Langlands Program

Job: organize reciprocal correspondences between Galois / motivic data and automorphic representations so that \(L\)-functions and \(\varepsilon\)-factors match — the organizing conjecture package of modern number theory.

Core

The Langlands philosophy posits a dictionary

\[ \bigl\{\text{``nice'' }n\text{-dim.\ Galois / Weil--Deligne representations}\bigr\} \;\longleftrightarrow\; \bigl\{\text{automorphic representations of }\mathrm{GL}_n\bigr\} \]

compatible with local-global principles, and generalized from \(\mathrm{GL}_n\) to reductive groups \(G\) via \(L\)-groups \({}^LG\). Special cases recover class field theory (\(n=1\)) and modularity of elliptic curves (\(n=2\) over \(\mathbb{Q}\)).

Layer Content
Global automorphic forms \(\leftrightarrow\) Galois reps; \(L\)-matching
Local Weil–Deligne \(\leftrightarrow\) irreducible smooth reps of \(G(F)\)
Geometric function fields / \(\ell\)-adic sheaves on \(\mathrm{Bun}_G\) (pointer)
Special cases CFT; modularity theorem; potential modularity

Reading map

Page Focus
Foundations global philosophy, \(L\)-functions, modularity as special case
Local Langlands \(\mathrm{GL}_n\) sketch; Weil–Deligne; Satake

Dependency order

  1. Galois theory and algebraic NT (idèles, class field).
  2. Analytic NT / \(L\)-functions and modular forms language.
  3. Representation theory (infinite-dimensional unitary / smooth reps).
  4. Local LLC for \(\mathrm{GL}_n\), then global for \(\mathrm{GL}_2\), then \(G\) reductive.

Interfaces

Area Contact
Class field theory \(n=1\): Artin reciprocity \(=\) Langlands for \(\mathrm{GL}_1\)
Analytic number theory \(L(s,\pi)\), functional equations, zeros
Representation theory \((\mathfrak{g},K)\)-modules; smooth reps of \(p\)-adic groups
Arithmetic geometry Galois reps from étale cohomology of varieties
Algebraic geometry moduli of elliptic curves; motives (pointer)

Notation (paper-standard)

  • \(G(\mathbb{A}_F)\), \(\pi=\bigotimes_v'\pi_v\), \(L(s,\pi)=\prod_v L(s,\pi_v)\)
  • \({}^LG= \widehat{G}\rtimes\mathrm{Gal}\); \(\rho:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}_n(\overline{\mathbb{Q}}_\ell)\)
  • \(\mathrm{WD}_F=\mathrm{Weil}(F)\ltimes\mathbb{C}\) (Weil–Deligne); Satake parameter \(t_{\pi_v}\in\widehat{G}_{\mathrm{ss}}/\!/\widehat{G}\)

See it

Modularity theorem as LLC special case. Every elliptic curve \(E/\mathbb{Q}\) is modular: \(L(E,s)=L(f,s)\) for a weight-\(2\) newform \(f\) of level \(N=\mathrm{cond}(E)\). Equivalently the associated Galois representation \(\rho_{E,\ell}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\mathbb{Z}_\ell)\) is automorphic. Wiles–Taylor–… prove this by deforming residual modular Galois reps (R=T). This is global Langlands for a thin but arithmetically central class of \(2\)-dimensional representations.

Check

Why does class field theory equal Langlands for \(\mathrm{GL}_1\)?

Continuous characters \(\mathrm{Gal}(F^{\mathrm{ab}}/F)\to\mathbb{C}^\times\) correspond, via the Artin map, to continuous characters of the idèle class group \(C_F=F^\times\backslash\mathbb{A}_F^\times\), i.e.\ automorphic representations of \(\mathrm{GL}_1(\mathbb{A}_F)\). \(L\)-functions match by construction of Artin \(L\) vs Hecke \(L\).

What does ‘local–global compatibility’ assert for an automorphic \(\pi\)?

If \(\pi\) is associated to a global Galois rep \(\rho\), then for almost all \(v\) (and conjecturally all \(v\), under careful normalizations) the local component \(\pi_v\) corresponds under local Langlands to the restriction/Weil–Deligne representation \(\rho|_{W_{F_v}}\).

Pitfalls

  • Automorphic form vs automorphic representation: forms are \(K\)-finite vectors in a rep of \(G(\mathbb{A})\); statements are about isomorphism classes of \(\pi\).
  • \(L\)-group is not \(G\): dual group \(\widehat{G}\) plus Galois action; essential already for non-split forms and endoscopic transfer.
  • Proven vs conjectural: LLC for \(\mathrm{GL}_n\) over \(p\)-adic fields is a theorem (Harris–Taylor, Henniart, Scholze); global for general \(n\) remains largely open.

Refs

  • Borel–Casselman, Automorphic Forms, Representations, and \(L\)-Functions (Corvallis)
  • Bump, Automorphic Forms and Representations
  • Gelbart, Automorphic Forms on Adele Groups; Arthur surveys
  • Kudla, From modular forms to automorphic representations (expository)

Next

Foundations · Local Langlands · Class field · L-functions · Algebraic NT · Representation theory · Arithmetic geometry