} / E)$ . This is the Kottwitz construction of Galois representations attached to automorphic representations of unitary similitude groups.

The local-global compatibility at primes of good reduction is the Eichler-Shimura-Langlands-Kottwitz formula: for a place $v$ of $E$ above a place $p$ of $E^{+}$ where the level $K$ is hyperspecial, $$ \mathrm{tr}(\mathrm{Frob}v^j \mid R\ell(\pi)) = \mathrm{tr}(c(\pi_v)^j) $$ for every $j \geq 1$ , with $c (π_{v})$ the Satake parameter of $π_{v}$ . This identifies the Galois-side $L$ -factor with the automorphic-side $L$ -factor at unramified places.

The local-global compatibility at ramified primes is more involved. Harris-Taylor study the Newton-stratification of the Shimura variety's reduction mod $p$ and identify the supercuspidal-summand contributions to the étale cohomology with the irreducible representations of $W_{E_{v}}$ via the Drinfeld-Carayol level structure on Rapoport-Zink spaces. The vanishing-cycle calculation on Rapoport-Zink spaces of EL type produces, for each supercuspidal representation $π_{v}$ of $GL_{n} (E_{v})$ , an $n$ -dimensional irreducible representation $σ_{v} (π_{v})$ of $W_{E_{v}}$ , with matching $L$ - and $ε$ -factors.

Local Langlands for $GL_{n} (Q_{p})$ is then extracted by a base-change argument: every supercuspidal representation of $GL_{n} (Q_{p})$ appears as a local component of a cuspidal automorphic representation on a suitable unitary similitude group at a place above $p$ , and the bijection $σ_{v}$ descends to a bijection $LL_{n}^{sc}$ on supercuspidals. The general case follows by parabolic induction (Bernstein-Zelevinsky 1977 classification, Langlands quotients) and uniqueness of the bijection imposed by the matching of $L$ - and $ε$ -factors.

The Henniart route. Henniart's argument is logically independent. He uses the numerical Langlands correspondence (a counting argument matching the number of supercuspidals of $GL_{n} (Q_{p})$ of given level / depth with the number of irreducible Weil-group representations of matching conductor) combined with the Bushnell-Kutzko-Harris theory of types for $GL_{n} (Q_{p})$ . The Bushnell-Kutzko types parametrise supercuspidals via compact-open subgroups and finite-image representations; matching these to Weil-Deligne parameters by their conductors and $ε$ -factors yields the bijection. The Cogdell-Piatetski-Shapiro converse theorem ^{[Bump 1997]} supplies the uniqueness side: an automorphic-side object is determined by its twisted $L$ -functions against all $GL_{m}$ ( $m < n$ ) automorphic representations, and the matching of $ε$ -factors forces the bijection. Henniart's proof is shorter and avoids Shimura varieties entirely.

Combining both routes. The bijection $LL_{n}$ is characterised by matching $L$ - and $ε$ -factors of all twists by $GL_{m}$ -characters for $m < n$ . Either route establishes existence; uniqueness from the characterisation gives that the two routes produce the same bijection. The bijection sends supercuspidals to irreducible Weil-group representations, generalised principal series to reducible-but-indecomposable Weil-Deligne representations with non-zero monodromy, and parabolic induction on the automorphic side matches induction of parameters on the Galois side. The detailed matching at each level of the Bernstein-Zelevinsky / Langlands classification is given in Harris-Taylor Ch. VII and Henniart §4. $□$

Bridge. The local Langlands correspondence at $GL_{n}$ builds toward global Langlands reciprocity by supplying the local pieces of the global automorphic-Galois pairing: a global cuspidal automorphic representation $π = \otimes_{v}^{'} π_{v}$ has each local component $π_{v}$ matched with a local Langlands parameter $ϕ_{v}$ by $LL_{n}$ , and the global Galois representation $ρ$ has each local restriction $ρ ∣_{G_{F_{v}}}$ matching $ϕ_{v}$ at every place. The central insight is exactly that local Langlands is dual to global Langlands at the level of local components: the local-global compatibility theorem (the foundational reason both proofs of local Langlands work) is that the global correspondence respects the local restriction at every place. This is exactly the local-global pattern that appears again in 21.06.01 (modularity theorem and BSD), where the global automorphic representation $π_{f_{E}}$ of $GL_{2} (A_{Q})$ is built from the local Hecke eigenvalues and the local Langlands correspondence at primes of good reduction matches $π_{v, f_{E}}$ with the local restriction $ρ_{E, ℓ} ∣_{G_{Q_{v}}}$ . Putting these together, local Langlands plus a global automorphic-to-Galois construction (Eichler-Shimura at $n = 2$ over $Q$ , Kottwitz on Shimura varieties at higher rank, Drinfeld / L. Lafforgue on shtukas over function fields) generalises the abelian Artin reciprocity of class field theory to the non-abelian setting and identifies arithmetic and analytic invariants at every place. The bridge is from the abelian one-dimensional Artin reciprocity (the case $n = 1$ ) to the conjectural non-abelian $n$ -dimensional reciprocity, with the local pillar identifying which automorphic representations and which Galois representations correspond.

Exercises Intermediate+

Exercise 2 (easy, numeric).

The local Langlands correspondence at $GL_{2} (Q_{p})$ pairs supercuspidal representations of $GL_{2} (Q_{p})$ with which kind of Weil-group representations?

Hint

Supercuspidals are the irreducible discrete-series representations of $GL_{2}$ . On the Weil-group side they correspond to irreducible representations of $W_{Q_{p}}$ of dimension matching $n$ , with no monodromy from $SL_{2} (C)$ .

Answer

$2$ -dimensional irreducible representations of $W_{Q_{p}}$ (with identity $SL_{2} (C)$ -component in the Weil-Deligne parameter). Equivalently, irreducible $2$ -dimensional Weil-Deligne representations with monodromy $N = 0$ . The supercuspidals correspond to these via the local Langlands correspondence, while the (generalised) principal series correspond to reducible Weil-Deligne representations and the Steinberg representation corresponds to the $2$ -dimensional Weil-Deligne representation with non-zero monodromy. Rubric: full credit for "irreducible $2$ -dimensional Weil-group representations" or equivalent.

Exercise 3 (medium, symbolic).

Show that class field theory is the $n = 1$ case of Langlands reciprocity.

Hint

A one-dimensional Galois representation is a continuous character $χ : G_{F} \to \overline{Q_{ℓ}}^{\times}$ , which factors through the abelianisation $G_{F}^{ab}$ . Class field theory identifies $G_{F}^{ab}$ with the idele class group $F^{\times} \ A_{F}^{\times}$ , and Hecke characters of $F$ are exactly continuous characters of the idele class group.

Answer

Galois side at $n = 1$ . A continuous one-dimensional $ℓ$ -adic Galois representation is a continuous character $ρ : G_{F} \to GL_{1} (\overline{Q_{ℓ}}) = \overline{Q_{ℓ}}^{\times}$ , factoring through the abelianisation $G_{F}^{ab}$ because $\overline{Q_{ℓ}}^{\times}$ is abelian.

Automorphic side at $n = 1$ . A cuspidal automorphic representation of $GL_{1} (A_{F}) = A_{F}^{\times}$ is, by the abelian structure, a continuous character $π : F^{\times} \ A_{F}^{\times} \to C^{\times}$ — that is, a Hecke character in the classical sense (Hecke 1918, 1920).

The bridge: Artin reciprocity. The reciprocity map of class field theory $$ \mathrm{rec}_F : F^\times \backslash \mathbb{A}F^\times ;\xrightarrow{\sim}; G_F^{\mathrm{ab}} $$ is a topological isomorphism (Takagi 1920 in the abelian-extension setting, Artin 1927 in the form above) that identifies the idele class group with the abelianised absolute Galois group. Pre-composition with $rec_{F}$ sets up the bijection $$ {\text{continuous characters } \chi : G_F^{\mathrm{ab}} \to \overline{\mathbb{Q}_\ell}^\times} ;\longleftrightarrow; {\text{Hecke characters } \pi : F^\times \backslash \mathbb{A}F^\times \to \mathbb{C}^\times} $$ where the codomain shift $\overline{\mathbb{Q}\ell}^\times \leftrightarrow \mathbb{C}^\times $i s han d l e d b y f i x in g ani so m or p hi s m$ \overline{\mathbb{Q}\ell} \cong \mathbb{C}$ (non-canonical but always available).

Matching $L$ -functions. At an unramified place $v$ of $F$ , the Galois-side local factor is $L_{v} (s, χ) = (1 - χ (Frob_{v}) q_{v}^{- s})^{- 1}$ with $Frob_{v}$ the geometric Frobenius. The automorphic-side local factor is $L_{v} (s, π) = (1 - π (ϖ_{v}) q_{v}^{- s})^{- 1}$ with $ϖ_{v}$ a local uniformiser. The reciprocity map sends $ϖ_{v} \mapsto Frob_{v}$ at unramified places, so $π (ϖ_{v}) = χ (Frob_{v})$ and the local factors agree. This gives $L (s, χ) = L (s, π)$ as Euler products, the $GL_{1}$ case of Langlands reciprocity. Rubric: full credit for identifying both sides, applying Artin reciprocity, and matching at least one local factor.

Exercise 4 (medium, symbolic).

State the Satake isomorphism for $GL_{n} (Q_{p})$ and explain how it produces a Satake parameter $c (π_{v}) \in GL_{n} (C)$ for an unramified representation $π_{v}$ .

Hint

The spherical Hecke algebra of $GL_{n} (Q_{p})$ at the level $GL_{n} (Z_{p})$ is the convolution algebra of compactly supported $GL_{n} (Z_{p})$ -bi-invariant functions on $GL_{n} (Q_{p})$ . The Satake isomorphism identifies it with a polynomial ring in $n$ variables (the symmetric functions in $n$ variables, mod the trace and determinant relations).

