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The Langlands programme: an orientation

Optional research direction: a map of one of the great unifying visions of modern mathematics, from quadratic reciprocity through class field theory to automorphic forms, Galois representations, functoriality, modularity and geometric Langlands — with a careful line between what is proved and what is conjectured.

math.number.langlands-programme

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Builds on

Algebraic number theory Analytic number theory Representation theory of finite groups

Required prerequisites, all the way down: 56

You will learn to

Trace reciprocity from quadratic reciprocity through class field theory, and state it for abelian extensions of ℚ (Kronecker–Weber).
Describe modular forms, automorphic representations and Galois representations, and what it means for them to correspond.
Explain functoriality and the modularity theorem as instances of the programme, separating proved cases from conjectures.

Definition

A modular form of weight k for SL(2, ℤ) is a holomorphic function f on the upper half-plane with f((az + b)/(cz + d)) = (cz + d)ᵏ f(z) for all matrices in SL(2, ℤ), holomorphic at infinity.
A Galois representation is a continuous homomorphism from the absolute Galois group of a number field to GLₙ of a topological field.
Class field theory describes the abelian extensions of a number field in terms of data intrinsic to the field.

Theorem

Quadratic reciprocity (Gauss): for distinct odd primes p and q, (p/q)(q/p) = (−1)^{((p−1)/2)((q−1)/2)}.
Kronecker–Weber: every finite abelian extension of ℚ is contained in a cyclotomic field.
Langlands reciprocity for GL(1) over number fields is equivalent to class field theory; for GL(2) over ℚ, important cases include the modularity theorem.
Functoriality in general is an open conjecture.

Common misconceptions

Thinking the Langlands programme is a single conjecture; it is a web of conjectures and theorems, many still open.
Believing geometric Langlands and arithmetic Langlands are the same statement; they are analogous but distinct, and progress on one does not settle the other.

Optional discoveries

Categorical representation theory The geometric Langlands programme is formulated as an equivalence of categories of sheaves; its categorical form was proved in 2024 by Gaitsgory, Raskin and collaborators (as announced; the proof spans several long papers).
Lie groups and Lie algebras The Langlands dual group, obtained by exchanging roots and coroots, governs functoriality.
← Lie groups and Lie algebras
← Analytic number theory
← Arithmetic geometry and elliptic curves

Learning pathways

Algebra and number theory