Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global

TL;DR

This paper completes the proof of an arithmetic Siegel--Weil formula for Kudla--Rapoport cycles on unitary Shimura varieties, linking degrees of special cycles to derivatives of Eisenstein series, with explicit local and global formulas.

Contribution

It provides precise normalizations for Siegel Eisenstein series, local special value formulas, and completes the arithmetic Siegel--Weil proof by integrating local results.

Findings

Explicit formulas for local Siegel--Weil values

Normalization of $U(m,m)$ Eisenstein series

Degrees of special cycles linked to Eisenstein series derivatives

Abstract

This is the fourth in a sequence of four papers, where we prove the arithmetic Siegel--Weil formula in co-rank $1$ for Kudla--Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel--Weil formula implies that degrees of Kudla--Rapoport arithmetic special $1$-cycles are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients. In this paper, we pin down precise normalizations for some $U(m,m)$ Siegel Eisenstein series, give local Siegel--Weil special value formulas with explicit constants, and record a geometric Siegel--Weil result for degrees of complex $0$-cycles. Using this, we complete the proof of our arithmetic Siegel--Weil results by patching together the local main theorems from our companion papers.