Answer

Satake isomorphism (Satake 1963; standard reference Bump 1997 Ch. 4). Let $G = GL_{n}$ , $F_{v} = Q_{p}$ , $K_{v} = GL_{n} (Z_{p})$ . The spherical Hecke algebra is $$ \mathcal{H}(G(F_v), K_v) = C_c^\infty(K_v \backslash G(F_v) / K_v, \mathbb{C}), $$ the convolution algebra of compactly supported $K_{v}$ -bi-invariant functions. The Satake isomorphism is a $C$ -algebra isomorphism $$ \mathcal{S} : \mathcal{H}(G(F_v), K_v) \xrightarrow{\sim} \mathbb{C}[X_1^{\pm 1}, \ldots, X_n^{\pm 1}]^{S_n} = \mathbb{C}[\widehat{T}]^{W} $$ with the right side the ring of Weyl-invariant Laurent polynomials in $n$ variables — equivalently, the representation ring of the dual maximal torus $T = (C^{\times})^{n}$ inside $G = GL_{n} (C)$ , quotiented by the Weyl group $W = S_{n}$ . The isomorphism is built from the Satake transform $S (f) (diag (X_{1}, \dots, X_{n})) = δ^{1/2} (diag) \int_{N} f (diag \cdot n) d n$ for $N$ the unipotent radical of the upper-triangular Borel, $δ$ the modular character.

Satake parameter. An unramified representation $π_{v}$ of $G (F_{v})$ is an irreducible smooth representation containing a non-zero $K_{v}$ -fixed vector. By Bernstein-Zelevinsky 1977 ^{[Bump 1997]}, such a $π_{v}$ is the unique irreducible subquotient of an unramified principal series $Ind_{B}^{G} (χ_{1} \otimes \dots \otimes χ_{n})$ for some unramified characters $χ_{i}$ of $F_{v}^{\times}$ , equivalently for some values $α_{i} = χ_{i} (ϖ_{v}) \in C^{\times}$ . The Satake parameter of $π_{v}$ is the diagonal matrix $$ c(\pi_v) = \mathrm{diag}(\alpha_1, \ldots, \alpha_n) \in \widehat{G}(\mathbb{C}) = \mathrm{GL}_n(\mathbb{C}), $$ well-defined up to Weyl-group permutation, hence up to conjugacy class.

Local $L$ -factor. For $r : G \to GL_{N} (C)$ the standard representation $r = std$ , the local $L$ -factor at $v$ is $$ L_v(s, \pi, \mathrm{std}) = \det(1 - r(c(\pi_v)) q_v^{-s})^{-1} = \prod_{i = 1}^n (1 - \alpha_i q_v^{-s})^{-1}. $$ The Satake parameter is the bridge between the automorphic-representation side and the $L$ -function side at unramified places. Rubric: full credit for stating the isomorphism, identifying the Satake parameter as a conjugacy class in $G$ , and writing the local $L$ -factor.

Exercise 5 (medium, symbolic).

State Langlands functoriality and give two substantive special cases.

Hint

Functoriality predicts that every morphism $^{L} H \to^{L} G$ of $L$ -groups produces a transfer of automorphic representations. Two famous special cases: base change (the inclusion $^{L} G_{F} \to^{L} G_{E}$ for $E / F$ an extension) and the symmetric-power transfer $^{L} GL_{2} \to^{L} GL_{k + 1}$ via $Sym^{k}$ .

Answer

Functoriality (Langlands 1967; Borel 1979 packaging). Let $H, G$ be split reductive groups over a global field $F$ , with $L$ -groups $^{L} H = H (C) \times Gal (\overline{F} / F)$ and $^{L} G = G (C) \times Gal (\overline{F} / F)$ , and let $r :^{L} H \to^{L} G$ be a morphism of $L$ -groups (a holomorphic homomorphism of dual reductive groups, compatible with Galois actions). The Langlands functoriality conjecture predicts a transfer of automorphic representations $$ \pi_H = \otimes'v \pi{H, v} ;\longmapsto; \pi_G = \otimes'v \pi{G, v} $$ matching Satake parameters at every unramified place by $c (π_{G, v}) = r (c (π_{H, v}))$ and matching $L$ -functions $L (s, π_{G}, std_{G}) = L (s, π_{H}, r \circ std_{G})$ for the standard representations.

Special case 1: base change. For $E / F$ a cyclic extension of degree $d$ and $G$ a split reductive group over $F$ , the base-change map $^{L} G_{F} \to^{L} G_{E}$ predicts a transfer $BC_{E / F} : Aut (G (A_{F})) \to Aut (G (A_{E}))$ , raising the conductor along the extension. Proved for $G = GL_{n}$ by Arthur-Clozel 1989 Simple algebras, base change, and the advanced theory of the trace formula (Ann. Math. Studies 120). Solvable base change for $GL_{2}$ was earlier proved by Langlands 1980 Base Change for $GL (2)$ (Ann. Math. Studies 96) and is the input to Langlands-Tunnell 1980 (modularity of mod- $3$ representations, hence Wiles's $3$ - $5$ switch).

Special case 2: symmetric-power transfer. For $G = GL_{k + 1}$ and $H = GL_{2}$ , the symmetric-power representation $Sym^{k} : GL_{2} (C) \to GL_{k + 1} (C)$ is a morphism of $L$ -groups, and functoriality predicts a transfer $Sym^{k} : Aut (GL_{2} (A_{F})) \to Aut (GL_{k + 1} (A_{F}))$ . Specialising: a cuspidal $π$ on $GL_{2}$ transfers to an automorphic $Sym^{k} π$ on $GL_{k + 1}$ whose Satake parameters are $Sym^{k} c (π_{v})$ at each unramified $v$ . Proved for $k = 2$ (Gelbart-Jacquet 1978 Ann. Sci. ENS 11), $k = 3$ (Kim-Shahidi 2002 Ann. Math. 155), $k = 4$ (Kim 2003 J. Amer. Math. Soc. 16), and in 2021 for all $k \geq 2$ on appropriate base extensions (Newton-Thorne 2021 Publ. Math. IHÉS 134). The symmetric-power transfer is the input to the Sato-Tate conjecture (equidistribution of Frobenius angles), proved as a corollary by Clozel-Harris-Shepherd-Barron-Taylor 2008-2011.

Why functoriality matters. Every morphism of $L$ -groups produces a candidate transfer, and most cases are open. Functoriality plus the unramification at all but finitely many places gives uniformity: the same automorphic-side input feeds into multiple Galois-side outputs (one per representation $r$ of $G$ ). Rubric: full credit for the conjecture statement, plus identification of two specific transfer cases (base change, symmetric-power, automorphic induction, theta-correspondence all acceptable).

Exercise 6 (medium, symbolic).

Sketch why the modularity theorem for elliptic curves over $Q$ is the $GL_{2} / Q$ case of Langlands reciprocity.

Hint

The modularity theorem says every elliptic curve over $Q$ corresponds to a weight- $2$ cusp newform $f_{E}$ with $L (E, s) = L (f_{E}, s)$ . The Galois side is $ρ_{E, ℓ}$ on the $ℓ$ -adic Tate module; the automorphic side is the cuspidal $GL_{2} (A_{Q})$ -representation generated by $f_{E}$ .

Answer

Galois side. For an elliptic curve $E / Q$ and a prime $ℓ$ , the $ℓ$ -adic Tate module $V_{ℓ} E = T_{ℓ} E \otimes_{Z_{ℓ}} Q_{ℓ}$ is a $2$ -dimensional $Q_{ℓ}$ -vector space with a continuous Galois action $ρ_{E, ℓ} : G_{Q} \to GL_{2} (Q_{ℓ})$ . It is unramified at every prime $p ∤ N_{E} ℓ$ (Néron-Ogg-Shafarevich), with $tr ρ_{E, ℓ} (Frob_{p}) = a_{p} = p + 1 - N_{p}$ and $det ρ_{E, ℓ} (Frob_{p}) = p$ . So the Galois-side $L$ -function is the Hasse-Weil $L$ -function $L (s, ρ_{E, ℓ}) = \prod_{p} det (1 - ρ_{E, ℓ} (Frob_{p}) p^{- s} ∣ V_{ℓ} E^{I_{p}})^{- 1} = L (E, s)$ .

Automorphic side. Let $f_{E}$ be the weight- $2$ cusp newform on $Γ_{0} (N_{E})$ corresponding to $E$ by the modularity theorem of Wiles 1995 / BCDT 2001 ^{[Manin-Panchishkin 2005]}. The form $f_{E}$ generates an irreducible cuspidal automorphic representation $π_{f_{E}}$ of $GL_{2} (A_{Q})$ via the adelisation procedure: the function on $H$ assembles, via strong approximation $GL_{2} (Q) \ GL_{2} (A_{Q}) / (Z (A) \cdot K_{\infty} \cdot K_{0} (N_{E})) ≅ Γ_{0} (N_{E}) \ H$ , into a smooth right- $K$ -finite function on $GL_{2} (Q) \ GL_{2} (A_{Q})$ generating an irreducible subrepresentation $π_{f_{E}}$ .

The local components are: at the archimedean place $\infty$ , $π_{f_{E}, \infty}$ is the discrete series of lowest weight $2$ representation $D_{2}^{+}$ of $GL_{2} (R)$ (or $PGL_{2} (R)$ ); at each finite prime $p ∤ N_{E}$ , $π_{f_{E}, p}$ is the unramified principal series with Satake parameter $c (π_{f_{E}, p}) = diag (α_{p}, β_{p})$ where $α_{p} β_{p} = p$ , $α_{p} + β_{p} = a_{p} (f_{E})$ . The automorphic-side $L$ -function is $L (s, π_{f_{E}}) = L (s, f_{E}) = \prod_{p} L_{p} (s, f_{E})$ .

The match. Modularity (21.06.01 modularity theorem) gives $a_{p} = a_{p} (f_{E})$ at every prime $p ∤ N_{E}$ , hence the Galois Frobenius traces equal the automorphic Hecke eigenvalues, hence $L (s, ρ_{E, ℓ}) = L (s, π_{f_{E}})$ . The local Langlands correspondence at each $p$ matches $ρ_{E, ℓ} ∣_{G_{Q_{p}}}$ with $π_{f_{E}, p}$ — at unramified primes by the Satake-parameter formula, at bad primes by the Eichler-Shimura / Carayol local-global compatibility (Carayol 1986 Ann. Sci. ENS 19). This is exactly the $GL_{2} / Q$ case of Langlands reciprocity. Rubric: full credit for identifying both sides, matching unramified Satake parameters, and citing modularity for the bridge.

