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arXiv · 2406.00986

Non-reductive special cycles and Twisted Arithmetic Fundamental Lemma

Zhiyu Zhang
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Abstract

We consider arithmetic analogs of the relative Langlands program and applications of new non-reductive geometry. Firstly, we introduce mirabolic special cycles, which produce special cycles on many Hodge type Rapoport-Zink spaces via pullbacks e.g. Kudla--Rapoport cycles. Secondly, we formulate arithmetic intersection problems for these cycles and formulate a method of arithmetic induction. As a main example, we formulate arithmetic twisted Gan--Gross--Prasad conjectures on unitary Shimura varieties and prove a key twisted arithmetic fundamental lemma using mirabolic special cycles, arithmetic inductions, and Weil type relative trace formulas.

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Zhiyu Zhang. 2024-06-03. Non-reductive special cycles and Twisted Arithmetic Fundamental Lemma. https://arxiv.org/abs/2406.00986

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