Higher Siegel--Weil formula for unitary groups II: corank one terms

TL;DR

This paper establishes a new higher Siegel--Weil formula for corank one terms, linking derivatives of Fourier coefficients of Eisenstein series to degrees of special cycles on moduli stacks, extending previous results to all r.

Contribution

It proves the higher Siegel--Weil formula for corank one terms for all r, regardless of Eisenstein series vanishing order, extending prior work on non-singular cases.

Findings

Proves the higher Siegel--Weil formula for corank one Fourier coefficients.

Relates derivatives of Eisenstein series to degrees of special cycles.

Extends the formula to all r, including cases with vanishing Eisenstein series.

Abstract

We prove the higher Siegel--Weil formula for \emph{corank one} terms, relating (1) the $r^{\rm th}$ central derivatives of corank one Fourier coefficients of Siegel--Eisenstein series, and (2) the degrees of special cycles of virtual dimension 0 on the moduli stack of Hermitian shtukas with $r$ legs. Notably, the formula holds for all $r$, regardless of the order of vanishing of the Eisenstein series. This extends earlier work of Feng--Yun--Zhang, who proved the higher Siegel--Weil formula for the non-singular (corank zero) terms.