The Marseilles Federative Colloquium of Mathematics is intended for all mathematicians of the Marseilles agglomeration. It takes the form of one-hour lectures taking place on certain Fridays at the FRUMAM, on the Saint-Charles campus of the University of Aix-Marseille. Moment of conviviality, it is followed by a pot allowing to pleasantly end the week.

Title: Functions L and self-propelled periods

Abstract: Self-propelled forms are generalizations of modular forms that play a central role in the Langlands program. More precisely, the global Langlands correspondence postulates the existence of still mysterious relations between these analytical objects and categories of arithmetic objects such as elliptical curves. A point of contact between these two worlds is the theory of L-functions and an advantage of this correspondence, when established, is that self-propelled L-functions are generally better understood than their arithmetic analogues (especially from the analytical point of view).

In the 1980s, Waldspurger discovered new and surprising relationships between the central value of certain self-propelled L functions and “toric periods” of self-propelled forms for GL(2). This Waldspurger formula had many applications (notably to the conjecture of Birch and Swinnerton-Dyer) and in the mid-2000s Gan-Gross-Prasad and then Ichino-Ikeda formulated higher-ranking generalizations for all classical groups (orthogonal, unitary and symplectic). These conjectures again link central values of functions L to “self-propelled periods” that is, explicit integrals of self-propelled forms. They are now essentially established for unit groups. In this talk I will propose an introduction to this circle of ideas and I will give a state of the art on the subject.