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arXiv · 2405.01426

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

Ryan C. Chen
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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.

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Ryan C. Chen. 2024-05-02. Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean. https://arxiv.org/abs/2405.01426

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