The modularity of special cycles on orthogonal Shimura varieties over totally real fields under the Beilinson-Bloch conjecture

TL;DR

This paper proves that, under the Beilinson-Bloch conjecture, the generating series of special cycles on orthogonal Shimura varieties over totally real fields form Hilbert-Siegel modular forms, generalizing Kudla's modularity conjecture.

Contribution

It establishes the modularity of generating series of special cycles in Chow groups over totally real fields assuming the Beilinson-Bloch conjecture, extending previous results.

Findings

Conditional proof of modularity of special cycle series

Generalization of Kudla's modularity conjecture to higher codimension

Connection between algebraic cycles and automorphic forms

Abstract

We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu-Yott and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson-Bloch conjecture on the injectivity of the higher Abel-Jacobi map, the generating series of special cycles of codimension $er$ in the Chow group is a Hilbert-Siegel modular form of genus $r$ and weight $1+n/2$. Our result is a generalization of \textit{Kudla's modularity conjecture}, solved by Yuan-Zhang-Zhang unconditionally when $e=1$.