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On the linear AFL: the non-basic case

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  23 June 2025

Qirui Li Open the ORCID record for Qirui Li [Opens in a new window]  and
Andreas Mihatsch Open the ORCID record for Andreas Mihatsch [Opens in a new window]
Qirui Li
Affiliation:
Mathematical Science Building, 77 Cheongam-Ro, Nam-Gu Pohang, Gyeongbuk 37673, Korea qiruili@postech.ac.kr
Andreas Mihatsch
Affiliation:
School of Mathematical Sciences, Zhejiang University, 866 Yuhangtang Rd, Hangzhou, 310058, P. R. China mihatsch@zju.edu.cn
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Abstract

The linear arithmetic fundamental lemma (AFL) is a conjectural identity of intersection numbers on Lubin–Tate deformation spaces and derivatives of orbital integrals. It was introduced for elliptic orbits by Li, and Howard and Li. For elliptic orbits, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the conjecture to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which relies on an analysis of the connected-étale sequence. Our results will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.

Keywords

arithmetic fundamental lemmaintersection numberLubin–Tate spaceorbital integralp-divisible group

MSC classification

Primary: 14G35: Modular and Shimura varieties 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 2 , February 2025 , pp. 385 - 425
Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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