Drinfel'd Module - Applications

Applications

The Langlands conjectures for functions fields state (very roughly) that there is a bijection between cuspidal automorphic representations of GL_n and certain representations of a Galois group. Drinfel'd used Drinfel'd modules to prove some special cases of the Langlands conjectures, and later proved the full Langlands conjectures for GL₂ by generalizing Drinfel'd modules to shtukas. The "hard" part of proving these conjectures is to construct Galois representations with certain properties, and Drinfel'd constructed the necessary Galois representations by finding them inside the l-adic cohomology of certain moduli spaces of rank 2 shtukas.

Drinfel'd suggested that moduli spaces of shtukas of rank r could be used in a similar way to prove the Langlands conjectures for GL_r; the formidable technical problems involved in carrying out this program were solved by Lafforgue after many years of effort.