Spans of special cycles of codimension less than 5

TL;DR

This paper proves that the span of special cycles in certain Chow groups of orthogonal Shimura varieties is finite dimensional for codimension less than 5, using new theory of Jacobi forms with rational index.

Contribution

It introduces a novel approach by developing the theory of Jacobi forms with rational index to analyze the span of special cycles.

Findings

Finite dimensionality of the span of special cycles for r<5

Development of Jacobi forms theory with rational index

Application to orthogonal Shimura varieties

Abstract

We show that the span of special cycles in the $r$th Chow group of a Shimura variety of orthogonal type is finite dimensional, if $r < 5$. As our main tool, we develop the theory of Jacobi forms with rational index $M \in \Mat{N}(\QQ)$.