Green forms, special cycles and modular forms.

Abstract: Shimura varieties attached to orthogonal groups (of which modular curves are examples) are interesting objects of study for many reasons, not least of which is the fact that they possess an abundance of “special” cycles. These cycles are at the centre of a conjectural program proposed by Kudla; roughly speaking, Kudla’s conjectures suggest that upon passing to an (arithmetic) Chow group, the special cycles behave like the Fourier coefficients of automorphic forms. These conjectures also include more precise identities; for example, the arithmetic Siegel-Weil formula relates arithmetic heights of special cycles to derivatives of Eisenstein series. In this talk, I’ll describe a construction (in joint work with Luis Garcia) of Green currents for these cycles, which are an essential ingredient in the “Archimedean” part of the story; I’ll also sketch a few applications of this construction to Kudla’s conjectures.

number theoryrepresentation theory

Audience: researchers in the topic

Geometry, Number Theory and Representation Theory Seminar