Maximal parahoric arithmetic transfers, resolutions and modularity
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913345078034432 |
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| author | Zhang, Zhiyu |
| author_facet | Zhang, Zhiyu |
| contents | For any unramified quadratic extension of $p$-adic local fields $F/F_0$ $(p>2)$, we formulate several arithmetic transfer conjectures at any maximal parahoric level, in the context of Zhang's relative trace formula approach to the arithmetic Gan--Gross--Prasad conjecture. The formulation involves a way to resolve the singularity of relevant moduli spaces via natural stratifications and modify derived fixed points. By a local-global method and double induction, we prove these conjectures for $F_0$ unramified over $\mathbb Q_p$, including the arithmetic fundamental lemma for $p>2$. Moreover, we prove new modularity results for arithmetic theta series at parahoric levels via a method of modification over $\mathbb F_q$ and $\mathbb C$. Along the way, we study the complex and mod $p$ geometry of Shimura varieties and special cycles. We introduce the relative Cayley map and also establish Jacquet--Rallis transfers at maximal parahoric levels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11994 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Maximal parahoric arithmetic transfers, resolutions and modularity Zhang, Zhiyu Number Theory Algebraic Geometry Representation Theory 11F67, 11G40, 14G35 For any unramified quadratic extension of $p$-adic local fields $F/F_0$ $(p>2)$, we formulate several arithmetic transfer conjectures at any maximal parahoric level, in the context of Zhang's relative trace formula approach to the arithmetic Gan--Gross--Prasad conjecture. The formulation involves a way to resolve the singularity of relevant moduli spaces via natural stratifications and modify derived fixed points. By a local-global method and double induction, we prove these conjectures for $F_0$ unramified over $\mathbb Q_p$, including the arithmetic fundamental lemma for $p>2$. Moreover, we prove new modularity results for arithmetic theta series at parahoric levels via a method of modification over $\mathbb F_q$ and $\mathbb C$. Along the way, we study the complex and mod $p$ geometry of Shimura varieties and special cycles. We introduce the relative Cayley map and also establish Jacquet--Rallis transfers at maximal parahoric levels. |
| title | Maximal parahoric arithmetic transfers, resolutions and modularity |
| topic | Number Theory Algebraic Geometry Representation Theory 11F67, 11G40, 14G35 |
| url | https://arxiv.org/abs/2112.11994 |