Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911863336337408 |
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| author | Chen, Ryan C. |
| author_facet | Chen, Ryan C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global Chen, Ryan C. Number Theory Algebraic Geometry Representation Theory 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. |
| title | Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2405.01429 |