Exercise 7 (hard, symbolic).

State Drinfeld's 1974 / 1980 theorem on the Langlands correspondence for $GL_{2}$ over function fields, and L. Lafforgue's 2002 generalisation to $GL_{n}$ .

Hint

Drinfeld proves the function-field analogue of Langlands reciprocity at $n = 2$ via the moduli of $F$ -bundles with shtuka structure (now called Drinfeld shtukas), a stack whose étale cohomology decomposes along automorphic representations and Galois representations simultaneously. Lafforgue generalises by working with $n$ -step shtuka chains.

Answer

Function-field setup. Let $X$ be a smooth projective geometrically connected curve over $F_{q}$ and $F = F_{q} (X)$ its function field. The places of $F$ correspond to closed points $v \in ∣ X ∣$ , with $F_{v}$ the completion (a local field of equal characteristic) and residue field $F_{q_{v}}$ . The adele ring $A_{F} = \prod_{v}^{'} F_{v}$ and $GL_{n} (A_{F})$ are constructed as in the number-field case.

Drinfeld's theorem ( $n = 2$ ; Drinfeld 1974 Funct. Anal. Appl. 8 + 1980 Soviet Math. Dokl. 21) ^{[Drinfeld 1974/1980]}. There is a bijection between irreducible continuous $ℓ$ -adic Galois representations $ρ : G_{F} \to GL_{2} (\overline{Q_{ℓ}})$ with determinant of finite order, unramified outside a finite set $S$ of places, and cuspidal automorphic representations $π$ of $GL_{2} (A_{F})$ with central character of finite order, unramified outside $S$ — matching Satake parameters at every unramified place and matching $L$ -functions $L (s, ρ) = L (s, π)$ .

Method. Drinfeld introduces the moduli space $Sht_{X}^{2}$ of shtukas of rank $2$ on $X$ : pairs $(F, ϕ)$ with $F$ a rank- $2$ vector bundle on $X$ over an $F_{q}$ -scheme $S$ and $ϕ : Frob^{*} F ⇢ F$ a meromorphic isomorphism with prescribed zeros and poles. The moduli space is a Deligne-Mumford stack of dimension $2$ with a finite étale map to $∣ X ∣ \times ∣ X ∣$ recording the two modifications. The middle-degree $ℓ$ -adic cohomology $H_{et}^{2} (Sht_{X}^{2} \times \overline{F_{q}}, \overline{Q_{ℓ}})$ carries commuting actions of $GL_{2} (A_{F})$ (via Hecke correspondences) and $Gal (\overline{F} / F)$ , and the isotypic decomposition along cuspidal $π$ realises each $ρ_{π}$ as the Galois representation paired with $π$ .

L. Lafforgue's theorem ( $n$ general; L. Lafforgue 2002 Invent. Math. 147) ^{[L. Lafforgue 2002]}. The bijection extends to all $n \geq 1$ : irreducible continuous $ℓ$ -adic Galois representations $ρ : G_{F} \to GL_{n} (\overline{Q_{ℓ}})$ with determinant of finite order, unramified outside a finite set $S$ , correspond bijectively to cuspidal automorphic representations $π$ of $GL_{n} (A_{F})$ with central character of finite order, unramified outside $S$ , with matching $L$ - and $ε$ -factors at every place (including ramified ones).

Method. Lafforgue replaces Drinfeld's two-modification shtukas with iterated shtukas: chains of vector bundles connected by an arbitrary number of meromorphic modifications. The moduli space $Sht_{X, I}^{n}$ of $n$ -shtukas with $I$ legs (modifications) is a Deligne-Mumford stack of dimension $∣ I ∣ \cdot (n - 1)$ , and its middle-degree cohomology carries the commuting actions. The Arthur-Selberg trace formula on $GL_{n} (A_{F})$ combined with the Lefschetz-Verdier trace formula on $Sht_{X, I}^{n}$ matches the two sides via a comparison of trace identities at every place.

Award. Drinfeld received the Fields Medal in 1990 (Kyoto ICM) for his $GL_{2}$ proof and the introduction of shtukas. L. Lafforgue received the Fields Medal in 2002 (Beijing ICM) for the $GL_{n}$ proof. The function-field case is the only known full Langlands reciprocity for general $n$ ; the number-field case at $n \geq 2$ is open in general and reduces, in $n = 2$ over totally real fields, to a sequence of modularity-lifting theorems (Wiles 1995, Skinner-Wiles, Khare-Wintenberger 2009, Newton-Thorne 2021). Rubric: full credit for statement of both theorems plus identification of shtukas as the underlying moduli, plus the Fields Medal attribution.

Exercise 8 (hard, symbolic).

State the geometric Langlands conjecture for $G = GL_{n}$ over a smooth projective curve $X$ over $C$ , in the categorical formulation.

Hint

Replace number-field arithmetic by complex algebraic geometry. The Galois side becomes the moduli of rank- $n$ local systems (equivalently, $n$ -dimensional flat connections) on $X$ . The automorphic side becomes $D$ -modules on the stack $Bun_{GL_{n}} (X)$ of rank- $n$ vector bundles. The conjecture identifies derived categories.

Answer

Setup (geometric Langlands over $C$ ; Beilinson-Drinfeld c. 1991-2004 ^{[Beilinson-Drinfeld]}). Let $X$ be a smooth projective connected curve over $C$ of genus $g \geq 2$ (for stability of $Bun_{G}$ at low rank, and to avoid degenerate cases at $g = 0, 1$ ). Let $G = GL_{n}$ , so $G = GL_{n} (C)$ .

Galois side: local systems. A local system of rank $n$ on $X$ is a $C$ -linear representation $π_{1} (X, x_{0}) \to GL_{n} (C)$ , equivalently a rank- $n$ vector bundle on $X$ with a flat algebraic connection (Riemann-Hilbert correspondence). The moduli stack $LocSys_{GL_{n}} (X)$ parametrises rank- $n$ local systems up to gauge.

Automorphic side: $D$ -modules on $Bun_{G}$ . The moduli stack of $G$ -bundles $Bun_{GL_{n}} (X)$ parametrises rank- $n$ vector bundles on $X$ , an algebraic stack of dimension $(g - 1) n^{2}$ . The category $D -mod (Bun_{GL_{n}} (X))$ of $D$ -modules on $Bun_{GL_{n}} (X)$ replaces the automorphic-form Hilbert space.

Hecke operators. For each $λ$ a dominant weight of $G$ , the Hecke correspondence $Hk_{λ} \subseteq Bun_{G} \times X \times Bun_{G}$ parametrises pairs of bundles $(F_{1}, F_{2})$ with a meromorphic isomorphism away from a single point $x \in X$ , modifying along $λ$ . The Hecke functor $H_{λ} : D -mod (Bun_{G}) \to D -mod (Bun_{G} \times X)$ is pull-push along $Hk_{λ}$ with the relevant twist.

Hecke eigensheaves. A $D$ -module $F$ on $Bun_{G}$ is a Hecke eigensheaf for a local system $E$ on $X$ if $H_{λ} (F) ≅ F ⊠ V_{λ} (E)$ for every $λ$ , where $V_{λ} (E)$ is the local system on $X$ associated to $E$ by the irreducible representation $V_{λ}$ of $G$ of highest weight $λ$ .

Geometric Langlands conjecture (categorical formulation). There is an equivalence of derived categories $$ \mathbb{L}G : D\text{-mod}(\mathrm{Bun}G(X)) ;\xrightarrow{\sim}; \mathrm{QCoh}(\mathrm{LocSys}{\widehat{G}}(X)) $$ *sending each Hecke eigensheaf $\mathcal{F}\mathcal{E} $(w i t h e i g e n v a l u e l oc a l sy s t e m$ \mathcal{E} $) t o t h es k y scr a p er s h e a f$ \mathcal{O}\mathcal{E} $a tt h e p o in t$ \mathcal{E} \in \mathrm{LocSys}{\widehat{G}}(X) $, an d in t er tw inin g t h eH ec k e f u n c t or s$ H_\lambda $o n t h e l e f tw i t h t h e t e n sor f u n c t or s$ \cdot \otimes p^* V_\lambda(\mathcal{E}) $o n t h er i g h t (w h er e$ p : \mathrm{LocSys}{\widehat{G}} \times X \to \mathrm{LocSys}{\widehat{G}}$ is the projection).*

Status (2024). The conjecture is proved for $G = GL_{2}$ by Gaitsgory-Raskin and collaborators in a 2024 series of arXiv preprints (starting with arXiv:2405.03648) ^{[Gaitsgory-Raskin 2024]}. The proof uses the fundamental local equivalence of D-modules on the affine Grassmannian, factorisation categories, and the chiral algebra at the critical level, all of which were developed in Beilinson-Drinfeld's original 1991-2004 preprint ^{[Beilinson-Drinfeld]}. The case $G = GL_{n}$ for $n \geq 3$ remains open, as do all non-type- $A$ cases.

Bridge to physics. Kapustin-Witten 2007 Commun. Number Theory Phys. 1 derived the conjecture from a topological twist of $N = 4$ super-Yang-Mills theory in four dimensions; the categorical equivalence is the categorical mirror symmetry between the Hitchin moduli space of $G$ -bundles with Higgs field and that of $G$ -bundles. Rubric: full credit for stating the categorical equivalence, identifying both sides, and citing the 2024 Gaitsgory-Raskin proof at $GL_{2}$ .

Lean formalization Intermediate+

Mathlib has no automorphic-representation framework. No part of Langlands reciprocity — neither the Galois side, the automorphic side, the matching $L$ -functions, nor the local-global compatibility — has a named declaration. The formalisation gap is structural and large; the present unit is therefore a survey of a research programme, not a formalisable theorem.

