An introduction to the Langlands Conjectures

Abstract

In this thesis, we take a look into a generalization of Local Class Field Theory (LCFT), called the Local Langlands Conjecture, which concerns about identifying special representations of the Weil group, a subgroup of the absolute Galois group of a field, with some representations of the general linear group with coefficients in that field. We study deeply the $n=1$ case, which corresponds to LCFT, and then we construct explicit elements of both sides of the bijection for the case $n=2$. Finally, we state the difficulties to formulate the analogous conjecture for global fields, the Global Langlands Conjecture.