Non-reductive special cycles and Twisted Arithmetic Fundamental Lemma

TL;DR

This paper introduces mirabolic special cycles on Rapoport-Zink spaces, formulates related intersection problems, and proves a twisted arithmetic fundamental lemma using these cycles and trace formulas, advancing the arithmetic Langlands program.

Contribution

It develops a new non-reductive geometric framework with mirabolic cycles and applies it to prove a twisted arithmetic fundamental lemma in the context of unitary Shimura varieties.

Findings

Established a method of arithmetic induction for special cycles.

Proved a key twisted arithmetic fundamental lemma.

Connected non-reductive geometry with arithmetic intersection theory.

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.