The minimum entry points required for any substantive Lean statement are sketched here as anticipated stubs:

import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.RepresentationTheory.Basic

/-- The adele ring of a global field — a restricted topological product
    of completions. Not yet a Mathlib type. -/
structure AdeleRing (F : Type*) [Field F] : Type _ :=
  -- restricted product of F_v over the places v
  -- with respect to the maximal compact subrings O_{F_v}
  carrier : Type*  -- placeholder
  -- topology, ring structure, locally compact, F embeds discretely
  -- All of this is a multi-month formalisation effort.

/-- A reductive group over F and its A_F-points. -/
structure ReductiveGroup (F : Type*) [Field F] : Type _ :=
  scheme : Type*  -- affine algebraic group scheme over F
  -- reductivity hypothesis: connected, smooth, unipotent radical = identity

/-- A cuspidal automorphic representation of G(A_F). -/
structure CuspidalAutomorphicRepresentation
    (F : Type*) [NumberField F]
    (G : ReductiveGroup F) : Type _ :=
  -- an irreducible unitary representation occurring in
  -- L²_cusp(G(F) \ G(A_F)) with fixed central character
  carrier : Type*  -- the underlying Hilbert space
  -- restricted-tensor decomposition pi = otimes' pi_v across places

/-- The Langlands dual group as a complex reductive group. -/
def langlandsDual (G : ReductiveGroup F) : Type _ :=
  sorry  -- complex reductive group with dual root datum

/-- The L-group as a semidirect product. -/
def LGroup (F : Type*) [Field F] (G : ReductiveGroup F) : Type _ :=
  sorry  -- langlandsDual G ⋊ Gal(F̄/F)

/-- A Langlands parameter for G at a local field F_v. -/
def LanglandsParameter (F_v : Type*) [LocalField F_v]
    (G : ReductiveGroup F_v) : Type _ :=
  sorry  -- continuous hom W_{F_v} × SL_2(C) → ^L G

/-- Langlands reciprocity for GL_n (statement only). -/
theorem langlandsReciprocityGLn (F : Type*) [NumberField F] (n : ℕ) :
    -- bijection between
    -- (irreducible continuous semisimple n-dim ell-adic Galois reps
    --  of G_F unramified outside finite S)
    -- and
    -- (cuspidal automorphic representations of GL_n(A_F)
    --  unramified outside S)
    -- matching L-functions L(s, ρ) = L(s, π) at every place
    True := sorry  -- statement-only stub

The Lean stubs above are theatrical: every sorry represents months of formalisation work even to state the theorem, and the proof in the function-field case (Drinfeld + L. Lafforgue) and the local case (Harris-Taylor + Henniart + Fargues-Scholze) is far beyond any current formalisation horizon. The status lean_status: none and the named human reviewer record that the correctness gate for this survey unit is mathematical review against the canonical references (Borel 1979, Bump 1997, Manin-Panchishkin 2005), not formalisation. The aggregated gap (adele ring, automorphic spectrum, $L$ -group, local Langlands, $L$ -function comparison) is the single most ambitious target in the arithmetic-geometry corner of the Mathlib roadmap.

Advanced results Master

Function-field Langlands: Drinfeld and L. Lafforgue

Theorem 1 (Drinfeld 1974 / 1980; Langlands correspondence for $GL_{2}$ over function fields). Let $X$ be a smooth projective geometrically connected curve over $F_{q}$ and $F = F_{q} (X)$ . There is a bijection between irreducible continuous $ℓ$ -adic Galois representations $ρ : G_{F} \to GL_{2} (\overline{Q_{ℓ}})$ with determinant of finite order, unramified outside a finite set $S$ , and cuspidal automorphic representations $π$ of $GL_{2} (A_{F})$ with central character of finite order, unramified outside $S$ , matching $L$ -functions $L (s, ρ) = L (s, π)$ and matching local Langlands parameters at every place.

The construction uses the moduli stack of rank- $2$ Drinfeld shtukas on $X$ and Hecke correspondences acting on its $ℓ$ -adic cohomology. Drinfeld received the Fields Medal in 1990 for this theorem and the introduction of shtukas.

Theorem 2 (L. Lafforgue 2002 Invent. Math. 147; Langlands correspondence for $GL_{n}$ over function fields). The bijection extends to every $n \geq 1$ : irreducible continuous $ℓ$ -adic Galois representations $ρ : G_{F} \to GL_{n} (\overline{Q_{ℓ}})$ with determinant of finite order, unramified outside $S$ , correspond bijectively to cuspidal automorphic representations $π$ of $GL_{n} (A_{F})$ with central character of finite order, unramified outside $S$ , matching $L$ - and $ε$ -factors at every place.

The proof uses iterated Drinfeld shtukas with $∣ I ∣$ legs and the Arthur-Selberg trace formula. L. Lafforgue received the Fields Medal in 2002 for this proof. The function-field case is the only known full Langlands reciprocity for general $n$ .

Theorem 3 (V. Lafforgue 2018 J. Amer. Math. Soc. 31; automorphic-to-Galois for reductive groups over function fields). Let $G$ be a connected split reductive group over $F = F_{q} (X)$ . There is a canonical decomposition of the cuspidal spectrum of $G (A_{F})$ along Langlands parameters: every cuspidal automorphic representation $π$ of $G (A_{F})$ determines a continuous semisimple $ℓ$ -adic Langlands parameter $σ_{π} : G_{F} \to G (\overline{Q_{ℓ}})$ , characterised by compatibility with Satake parameters at unramified places.

The construction proceeds via excursion operators: a family of operators on the $ℓ$ -adic cohomology of the moduli stack of $G$ -shtukas, parametrised by tuples of representations of $G$ and elements of the Galois group, satisfying explicit commutation relations. The simultaneous eigenvalues of the excursion operators on an isotypic component for $π$ assemble into the Langlands parameter $σ_{π}$ . V. Lafforgue's theorem realises the automorphic-to-Galois direction of Langlands functoriality for arbitrary reductive groups in the function-field case.

Local Langlands and the Fargues-Scholze geometrisation

Theorem 4 (Harris-Taylor 2001; Henniart 2000; local Langlands for $GL_{n}$ over $p$ -adic fields). Stated in the Key theorem section above. The bijection $LL_{n} : Irr^{adm} (GL_{n} (Q_{p})) \sim Param_{n} (W_{Q_{p}})$ is characterised by matching $L$ - and $ε$ -factors of all twists.

Theorem 5 (Fargues-Scholze 2024 Astérisque 449; geometrisation of local Langlands). Let $G$ be a connected reductive group over $Q_{p}$ . There is a canonical functor $$ \mathrm{Rep}^{\mathrm{sm}}(G(\mathbb{Q}p)) ;\longrightarrow; \mathrm{Coh}(\mathrm{LocSys}{\widehat{G}}^{\square}), $$ from smooth representations of $G (Q_{p})$ to coherent sheaves on the (formal) stack of $G$ -valued Langlands parameters, sending each irreducible smooth $π$ to the structure sheaf of the corresponding parameter $ϕ_{π}$ . The construction proceeds via the $ℓ$ -adic cohomology of the moduli of $G$ -bundles on the Fargues-Fontaine curve.

The Fargues-Fontaine curve $X_{FF}$ is a curve-like object in the world of perfectoid spaces / diamonds (Scholze 2012-2017): a sousmaltent of the perfectoid disc punctured at $0$ , modded out by a Frobenius operator. Its $G$ -bundle moduli stack $Bun_{G} (X_{FF})$ has a stratification by Newton points parametrising isocrystals; the Fargues-Scholze functor extracts, from this stratification and from $ℓ$ -adic sheaves on $Bun_{G} (X_{FF})$ , the local Langlands correspondence as a categorical rather than set-theoretic statement. The construction recovers the Harris-Taylor / Henniart bijection on irreducible smooth representations of $GL_{n} (Q_{p})$ and extends to arbitrary connected reductive $G$ .

Geometric Langlands over $C$

Theorem 6 (Beilinson-Drinfeld c. 1991-2004 — construction of Hecke eigensheaves; Gaitsgory-Raskin et al. 2024 — proof for $GL_{2}$ ). Let $X$ be a smooth projective connected curve over $C$ of genus $g \geq 2$ and $G = GL_{2}$ . There is an equivalence of derived categories $$ D\text{-mod}(\mathrm{Bun}_{\mathrm{GL}2}(X)) ;\xrightarrow{\sim}; \mathrm{IndCoh}(\mathrm{LocSys}{\mathrm{GL}_2}(X)), $$ identifying $D$ -modules on the moduli stack of rank- $2$ vector bundles on $X$ with ind-coherent sheaves on the moduli stack of rank- $2$ local systems on $X$ . The equivalence sends each Hecke eigensheaf for a local system $E$ to the skyscraper sheaf at $E$ , intertwining the Hecke functors with the tensor functors.

The Beilinson-Drinfeld construction produces Hecke eigensheaves from opers — flat connections on $X$ in the principal $sl_{2}$ -subalgebra. The Gaitsgory-Raskin 2024 proof completes the equivalence at $GL_{2}$ via the fundamental local equivalence of $D$ -modules on the affine Grassmannian, factorisation categories, and the chiral algebra at the critical level. The cases $G = GL_{n}$ for $n \geq 3$ and non-type- $A$ reductive $G$ remain open as of 2026.

Functoriality consequences

Theorem 7 (Newton-Thorne 2021 Publ. Math. IHÉS 134; symmetric-power lifts for $GL_{2}$ Hecke eigenforms). Let $f$ be a non-CM weight- $k$ Hecke eigenform on $Γ_{0} (N)$ . For every $k \geq 1$ , the symmetric power $Sym^{k} π_{f}$ is a cuspidal automorphic representation of $GL_{k + 1} (A_{Q})$ .

The case $k \leq 4$ was earlier work of Gelbart-Jacquet, Kim-Shahidi, Kim. Newton-Thorne 2021 prove all $k \geq 5$ by a modularity-lifting / automorphy-lifting argument over a sequence of soluble extensions, combined with a careful study of the residual mod- $ℓ$ representations. The theorem is the input to the Sato-Tate conjecture for non-CM Hecke eigenforms (Clozel-Harris-Shepherd-Barron-Taylor 2008-2011 had earlier proved the elliptic-curve case).

