Overview

Building on several landmark results relating cycles to Fourier coefficients of modular forms and special values of L-functions or their derivatives, as exemplified by the work of Gross-Zagier, Hirzebruch-Zagier, Gross-Keating, Kudla-Millson and others, Steven Kudla formulated a general programme in the 1990's aimed at studying the arithmetic properties of the first derivative of incoherent Siegel-Eisenstein series at their central point and connecting these with a specific class of arithmetic cycles and special values of Rankin-Selberg L-functions. In that context Kudla also introduced the ideas of arithmetic theta lift and arithmetic Siegel-Weil formula. In the ensuing two decades, important cases of this programme have been established, spurring the study of low-dimensional Shimura varieties, such as Shimura curves, Siegel threefolds, Hilbert-Blumenthal surfaces and Picard modular surfaces.

Kudla's programme has also served as motivation for ongoing developments in the theory of arithmetic models and their compactifications and Arakelov geometry for Shimura varieties. In the perspective of the Programme these are essential pre-requisites. At this point in time, many of these key components are available thanks to tremendous efforts by an active and growing community of researchers.

The Kudla programme is now being investigated more intensively than ever, with the result that the published literature lags significantly behind the most recent developments. It is the purpose of this workshop to provide some perspective and overview of the programme and have some of the main contributors report on the new developments in this rapidly growing area.