On geometric Langlands theory and stacks

Abstract

R.Langlands conjectured the existence of a bridge between two parts of number theory. This correspondence, called 'Langlands conjecture' was proved by L. Lafforgue who obtained a Fields medal for his work. G. Laumon gave a geometric translation of a part of the theorem, called 'geometric Langlands correspondence'. The latter was proved by E. Frenkel, D. Gaitsgory and K. Vilonen for the case of finite fields and the group GL-n- of invertible matrices. A statement and a proof of the complex case should be obtained by mimicking the proof for finite fields. However, it does not appear anywhere in the literature. In this thesis, we chose a different approach. Our first aim was to make a correct statement and proof for simple cases. More precisely, we deal with cases where the two difficult objects "stacks" and "perverse sheaves", which appear in the original statement, are not involved. We also state and prove the correspondence in a new context, namely the case of singular curves. A large part of the thesis is devoted to making the objects appearing in the geometric statement, such as stacks, mor explicit. These objects are categories with glueing properties. They are now widely used but references are rare and often difficult to read. We thus give numerous examples of stacks and treat in detail explicit ones. We chose for instance to work on the case of trangles in order to get a better understanding of the interest of stacks. Another important example is the stack Bun-n-x of vector bundles or rank -n- over curve x. This is an interesting stack which plays an important role in the geometric Langlands correspondance