Modularity of arithmetic special divisors for unitary Shimura varieties (with an appendix by Yujie Xu)

TL;DR

This paper constructs explicit generating series of arithmetic special divisors on unitary Shimura varieties, proving their modularity and addressing Kudla's modularity problem using advanced arithmetic and automorphic techniques.

Contribution

It introduces a new explicit construction of arithmetic generating series of special divisors and proves their modularity in the context of unitary Shimura varieties.

Findings

Generated series are modular forms valued in arithmetic Chow groups.

Provides a partial solution to Kudla's modularity conjecture.

Utilizes an arithmetic mixed Siegel-Weil formula in the proof.

Abstract

We construct explicit generating series of arithmetic extensions of Kudla's special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla's modularity problem. The main ingredient in our construction is S.~Zhang's theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula.