Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms

Abstract

The purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms.

The Noether-Lefschetz conjecture and generalizations

“…Note that there is perfect pairing from Poincaré duality (8.6) ⟨, ⟩ ∶ H rq (X K , C) × H (p−r)q c (X K , C) → C, coming from Poincaré duality, where H * c (−) denotes the de Rham cohomology group with compact support. We now recall the main result of [34] (see also [20]): Proposition 8.6. [34, Theorem 2] As a function of g ′ ∈ Mp 2r (A), the cohomology class [θ rq (g ′ , ϕ)] is a holomorphic Siegel modular form of weight m 2 for some congruence subgroup with coefficients inH rq (X K , C).…”

The Secondary Spherical Functions and Automorphic Green Currents for Certain Symmetric Pairs

“…Though the relation is somewhat indirect from technical point of view, there are some results strongly related from the geometric point of view, which are worked in the context of Weil representation and the theta correspondence ( [23], [24], [14], [15], [4], [16], [17]). To explain the connection in detail takes some space; therefore, we leave that to the readers.…”

A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

“…The projection onto V 2m is the map denoted P in Section 5 [Be], and the CM cycles defined there are the images of the fundamental classes of our m-dimensional CM under P . We note that related modularity results, with the local system V 2m (but using other means like the Shintani and Kudla-Millson lifts), are presented, for example, in [FM1] and [FM2]. However, our result applies for the full local system Sym m V 2 .…”