Special cycles on unitary Shimura varieties II: global theory

TL;DR

This paper develops a global theory linking intersection numbers of arithmetic cycles on unitary Shimura varieties to Fourier coefficients of derivatives of Eisenstein series, extending the understanding of these special cycles and their arithmetic properties.

Contribution

It introduces a global framework connecting intersection theory on unitary Shimura varieties with automorphic forms, building on previous local and formal analyses.

Findings

Proved a relation between intersection numbers and Fourier coefficients of Eisenstein series derivatives.

Established a link between arithmetic cycles and representation densities of hermitian forms.

Extended the theory of special cycles to a global setting with explicit formulas.

Abstract

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the group U(n, n). This is done by relating the arithmetic cycles to their formal counterpart from Part I via non-archimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of Part I and a counting argument.