Automorphic forms

Automorphic forms were discovered at the turn of the century XX^th century by Henri Poincaré as a non-commutative generalization of periodic functions. Modular forms, which are special cases of automorphic forms, were studied intensively in Germany before and after Poincaré's discovery, in connection with arithmetic and, in particular, with the theory of class bodies. The 1960s saw a proliferation of research into automorphic forms in many directions, including Israel Gelfand's school, which linked them to infinite-dimensional representations of Lie groups, and the Shimura-Taniyama-Weil conjecture, which linked modular forms to the arithmetic of elliptic curves. It was in the late 1960s that Robert Langlands' program brought the many facets of automorphic forms into a deep and fascinating perspective, linking the arithmetic of algebraic varieties to automorphic representations via the L-functions, which are generalizations of Riemann's zeta function, in the form of a vast non-Abelian generalization of class field theory (reciprocity conjecture) and a major organizing principle in representation theory (functoriality conjecture). The Langlands program underwent another revolution in the 1990s with the emergence of the geometric Langlands program initiated by Vladimir Drinfeld and Gérard Laumon, which links the study of automorphic forms to the geometry of fibered moduli spaces.

The lectures taught in this Chair will address various automorphic topics, including representation theory and harmonic analysis on real or p-adic reductive groups, trace formulas and orbital integrals, geometry of moduli spaces, Shimura varieties...