J. Bruinier et J. Ignacio Burgos Gil - Arakelov theory on Shimura varieties (part1)
Résumé
A Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties have a very rich geometric and arithmetic structure. For instance they are defined over a number field (the reflex field), they have line bundles provided with hermitian metrics that come from a representation of a maximal compact subgroup and sometimes they have models over a localization of a ring of integers coming from a modular interpretation. Open Shimura varieties admit toroidal compactifications, but the mentioned metrized line bundles do not extend to a smoothly metrized line bundle in the compactification, but to a line bundles with logarithmic singular metric. Thus the usual Arakelov geometry can not be applied to them. In this course we will explain how to extend Arakelov theory to cover this class of singular metrics.
Important applications of this extended Arakelov theory arise in the context of the Kudla program, which predicts deep connections between the arithmetic geometry of arithmetic special cycles on integral models of orthogonal and unitary Shimura varieties and the theory of Siegel modular forms. These connections lead to (often conjectural) generalizations of results of Gross, Kohnen and Zagier on Heegner divisors on modular curves. We will give an introduction to the Kudla program and discuss some cases where the predictions have been proved.
