RTG: Arithmetic Modularity and Arithmetic Chow Groups

Abstract: We study the analogue of Kudla-Millson's work in the arithmetic setting. The arithmetic modularity conjecture states that Kudla's generating series defines a holomorphic automorphic form valued in the Chow group of the unitary Shimura variety. We also discuss known progress towards this conjecture. We next define the arithmetic Chow group, a refinement of the usual Chow group of an algebraic variety defined by Gillet-Soule in the spirit of Arakelov theory, which will be useful for defining further refinements of Kudla's conjecture.