Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917655797039104 |
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| author | Chen, Ryan C. |
| author_facet | Chen, Ryan C. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_01426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean Chen, Ryan C. Number Theory Algebraic Geometry Representation Theory 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. |
| title | Co-rank $1$ Arithmetic Siegel--Weil I: Local non-Archimedean |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2405.01426 |