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

Ryan C. Chen

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arXiv · 2405.01429

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

Ryan C. Chen

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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.

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Ryan C. Chen. 2024-05-02. Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global. https://arxiv.org/abs/2405.01429

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