Formal Fourier Jacobi Expansions and Special Cycles of Codimension 2

TL;DR

This paper proves that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms, leading to new insights into special cycles, modularity, and computational algorithms for these forms.

Contribution

It establishes the modularity of formal Fourier Jacobi expansions of degree 2 and applies this to the study and computation of special cycles and Fourier expansions.

Findings

Formal Fourier Jacobi expansions of degree 2 are Siegel modular forms.

The modularity of the generating function of special cycles of codimension 2 is proven.

An algorithm for computing Fourier expansions of degree 2 Siegel modular forms terminates successfully.

Abstract

We prove that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension 2, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree 2. Combining both results enables us to compute relations of special cycles in the second Chow group.