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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
Se basa en
Requisitos previos obligatorios, hasta el fondo: 56
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- 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.
Descubrimientos opcionales
- 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
Itinerarios de aprendizaje
Lecturas complementarias
- Joseph Bernstein and Stephen Gelbart (editors), An Introduction to the Langlands Program (2003)
- Edward Frenkel, Love and Math (2013)