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

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.