CYCLADES

Periods are integrals of algebraic differential forms on a topological cycle of an algebraic variety. They admit several incarnations. They can be numbers (if the variety is defined over a number field) or functions (if one considers a family of such varieties). They exist in real and complex as well as p-adic contexts. In this project, we propose to study and deepen the links between these different points of view, by exploiting the relations between complex and p-adic period maps, and between numerical periods and special values of functional periods. The properties of periods, notably their transcendence properties, are conjecturally controlled by algebraic cycles, thanks to the Grothendieck period conjecture and its variants. Conversely, functional as well as numerical periods are precious tools to construct algebraic cycles, by means of the complex and p-adic Hodge theories. We wish to develop the interactions between these two themes, and to extend them to the new framework of exponential periods.