Some Cases of Kudla’s Modularity Conjecture for Unitary Shimura Varieties

Abstract: We present three papers about modular forms, number theory, and geometry. A common theme in all three papers is the interplay of symmetry of rigidity in mathematics. In Article I, we establish the existence of rational geometric designs for rational polytopes, and discuss the existence of rational spherical designs which relates to the Lehmer's conjecture on the Ramanujan tau function. In Article II, we break the barrier for the first time on expressing weight-2 modular forms of higher level whose central L-values vanish by products of at most two Eisenstein series. This work shows the power of Rankin--Selberg method and also contributes to the computation of elliptic modular forms. In Preprint III, we prove unconditionally the first few cases of Kudla's conjecture on the modularity of generating functions of special cycles on unitary Shimura varieties, for norm-Euclidean imaginary quadratic fields. Our method is based on a result of Liu and work of Bruinier--Raum, who verified the conjecture in the orthogonal case over Q.