Theorem 8 (Sato-Tate, classical case; Clozel-Harris-Shepherd-Barron-Taylor 2008-2011). Let $E / Q$ be a non-CM elliptic curve and write $a_{p} = 2 p cos θ_{p}$ with $θ_{p} \in [0, π]$ . The angles ${θ_{p}}_{p}$ are equidistributed in $[0, π]$ with respect to the Sato-Tate measure $(2/ π) sin^{2} θ d θ$ as $p \to \infty$ .

The proof uses the symmetric-power lifts $Sym^{k} π_{E}$ for every $k$ and the analytic Tauberian theorem applied to the partial $L$ -function $L (s, Sym^{k} π_{E})$ at $s = 1$ . The Sato-Tate conjecture is the simplest consequence of full functoriality at $GL_{2}$ , and its proof was one of the first applications of Galois deformation theory to functoriality outside modularity itself.

Synthesis. The Langlands philosophy is the foundational reason number theory and harmonic analysis are dual languages for the same arithmetic. The central insight is exactly that the absolute Galois group $G_{F}$ and the adelic group $G (A_{F})$ pair up at the level of irreducible representations: every irreducible $n$ -dimensional $ℓ$ -adic Galois representation of $G_{F}$ is conjecturally matched with a cuspidal automorphic representation of $GL_{n} (A_{F})$ , and the matching is exactly the equality of $L$ -functions. Putting these together, the abelian case (class field theory, $n = 1$ ) is a theorem of Artin and Takagi, the $GL_{2} / Q$ case is the modularity theorem of Wiles and BCDT, the function-field case at every $n$ is the theorem of Drinfeld and L. Lafforgue, and the local case is the theorem of Harris-Taylor and Henniart, geometrised by Fargues-Scholze. The bridge is from the Galois-theoretic encoding of arithmetic to the automorphic-representation encoding of analysis, and the recursion through Satake parameters and local Langlands gives a uniform language at every place. This pattern recurs in 21.06.01 (modularity theorem and BSD), where the elliptic-curve $L$ -function equals the modular-form $L$ -function under modularity, and in 21.04.03 (Eichler-Shimura), where the weight- $2$ cusp eigenform attaches a Galois representation, and in 21.03.03 (Dedekind-Hecke-Artin $L$ -functions), where Brauer induction realises every Artin $L$ -function inside the framework that Langlands functoriality predicts to be the $GL_{n}$ automorphic spectrum.

The geometric Langlands programme transposes the picture into complex algebraic geometry. The Galois side becomes the moduli stack of $G$ -local systems on a curve $X$ over $C$ , and the automorphic side becomes the derived category of $D$ -modules on $Bun_{G} (X)$ . The Beilinson-Drinfeld construction realises Hecke eigensheaves via opers, and the Gaitsgory-Raskin 2024 proof for $GL_{2}$ identifies the categorical equivalence at the level of derived categories. This is exactly the geometrisation of Langlands reciprocity: the number-theoretic correspondence between Galois representations and automorphic representations becomes a geometric correspondence between local systems and Hecke eigensheaves on moduli stacks. The Fargues-Scholze 2024 geometrisation of local Langlands realises the same pattern locally: smooth representations of $G (Q_{p})$ identify with coherent sheaves on the moduli of local Langlands parameters, mediated by the Fargues-Fontaine curve in the perfectoid world.

The bridge is exactly the recognition that Galois representations and automorphic representations are encoding the same arithmetic data in dual languages, and that this duality persists across local and global, number-field and function-field, and arithmetic and geometric settings. The Langlands philosophy is dual to itself in a precise sense: the reciprocity conjecture is symmetric in Galois and automorphic data, and the same statement read backwards predicts that every automorphic representation comes from a Galois representation. The remaining open cases — the number-field case at $n \geq 2$ beyond modularity, the geometric case at $GL_{n}$ for $n \geq 3$ , the local geometric case for non-type- $A$ groups — define the programme for the coming decades. The pattern recurs in the Beilinson-Bloch-Kato conjectures generalising BSD to higher-dimensional varieties, and in the Langlands-Kottwitz method on Shimura varieties unifying the modular-curve Eichler-Shimura with the higher-rank Shimura-variety constructions.

Full proof set Master

The Langlands reciprocity conjecture itself is a survey statement; its proof in the function-field case (Drinfeld, L. Lafforgue, V. Lafforgue) and the local case (Harris-Taylor, Henniart, Fargues-Scholze) is the deep content. What can be proved at survey level is the precise statement of the conjecture at $GL_{n}$ , the equivalence of formulations via Satake parameters, and the explicit cases that follow from class field theory and the Satake isomorphism. The harder theorems are stated without proof, with primary citations to the original papers.

Proposition 14 (Satake isomorphism; Satake 1963 Publ. Math. IHÉS 18). Let $G$ be a connected reductive group over a non-archimedean local field $F_{v}$ with maximal unramified extension $\tilde{F}_{v}$ , and let $K_{v} = G (O_{F_{v}})$ be a hyperspecial maximal compact subgroup. There is a $C$ -algebra isomorphism $$ \mathcal{S} : \mathcal{H}(G(F_v), K_v) \xrightarrow{\sim} \mathbb{C}[\widehat{T}]^{W} = R(\widehat{G}) $$ from the spherical Hecke algebra to the representation ring of the dual reductive group $G$ , equivalently to the Weyl-invariant Laurent polynomials on the dual maximal torus.

Proof. The Satake transform is defined by $$ \mathcal{S}(f)(t) = \delta_B(t)^{1/2} \int_{N(F_v)} f(t n) dn $$ for $f \in C_{c}^{\infty} (K_{v} \ G (F_{v}) / K_{v})$ , where $B = T N$ is a Borel containing $T$ , $δ_{B}$ is the modular character, and the integral is over the unipotent radical. Bruhat decomposition $G (F_{v}) = ⨆_{w} B w B$ refined by the Iwasawa decomposition $G (F_{v}) = K_{v} B (F_{v})$ identifies the image of $S$ with the Weyl-invariants in $C [T]$ . The injectivity uses the Cartan decomposition $G (F_{v}) = K_{v} T (Z_{\geq 0}) K_{v}$ (sum over dominant cocharacters $λ$ ), and the surjectivity is checked on the characters $ch_{λ}$ of irreducible representations $V_{λ}$ of $G$ . Detailed proof in Bump 1997 §3.5 ^{[Bump 1997]}. $□$

Proposition 15 (Satake parameter of an unramified representation). Let $π_{v}$ be an irreducible smooth admissible representation of $G (F_{v})$ with non-zero $K_{v}$ -fixed vector. Then the spherical Hecke algebra $H (G (F_{v}), K_{v})$ acts on the $K_{v}$ -fixed line $π_{v}^{K_{v}}$ by a character $χ_{π_{v}} : H (G (F_{v}), K_{v}) \to C$ . Via the Satake isomorphism, $χ_{π_{v}}$ corresponds to a semisimple conjugacy class $c (π_{v}) \in G (C) / conj$ , the Satake parameter of $π_{v}$ .

Proof. The action of $H (G (F_{v}), K_{v})$ on $π_{v}^{K_{v}}$ is well-defined: convolution against a bi- $K_{v}$ -invariant function preserves $K_{v}$ -fixed vectors. Schur's lemma applied to the algebra action on the one-dimensional $π_{v}^{K_{v}}$ (one-dimensionality follows from $π_{v}$ being irreducible and unramified) gives that the algebra acts by a character. The Satake isomorphism translates the character into a $C$ -algebra homomorphism $R (G) \to C$ , equivalently a point of $Spec R (G) = G (C) / / G$ (the GIT quotient by conjugation), realised as a semisimple conjugacy class in $G (C)$ . $□$

Proposition 16 (matching local $L$ -factors at unramified places). Let $π$ be a cuspidal automorphic representation of $GL_{n} (A_{F})$ and let $ρ$ be a continuous semisimple $n$ -dimensional $ℓ$ -adic Galois representation of $G_{F}$ , both unramified at a place $v$ of $F$ . Then $L (s, ρ) = L (s, π)$ at the place $v$ if and only if the Frobenius conjugacy class $ρ (Frob_{v}) \in GL_{n} (\overline{Q_{ℓ}})$ corresponds (under a fixed isomorphism $ι : \overline{Q_{ℓ}} ≅ C$ ) to the Satake parameter $c (π_{v}) \in GL_{n} (C)$ .

Proof. The Galois-side local factor at an unramified place $v$ is $$ L_v(s, \rho) = \det(1 - \rho(\mathrm{Frob}_v) q_v^{-s})^{-1}. $$ The automorphic-side local factor at $v$ via the standard representation $std : GL_{n} (C) \to GL_{n} (C)$ is $$ L_v(s, \pi) = \det(1 - \mathrm{std}(c(\pi_v)) q_v^{-s})^{-1} = \det(1 - c(\pi_v) q_v^{-s})^{-1}. $$ These are equal as Euler factors precisely when $det (1 - ρ (Frob_{v}) X) = det (1 - c (π_{v}) X)$ as polynomials in $X$ , equivalently when the characteristic polynomials of $ρ (Frob_{v})$ and $c (π_{v})$ are equal, equivalently when $ρ (Frob_{v})$ and $c (π_{v})$ have the same eigenvalues (with multiplicity). Since both are semisimple by hypothesis, equal characteristic polynomials means equal conjugacy classes. $□$

Proposition 17 (class field theory as $n = 1$ Langlands). For $G = GL_{1}$ and any global field $F$ , Langlands reciprocity reduces to class field theory. The bijection $ρ \leftrightarrow π$ is the Artin reciprocity correspondence between continuous characters $χ : G_{F}^{ab} \to \overline{Q_{ℓ}}^{\times}$ and Hecke characters $π : F^{\times} \ A_{F}^{\times} \to C^{\times}$ , mediated by the Artin reciprocity map $rec_{F} : F^{\times} \ A_{F}^{\times} \sim G_{F}^{ab}$ .

