We give a purely local proof, in the depth 0 case, of the result by HarrisTaylor which asserts that the local Langlands correspondence for GLn is realized in the vanishing cycle cohomology of the deformation spaces of one-dimensional formal modules of height n. Our proof is given by establishing the direct geometric link with the Deligne-Lusztig theory for GLn(Fq).

It is surprising that the following question has not been addressed in the literature: what is the contragredient in terms of Langlands parameters? Thus suppose G is a connected, reductive algebraic group defined over a local field F , and G(F ) is its F -points. According to the local Langlands conjecture, associated to an admissible homomorphism φ from the WeilDeligne group of F into the L-gr...

2 Modular forms and Automorphic forms 2 2.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The Adele group of GL2 over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Hecke act...

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group G. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundl...

We complete the calculations begun in [BG09], using the p-adic local Langlands correspondence for GL2(Qp) to give a complete description of the reduction modulo p of the 2-dimensional crystalline representations of GQp of slope less than 1, when p > 2.

Drinfeld in 1974, in his seminal paper [10], revolutionized the contribution to arithmetic of the area of global function fields. He introduced a function field analog of elliptic curves over number fields. These analogs are now called Drinfeld modules. For him and for many subsequent developments in the theory of automorphic forms over function fields, their main use was in the exploration of ...

Let ∧ : GLn(C) −→ GLN (C), where N = n(n−1) 2 , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of GLn to automorphic representations of GLN , which satisfies certain canonical properties. To explain, let F be a number field, and let A be its ring of adeles. Let π = ⊗ v πv be a cuspidal (automorphic) representatio...

Let F be a totally ramified extension of Qp. We consider supersingular representations of GL2(F ) whose socles as GL2(OF )-modules are of a certain form that is expected to appear in the mod p local Langlands correspondence and establish a condition under which they are irreducible.

We give a survey, for non-experts, of the non-abelian Lubin-Tate theory, a cohomological realization of the local Langlands correspondence over p-adic fields. As its proof at the moment (mostly given by the work of Harris-Taylor [HT]) requires global techniques using certain Shimura varieties, we will treat the arithmetic geometry of these Shimura varieties, including some of the recent develop...