The local Langlands correspondence is a conjectural parametrization of irreducible representations of a p-adic reductive group by local Galois representations. It is a vast generalization of the local class field theory, which determines the abelianization of the absolute Galois group of a p-adic field.
It is known that the local class field theory can be proved by using a formal group over a p-adic field (the Lubin-Tate theory). Around 2000, Harris and Taylor established the local Langlands correspondence for general linear groups by generalizing the Lubin-Tate theory; they constructed the correspondence by using the etale cohomology of a p-adic analytic space which is related with deformations of a formal group.
Recently the local Langlands correspondence for several classical groups has also been obtained by automorphic representation technique, but I am still interested in how the correspondence is related to p-adic geometry. In this talk, after reviewing the current status of the local Langlands correspondence, I will explain my recent work connecting the local Langlands correspondence for GSp(4) with p-adic geometry.