Proof. The Galois side at $n = 1$ consists of continuous one-dimensional $ℓ$ -adic representations of $G_{F}$ , which factor through the abelianisation $G_{F}^{ab}$ because the codomain $\overline{Q_{ℓ}}^{\times}$ is abelian. The automorphic side at $n = 1$ consists of cuspidal automorphic representations of $GL_{1} (A_{F}) = A_{F}^{\times}$ , which are by abelianness continuous characters $π : F^{\times} \ A_{F}^{\times} \to C^{\times}$ (Hecke characters in the classical Hecke 1918 / 1920 sense). The Artin reciprocity map $rec_{F} : F^{\times} \ A_{F}^{\times} \sim G_{F}^{ab}$ of class field theory is a topological isomorphism; pre-composition with $rec_{F}$ sets up a bijection between continuous characters of $G_{F}^{ab}$ and Hecke characters of $F$ . Matching local factors: at an unramified place $v$ , $rec_{F} (ϖ_{v}) = Frob_{v}$ (under the geometric Frobenius normalisation), so $π (ϖ_{v}) = χ (Frob_{v})$ and the local $L$ -factors agree. $□$

Proposition 18 (functoriality preserves $L$ -functions). Let $r :^{L} H \to^{L} G$ be a morphism of $L$ -groups for split reductive $H, G$ over a global field $F$ . If the functorial transfer $Trans_{r} : Aut (H (A_{F})) \to Aut (G (A_{F}))$ exists and matches Satake parameters at every unramified place by $c (Trans_{r} (π_{H})_{v}) = r (c (π_{H, v}))$ , then for every representation $R :^{L} G \to GL_{N} (C)$ , $$ L(s, \mathrm{Trans}_r(\pi_H), R) = L(s, \pi_H, R \circ r). $$

Proof. At an unramified place $v$ , the Satake parameters satisfy $c (Trans_{r} (π_{H})_{v}) = r (c (π_{H, v}))$ by hypothesis. The local $L$ -factor of $Trans_{r} (π_{H})$ at $v$ under $R$ is $$ L_v(s, \mathrm{Trans}r(\pi_H), R) = \det(1 - R(r(c(\pi{H, v}))) q_v^{-s})^{-1} = \det(1 - (R \circ r)(c(\pi_{H, v})) q_v^{-s})^{-1} = L_v(s, \pi_H, R \circ r). $$ The global $L$ -functions agree at every unramified place, hence agree as Euler products (after handling the finitely many ramified places by compatible ramified local Langlands at $H$ and $G$ ). $□$

Stated without proof: Drinfeld 1974 / 1980, L. Lafforgue 2002, V. Lafforgue 2018, Harris-Taylor 2001, Henniart 2000, Fargues-Scholze 2024, Beilinson-Drinfeld 1991-2004 + Gaitsgory-Raskin 2024. These are theorem-level results whose proofs are book-length and rely on deep machinery (shtukas, Shimura varieties, perfectoid spaces / diamonds, chiral algebras at the critical level). Each is cited with primary reference in the Bibliography. The detailed proofs are surveyed in: Lafforgue's ICM 2002 lecture (Beijing); Harris-Taylor's Annals of Math. Studies 151 monograph; Fargues-Scholze's 600-page Astérisque 449; the Gaitsgory-Raskin 2024 arXiv series.

Connections Master

$ℓ$ -adic Galois representations 21.05.01. The Galois side of Langlands reciprocity is exactly the $ℓ$ -adic Galois-representation framework developed in 21.05.01: continuous semisimple representations $ρ : G_{F} \to GL_{n} (\overline{Q_{ℓ}})$ unramified outside a finite set, with their Frobenius traces at unramified places, their de Rham / crystalline structure at places above $ℓ$ (Fontaine 1994), and their geometric subcategory (Fontaine-Mazur 1995). Langlands reciprocity is the conjecture that this category is exactly the category of cuspidal automorphic representations of $GL_{n} (A_{F})$ on the other side of the bridge.
Eichler-Shimura correspondence 21.04.03. The originating example of automorphic-to-Galois construction. Eichler 1954 + Shimura 1958 attached two-dimensional $ℓ$ -adic Galois representations to weight- $2$ cusp newforms via the Jacobian of the modular curve $X_{0} (N)$ . This is the $GL_{2} / Q$ holomorphic-cuspidal case of Langlands reciprocity, generalised to weight $k \geq 2$ by Deligne 1971 via Kuga-Sato varieties and to weight $1$ by Deligne-Serre 1974 via congruences. The Eichler-Shimura construction is the template that the higher-rank cases (Harris-Taylor on Shimura varieties, Kottwitz on unitary similitude groups) generalise.
Modularity theorem and BSD 21.06.01. The $GL_{2} / Q$ case of Langlands reciprocity for elliptic curves. The modularity theorem of Wiles 1995 and BCDT 2001 identifies every elliptic curve $E$ over $Q$ with a weight- $2$ cusp newform $f_{E}$ , hence with a cuspidal automorphic representation $π_{f_{E}}$ of $GL_{2} (A_{Q})$ . The matching $L (E, s) = L (f_{E}, s) = L (s, π_{f_{E}})$ is exactly the Langlands $L$ -function identity. The BSD conjecture, predicting $ord_{s = 1} L (E, s) = rank E (Q)$ , is the $GL_{2}$ case of the Bloch-Kato refined conjecture on special values of motivic $L$ -functions.
Dedekind / Hecke / Artin $L$ -functions 21.03.03. The $L$ -function family that the Langlands $L$ -function generalises. Dedekind $ζ_{K} (s)$ , Hecke $L (s, χ)$ , and Artin $L (s, ρ, L / K)$ are the $GL_{1}$ cases of Langlands reciprocity (via class field theory, Proposition 17). The Brauer 1947 induction theorem realises every Artin $L$ -function as a quotient of products of Hecke $L$ -functions, which Langlands functoriality predicts is a manifestation of the broader $GL_{n}$ automorphic framework: every Artin $L$ -function should be an automorphic $L$ -function on $GL_{n}$ for $n = dim ρ$ . This is the Artin holomorphy conjecture as a corollary of functoriality.
Hecke operators and Hecke algebra 21.04.02. The operator-theoretic substrate. The Hecke algebra at a place $v$ is the spherical Hecke algebra $H (G (F_{v}), G (O_{F_{v}}))$ whose Satake transform identifies it with the representation ring of $G$ . The Hecke eigenvalues of an automorphic representation $π$ on its $K_{v}$ -fixed vectors are exactly the Satake parameters $c (π_{v})$ , and the Langlands $L$ -function is built from these parameters. The classical Hecke operator $T_{p}$ on weight- $k$ modular forms is the $GL_{2} / Q$ case at an unramified prime.
Modular forms on $SL_{2} (Z)$ 21.04.01. The $GL_{2} / Q$ level- $1$ analytic theory. Hecke eigenforms on $SL_{2} (Z)$ are the simplest examples of cuspidal automorphic representations of $GL_{2} (A_{Q})$ , with identity central character and no ramification at any finite prime. The Atkin-Lehner newform theory at level $N$ extends to ramified primes, and the resulting cuspidal automorphic representations of $GL_{2} (A_{Q})$ are matched by Eichler-Shimura with two-dimensional $ℓ$ -adic Galois representations.
Riemann zeta function $ζ (s)$ 21.03.01. The simplest $L$ -function on the bridge: $ζ (s) = L (s, 1)$ for $1$ the identity Hecke character on $Q$ , equivalently for $1$ the identity Galois representation of $G_{Q}$ . Langlands reciprocity at $n = 1, F = Q, ρ = 1, π = 1$ recovers the analytic continuation and functional equation of $ζ (s)$ as a special case. Tate's 1950 thesis gave the adelic / $GL_{1}$ derivation that is the prototype for the entire automorphic $L$ -function framework.
Dirichlet $L$ -functions $L (s, χ)$ 21.03.02. The $GL_{1} / Q$ case beyond the identity character. Dirichlet characters $χ : (Z / m)^{\times} \to C^{\times}$ extend by class field theory to Hecke characters of $Q$ , hence to cuspidal automorphic representations of $GL_{1} (A_{Q})$ . The matching Galois representations are one-dimensional characters $G_{Q} \to C^{\times}$ factoring through $Gal (Q (ζ_{m}) / Q) = (Z / m)^{\times}$ . The Dirichlet $L$ -function $L (s, χ)$ is the Langlands $L$ -function in this case.
Iwasawa $Z_{p}$ -extensions 21.07.01 and $p$ -adic $L$ -functions 21.07.02. The $p$ -adic refinement of the Langlands $L$ -function. The Mazur-Swinnerton-Dyer $p$ -adic $L$ -function of an elliptic curve, the Iwasawa main conjecture (Mazur-Wiles 1984 for $GL_{1} / Q$ , Kato + Skinner-Urban for $GL_{2} / Q$ ), and the cyclotomic-tower Selmer-group / $p$ -adic- $L$ -function comparison are all $p$ -adic refinements of the Langlands $L$ -function framework. The Bloch-Kato refined conjecture on special values generalises to motives and feeds back into Langlands reciprocity via the equivariant Tamagawa number conjecture (Burns-Flach 2001).
Elliptic curves 04.04.03. The arithmetic-geometric foundation. Elliptic curves over $Q$ supply the simplest non-abelian Galois representations of arithmetic origin ( $ρ_{E, ℓ}$ on the $ℓ$ -adic Tate module), and their automorphic counterparts (the weight- $2$ cusp newforms $f_{E}$ ) are the simplest non-abelian automorphic representations of arithmetic origin. The modularity bridge is the $GL_{2} / Q$ Langlands reciprocity for these objects, with downstream consequences (Fermat, Sato-Tate, BSD in analytic rank $\leq 1$ ).

Historical & philosophical context Master

The Langlands philosophy originates in a seventeen-page handwritten letter from Robert Langlands to André Weil, written at Yale in January 1967 ^{[Langlands 1967]}. The letter opens with "if you are willing to read it as pure speculation I would appreciate that; if not — I am sure you have a waste basket handy" and proceeds to sketch what is now called the Langlands functoriality conjecture, the construction of the $L$ -group $^{L} G = G (C) ⋊ Gal (\overline{F} / F)$ , and the reciprocity conjecture relating $n$ -dimensional Galois representations to automorphic representations of $GL_{n} (A_{F})$ . The letter was circulated in mimeographed form among a small circle of correspondents (Borel, Harish-Chandra, Tate) through the late 1960s; the first published version appeared in 1970 as Langlands's contribution Problems in the theory of automorphic forms to the Yale Lecture Notes proceedings Lectures in Modern Analysis and Applications III (LNM 170, Springer, 18-86).

