The unramified two-dimensional Langlands correspondence

Abstract

In this paper we describe the unramified Langlands correspondence for twodimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this description is the construction of a central extension. For this (and other) central extension we prove noncommutative reciprocity laws (i.e. the splitting of the central extensions over some subgroups) for arithmetic surfaces and projective surfaces over a finite field. These reciprocity laws connect central extensions which are constructed locally and globally.

Noncommutative reciprocity laws on algebraic surfaces: the case of tame ramification

“…This central extension splits over the subgroup A * ∆ ⊂ GL n (A ∆ ) . We note that similar to remark 1 we have that the embedding of the group A * ∆ into the other place of the diagonal of the matrix group GL n (A ∆ ) does not change the isomorphism class of the central extension GL n,a (A ∆ ) (compare also with remark 3 of [8]).…”

“…Remark 2 When n = 2 and k = F q , starting from a symbol given by map ( 2) we obtain a central extension which correspond to the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov, see [4] and also [8].…”

“…We recall some calculations from [8, § 2.2]. For any n ≥ 2 we have the following formula from remark 2 of [8]:…”

“…If P = Pic Z and L ∈ Ob(H 2 (BG, P)) , then it is easy to see from formula (8) that after application of the functor ψ : Pic Z −→ Z we obtain…”

“…Proof The method for the proof of this theorem is similar to the method for the proof of theorem 1 from [8].…”