Lafforgue's theorem

The Langlands conjectures for GL1(K) follow from (and are essentially equivalent to) class field theory.

Here F is a global field of some positive characteristic p, and ℓ is some prime not equal to p. Lafforgue's theorem states that there is a bijection σ between: that preserves the L-function at every place of F. The proof of Lafforgue's theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π.

A major problem is that the moduli stack is not of finite type, which means that there are formidable technical difficulties in studying its cohomology.

Lafforgue's theorem implies the Ramanujan–Petersson conjecture that if an automorphic form for GLn(F) has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1.

Lafforgue's theorem implies the conjecture of Deligne (1980, 1.2.10) that an irreducible finite-dimensional l-adic representation of the absolute Galois group with determinant character of finite order is pure of weight 0.