The 1979 Corvallis conference Automorphic forms, representations and $L$ -functions (Proc. Symp. Pure Math. 33, two volumes) ^{[Borel 1979]} codified the basic framework. Borel's article Automorphic $L$ -functions (Vol. 2, 27-61) supplied the standard formulation of the local Langlands desiderata at $GL_{n}$ over a non-archimedean local field, the global functoriality conjecture, and the matching of $L$ - and $ε$ -factors. The conference proceedings established the vocabulary (the $L$ -group, the $A$ -packet, the Arthur conjecture for the discrete spectrum) that the subsequent forty years of work develops.

The function-field case progressed first because the arithmetic of function fields over finite fields admits the powerful machinery of moduli of bundles and shtukas. Drinfeld's 1974 Funct. Anal. Appl. 8 (the first announcement) and 1980 Soviet Math. Dokl. 21 (the published proof) ^{[Drinfeld 1974/1980]} proved the $GL_{2}$ case via the moduli of rank- $2$ shtukas, an idea that grew out of his earlier work on elliptic modules (Drinfeld modules, the function-field analogue of $CM$ elliptic curves). Vladimir Drinfeld received the Fields Medal in 1990 at the Kyoto ICM. Laurent Lafforgue 2002 Invent. Math. 147 ^{[L. Lafforgue 2002]} generalised to $GL_{n}$ by working with iterated shtukas (chains of bundles connected by an arbitrary number of meromorphic modifications); the proof is approximately 250 pages and uses the Arthur-Selberg trace formula on $GL_{n} (A_{F})$ matched with the Lefschetz-Verdier formula on the moduli of shtukas. Laurent Lafforgue received the Fields Medal in 2002 at the Beijing ICM. Vincent Lafforgue 2018 J. Amer. Math. Soc. 31 extended the automorphic-to-Galois direction to arbitrary split reductive groups over function fields, introducing the excursion operators on the cohomology of moduli of shtukas; the construction realises the Langlands parameter $σ_{π}$ associated to a cuspidal $π$ as the simultaneous eigenvalue datum of the excursion operators on the $π$ -isotypic component.

The local case at $GL_{n}$ progressed in parallel. Harris and Taylor 2001 ^{[Harris-Taylor 2001]} proved local Langlands for $GL_{n} (Q_{p})$ via the étale cohomology of unitary similitude Shimura varieties of signature $(1, n - 1)$ , using the Kottwitz construction of Galois representations attached to cuspidal automorphic representations of the unitary similitude group and a Newton-stratification analysis of the reduction modulo $p$ . Guy Henniart 2000 Invent. Math. 139 gave an independent proof via the numerical Langlands correspondence and the Bushnell-Kutzko-Harris theory of types, combined with the Cogdell-Piatetski-Shapiro converse theorem. The two proofs are logically independent and yield the same bijection (by uniqueness of the matching of $L$ - and $ε$ -factors). Laurent Fargues and Peter Scholze 2024 Astérisque 449 ^{[Fargues-Scholze 2024]} geometrised the local Langlands correspondence: the bijection between irreducible smooth representations of $G (Q_{p})$ and Langlands parameters becomes a categorical equivalence between smooth representations and coherent sheaves on the stack of Langlands parameters, mediated by the moduli of $G$ -bundles on the Fargues-Fontaine curve (a curve-like object in the world of diamonds, after Scholze 2012-2017).

The geometric Langlands programme over $C$ originates in Beilinson and Drinfeld's circa-1991 preprint Quantization of Hitchin's integrable system and Hecke eigensheaves ^{[Beilinson-Drinfeld]}, a 250-page document circulated and refined over the 1990s and 2000s. The preprint constructs Hecke eigensheaves on the moduli stack $Bun_{G}$ via opers (flat connections in the principal $sl_{2}$ -subalgebra of $g$ ), opening the route to the categorical equivalence between $D$ -modules on $Bun_{G}$ and quasi-coherent sheaves on $LocSys_{G}$ . Edward Frenkel 2007 ^{[Frenkel 2007]} supplied the bridge between geometric Langlands and conformal field theory via the chiral algebra at the critical level. Anton Kapustin and Edward Witten 2007 derived the conjecture from a topological twist of $N = 4$ super-Yang-Mills theory in four dimensions, exhibiting geometric Langlands as categorical mirror symmetry between Hitchin moduli spaces. Dennis Gaitsgory, Sam Raskin, and collaborators 2024 (arXiv:2405.03648 and companion papers) ^{[Gaitsgory-Raskin 2024]} proved the categorical geometric Langlands conjecture for $G = GL_{2}$ over $C$ , using the fundamental local equivalence of $D$ -modules on the affine Grassmannian and factorisation categories developed by Beilinson-Drinfeld. The cases $G = GL_{n}$ for $n \geq 3$ and non-type- $A$ groups remain open.

Over number fields the Langlands reciprocity conjecture remains open beyond the $n = 1$ class field theory case and specific families at $n = 2$ . The modularity theorem of Wiles 1995 + BCDT 2001 ^{[Manin-Panchishkin 2005]} is the $GL_{2} / Q$ case for elliptic curves; Khare-Wintenberger 2009 proved Serre's modularity conjecture (every odd irreducible mod- $ℓ$ Galois representation of $G_{Q}$ comes from a modular form); Newton-Thorne 2021 proved symmetric-power lifts for non-CM Hecke eigenforms (hence Sato-Tate over arbitrary totally real fields). The full $GL_{n} / Q$ reciprocity for $n \geq 3$ requires constructions on Shimura varieties of higher rank, where the Langlands-Kottwitz method has produced partial results but the full automorphic-Galois bijection remains conjectural.

Bibliography Master

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  number  = {2},
  year    = {2000},
  pages   = {439--455}
}

@article{VLafforgue2018,
  author  = {Lafforgue, Vincent},
  title   = {Chtoucas pour les groupes r{\'e}ductifs et param{\'e}trisation de {L}anglands globale},
  journal = {Journal of the American Mathematical Society},
  volume  = {31},
  number  = {3},
  year    = {2018},
  pages   = {719--891}
}

@book{FarguesScholze2024,
  author    = {Fargues, Laurent and Scholze, Peter},
  title     = {Geometrization of the local {L}anglands correspondence},
  series    = {Ast{\'e}risque},
  volume    = {449},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  year      = {2024},
  note      = {arXiv:2102.13459}
}

@unpublished{BeilinsonDrinfeld,
  author = {Beilinson, Alexander and Drinfeld, Vladimir},
  title  = {Quantization of {H}itchin's integrable system and {H}ecke eigensheaves},
  note   = {Unpublished preprint, c. 1991--2004; ~250 pages; available at http://www.math.uchicago.edu/$\sim$drinfeld/langlands/}
}

@misc{GaitsgoryRaskin2024,
  author = {Gaitsgory, Dennis and Raskin, Sam and others},
  title  = {Proof of the geometric {L}anglands conjecture for {$\mathrm{GL}_2$}},
  year   = {2024},
  note   = {arXiv:2405.03648 and companion papers}
}

@incollection{Frenkel2007,
  author    = {Frenkel, Edward},
  title     = {Lectures on the {L}anglands program and conformal field theory},
  booktitle = {Frontiers in Number Theory, Physics, and Geometry II},
  publisher = {Springer-Verlag},
  year      = {2007},
  pages     = {387--533}
}

@book{Frenkel2013,
  author    = {Frenkel, Edward},
  title     = {Love and Math: The Heart of Hidden Reality},
  publisher = {Basic Books},
  year      = {2013}
}

@article{Satake1963,
  author  = {Satake, Ichiro},
  title   = {Theory of spherical functions on reductive algebraic groups over {$\mathfrak{p}$}-adic fields},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {18},
  year    = {1963},
  pages   = {5--69}
}

@book{ArthurClozel1989,
  author    = {Arthur, James and Clozel, Laurent},
  title     = {Simple algebras, base change, and the advanced theory of the trace formula},
  series    = {Annals of Mathematics Studies},
  volume    = {120},
  publisher = {Princeton University Press},
  year      = {1989}
}

@book{Langlands1980BaseChange,
  author    = {Langlands, Robert P.},
  title     = {Base Change for {$\mathrm{GL}(2)$}},
  series    = {Annals of Mathematics Studies},
  volume    = {96},
  publisher = {Princeton University Press},
  year      = {1980}
}

@article{NewtonThorne2021,
  author  = {Newton, James and Thorne, Jack A.},
  title   = {Symmetric power functoriality for holomorphic modular forms},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {134},
  year    = {2021},
  pages   = {1--116}
}

@article{ClozelHarrisShepherdBarronTaylor2008,
  author  = {Clozel, Laurent and Harris, Michael and Shepherd-Barron, Nick and Taylor, Richard},
  title   = {Automorphy for some {$\ell$}-adic lifts of automorphic mod {$\ell$} {G}alois representations},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {108},
  year    = {2008},
  pages   = {1--181}
}

@article{KhareWintenberger2009,
  author  = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre},
  title   = {Serre's modularity conjecture {I-II}},
  journal = {Inventiones Mathematicae},
  volume  = {178},
  year    = {2009},
  pages   = {485--504, 505--586}
}

@article{KapustinWitten2007,
  author  = {Kapustin, Anton and Witten, Edward},
  title   = {Electric-magnetic duality and the geometric {L}anglands program},
  journal = {Communications in Number Theory and Physics},
  volume  = {1},
  number  = {1},
  year    = {2007},
  pages   = {1--236}
}

