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

2024/05/02 by Chen, Ryan C.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2405.01429

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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• Co-rank 1 Arithmetic Siegel--Weil I: Local non-Archimedean
• Co-rank 1 Arithmetic Siegel--Weil II: Local Archimedean
• On the arithmetic Siegel--Weil formula for GSpin Shimura varieties
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