Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean

TL;DR

This paper establishes local non-Archimedean theorems crucial for proving the arithmetic Siegel--Weil formula in co-rank 1, linking local representation theory with geometric cycles on Shimura varieties.

Contribution

It introduces a new local limiting method at all places and proves key local theorems connecting local Whittaker functions and geometric cycle degrees.

Findings

Proved local theorems at all non-Archimedean places.

Established relations between local Whittaker functions and geometric cycles.

Linked local contributions to heights of cycles in mixed characteristic.

Abstract

This is the first 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. The crucial input is a new local limiting method at all places. In this paper, we formulate and prove the key local theorems at all non-Archimedean places. On the analytic side, the limit relates local Whittaker functions on different groups. On the geometric side at nonsplit non-Archimedean places, the limit relates degrees of $0$-cycles on Rapoport--Zink spaces and local contributions to heights of $1$-cycles in mixed characteristic.