Prerequisites

21.03.03
21.04.03
21.05.01
21.06.01

Tier anchors

beginner: Gelbart 1984 *An elementary introduction to the Langlands program* (Bull. AMS 10 (2), 177-219) — the canonical informal introduction; Frenkel *Love and Math* (Basic Books 2013) Ch. 9-10 popular exposition
intermediate: Bump 1997 *Automorphic Forms and Representations* (Cambridge Studies in Advanced Mathematics 55) Ch. 1-3 (analytic theory at $\mathrm{GL}_n$); Manin-Panchishkin 2005 *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed.) Ch. 6 §§5-6 (number-theory survey); Cogdell 2003 *Lectures on L-functions, converse theorems, and functoriality* (Park City lecture notes)
master: Langlands 1967 *Problems in the theory of automorphic forms* (Letter to A. Weil, Yale, January 1967; published *Lectures in Modern Analysis and Applications III*, LNM 170, Springer 1970, 18-86) — originator paper containing functoriality, the $L$-group, and the reciprocity conjecture; Borel 1979 *Automorphic forms, representations and L-functions* (Proc. Symp. Pure Math. 33, Corvallis, two volumes) — formal packaging at Corvallis; Bump 1997 (Cambridge Studies 55) — modern $\mathrm{GL}_n$ treatment; Drinfeld 1974 *Funct. Anal. Appl.* 8 + 1980 *Soviet Math. Dokl.* 21 (function-field $\mathrm{GL}_2$, shtukas; Fields Medal 1990); Deligne 1973 *Formes modulaires et représentations de $\mathrm{GL}_2$* (in Antwerp II, LNM 349); L. Lafforgue 2002 *Invent. Math.* 147 (function-field $\mathrm{GL}_n$, Fields Medal 2002); Harris-Taylor 2001 *The Geometry and Cohomology of Some Simple Shimura Varieties* (Ann. Math. Studies 151) — local Langlands $\mathrm{GL}_n(\mathbb{Q}_p)$; Henniart 2000 *Invent. Math.* 139 (independent proof, numerical Langlands); V. Lafforgue 2018 *J. Amer. Math. Soc.* 31 (automorphic-to-Galois for reductive groups over function fields via excursion operators); Fargues-Scholze 2024 *Geometrization of the local Langlands correspondence* (Astérisque 449; arXiv:2102.13459); Beilinson-Drinfeld c. 1991-2004 *Quantization of Hitchin's integrable system and Hecke eigensheaves* (preprint); Gaitsgory-Raskin et al. 2024 (arXiv:2405.03648 and companion papers — geometric Langlands for $\mathrm{GL}_2$ over $\mathbb{C}$); Frenkel 2007 *Lectures on the Langlands Program and Conformal Field Theory* (in *Frontiers in Number Theory, Physics, and Geometry II*, Springer, 387-533); Manin-Panchishkin 2005 Ch. 6 of Part II

References

Langlands 1967 — Problems in the theory of automorphic forms (Letter to A. Weil) · Yale, January 1967; first published in *Lectures in Modern Analysis and Applications III*, Lecture Notes in Mathematics 170, Springer 1970, 18-86. The originator paper containing the functoriality conjecture, the $L$-group construction, and the reciprocity conjecture in essentially the form they are stated today. The letter to Weil sketched, on seventeen handwritten pages, the framework that organises the modern Langlands programme.
Borel 1979 — Automorphic forms, representations and L-functions · *Proceedings of Symposia in Pure Mathematics* 33 (Corvallis 1977), American Mathematical Society 1979, two volumes. The canonical packaging of the basic conjectures, local Langlands desiderata, and the $L$-group formalism; Borel's article *Automorphic L-functions* (Vol. 2, 27-61) is the most cited single statement of the local desiderata.
Bump 1997 — Automorphic Forms and Representations · Cambridge Studies in Advanced Mathematics 55, Cambridge University Press. The canonical modern textbook on the analytic theory at $\mathrm{GL}_n$: Ch. 1 (Hecke characters and $\mathrm{GL}_1$ Tate thesis), Ch. 2 (modular forms and automorphic representations at $\mathrm{GL}_2$), Ch. 3 (automorphic forms on $\mathrm{GL}_n$, the Hecke algebra, and the spherical representation theory at unramified places).
Manin-Panchishkin 2005 — Introduction to Modern Number Theory · Encyclopaedia of Mathematical Sciences 49, 2nd edition, Springer. Ch. 6 §§5-6 of Part II surveys the Langlands picture from the number-theory side: $\ell$-adic Galois representations, automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$, reciprocity, and functoriality.
Gelbart 1984 — An elementary introduction to the Langlands program · *Bulletin of the American Mathematical Society* 10 (2), 177-219. The standard informal exposition: the abelian case ($n = 1$, class field theory) as the prototype, the $\mathrm{GL}_2$ case via modular forms, the philosophical content of functoriality. Read as the bridge between the Beginner / Intermediate exposition here and the technical machinery.
Drinfeld 1974 / 1980 — Langlands correspondence for $\mathrm{GL}_2$ over function fields · *Funct. Anal. Appl.* 8 (1974), 274-275 + *Soviet Math. Dokl.* 21 (1980), 565-568. The construction of $\mathrm{GL}_2$ shtukas and the proof of the global Langlands correspondence for $\mathrm{GL}_2$ over function fields of curves over $\mathbb{F}_q$. Fields Medal 1990.
Deligne 1973 — Formes modulaires et représentations de $\mathrm{GL}_2$ · in *Modular Functions of One Variable II* (Antwerp 1972), Lecture Notes in Mathematics 349, Springer 1973, 55-105. The Deligne perspective on the local-global compatibility between weight-$k$ Hecke eigenforms and two-dimensional $\ell$-adic representations; the local-global package that became the template for the Langlands philosophy at $\mathrm{GL}_2$.
L. Lafforgue 2002 — Chtoucas de Drinfeld et correspondance de Langlands · *Inventiones Mathematicae* 147 (1), 1-241. The proof of the global Langlands correspondence for $\mathrm{GL}_n$ over the function field of a smooth projective curve over $\mathbb{F}_q$, via the moduli of Drinfeld shtukas. Fields Medal 2002.
Harris-Taylor 2001 — The Geometry and Cohomology of Some Simple Shimura Varieties · *Annals of Mathematics Studies* 151, Princeton University Press. The proof of the local Langlands correspondence for $\mathrm{GL}_n(\mathbb{Q}_p)$ via the étale cohomology of certain simple Shimura varieties; identifies the supercuspidal representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ with the irreducible $n$-dimensional representations of the Weil group $W_{\mathbb{Q}_p}$.
Henniart 2000 — Une preuve simple des conjectures de Langlands pour $\mathrm{GL}(n)$ sur un corps p-adique · *Inventiones Mathematicae* 139 (2), 439-455. An independent proof of local Langlands for $\mathrm{GL}_n(\mathbb{Q}_p)$ via the numerical Langlands correspondence, using the Bushnell-Kutzko-Harris theory of types together with the converse theorems of Cogdell-Piatetski-Shapiro.
V. Lafforgue 2018 — Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale · *Journal of the American Mathematical Society* 31 (3), 719-891. The automorphic-to-Galois direction of the global Langlands correspondence for arbitrary split reductive groups over function fields, via the construction of excursion operators on the cohomology of moduli of shtukas. The decomposition of the cuspidal spectrum of $G(\mathbb{A}_F)$ along Langlands parameters into $\widehat{G}(\overline{\mathbb{Q}_\ell})$ for $F$ a function field.
Fargues-Scholze 2024 — Geometrization of the local Langlands correspondence · *Astérisque* 449, Société Mathématique de France (arXiv:2102.13459 first version 2021; revised 2024). The construction of the local Langlands correspondence for $G(\mathbb{Q}_p)$ as a categorical equivalence between smooth representations and coherent sheaves on the stack of $L$-parameters, mediated by the Fargues-Fontaine curve and diamonds. Generalises and refines Harris-Taylor / Henniart for $\mathrm{GL}_n$ and extends to arbitrary reductive groups.
Beilinson-Drinfeld c. 1991-2004 — Quantization of Hitchin's integrable system and Hecke eigensheaves · Unpublished preprint of approx. 250 pages, circulated 1991-2004; available at http://www.math.uchicago.edu/~drinfeld/langlands/. The construction of Hecke eigensheaves on $\mathrm{Bun}_G$ via opers, opening the geometric Langlands programme over $\mathbb{C}$ for split reductive $G$ at the level of categorical equivalences.
Gaitsgory-Raskin et al. 2024 — Proof of the geometric Langlands conjecture for $\mathrm{GL}_2$ · Series of arXiv preprints starting with arXiv:2405.03648 (2024) and companion papers; final form to appear. The proof of the categorical geometric Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{C}$, identifying the derived category of $D$-modules on $\mathrm{Bun}_{\mathrm{GL}_2}(X)$ with the derived category of quasi-coherent sheaves on the moduli of rank-$2$ flat connections on $X$, with the Hecke eigensheaves matching the structure sheaves of points.
Frenkel 2007 — Lectures on the Langlands Program and Conformal Field Theory · in *Frontiers in Number Theory, Physics, and Geometry II*, Springer, 387-533 (arXiv:hep-th/0512172). The bridge between geometric Langlands and conformal field theory: the Beilinson-Drinfeld construction of Hecke eigensheaves via the Wakimoto modules and the chiral algebra at the critical level; the Kapustin-Witten 2007 derivation from $\mathcal{N} = 4$ super-Yang-Mills.
Frenkel 2013 — Love and Math · Basic Books, New York. Ch. 9-10 give the popular-mathematics exposition of the Langlands philosophy, suitable as the Beginner-tier anchor; explains the abelian case (class field theory), the modular-form-elliptic-curve bridge, and the geometric programme at a level requiring only undergraduate algebra.

Estimated time

beginner: 25m
intermediate: 55m
master: 105m

Intuition Beginner

Visual Beginner

Worked example Beginner

Check your understanding Beginner

Formal definition Intermediate+

Counterexamples to common slips

Key theorem with proof Intermediate+

Exercises Intermediate+

Lean formalization Intermediate+

Advanced results Master

Function-field Langlands: Drinfeld and L. Lafforgue

Local Langlands and the Fargues-Scholze geometrisation

Geometric Langlands over C

Functoriality consequences

Full proof set Master

Connections Master

Historical & philosophical context Master

Bibliography Master

Geometric Langlands over $C$