Abstracts Mathematics

Geometric pullback formula for unitary Shimura varieties

by Nguyen Chi Dung

In this thesis we study Kudla’s special cycles of codimension 𝑟 on a unitary Shimura variety Sh(U(n − 1,1)) together with an embedding of a Shimura subvariety Sh(U(m − 1,1)). We prove that when 𝑟 = 𝑛 − 𝑚, for certain cuspidal automorphic representations 𝜋 of the quasi-split unitary group U(𝑟,𝑟) and certain cusp forms ⨍ ∈ 𝜋, the geometric volume of the pullback of the arithmetic theta lift of ⨍ equals the special value of the standard 𝐿-function of 𝜋 at 𝑠 = (𝑚 − 𝑟 + 1)/2. As ingredients of the proof, we also give an exposition of Kudla’s geometric Siegel-Weil formula and Yuan-Zhang-Zhang’s pullback formula in the setting of unitary Shimura varieties, as well as Qin’s integral representation result for 𝐿-functions of quasi-split unitary groups.