Kudla's Modularity Conjecture and Formal Fourier-Jacobi Series

TL;DR

This paper proves the modularity of formal Fourier-Jacobi series of Jacobi forms under symmetry conditions, leading to a proof of Kudla's conjecture on generating series of special cycles in orthogonal Shimura varieties.

Contribution

It establishes the modularity of formal Jacobi series satisfying symmetry, confirming Kudla's conjecture for all orthogonal Shimura varieties.

Findings

Proved modularity of formal Jacobi series with symmetry

Confirmed Kudla's conjecture on special cycles

Extended results to all orthogonal Shimura varieties

Abstract

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogues of Fourier-Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.