Mini-course: The Langlands program
Lecturers: Jim Cogdell and Solomon Friedberg. TA: Ameya Pitale.
Description
A classical modular form for SL(2, Z) on the complex upper half-plane can be lifted to an automorphic form on GL(2, A) (where A is the ring of adeles over Q), and the functions it generates under right translation by GL(2, A) yields an automorphic representation of GL(2, A). This formalism then easily extends to automorphic forms and representations for other groups G, such as the symplectic group Sp(n). The theory of L-functions also extends to automorphic representations of G(A). These L-functions are conjectured to have meromorphic continuation and functional equation and so provide a vast generalization of the Riemann zeta function. One of Langlands' most important concepts was the introduction of the Langlands dual group of G and the related L-group, LG. The L-group mediates between the analytic theory of automorphic representations and L-functions, and the arithmetic theory of Galois representations and their L-functions. The precise nature of this connection is given in the local and global Langlands Conjectures and leads to Langlands' Principal of Functoriality.
Syllabus
- Automorphic forms;
- automorphic representations;
- automorphic L-functions;
- the L-group;
- the Langlands Conjectures;
- the Principal of Functoriality.
Prerequisites
Knowledge of classical modular forms, familiarity with the adeles and ideles.
Notes and problems
Jim Cogdell: Lectures 1, 2, and 3; and the exercises in a separate document.
Solomon Friedberg lectures and exercises..