Hostname: page-component-6bf8c574d5-mcfzb Total loading time: 0 Render date: 2025-03-07T23:24:28.387Z Has data issue: false hasContentIssue false
  • English
  • Français

A linear AFL for quaternion algebras

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  04 March 2025

Nuno Hultberg Open the ORCID record for Nuno Hultberg [Opens in a new window]  and
Andreas Mihatsch Open the ORCID record for Andreas Mihatsch [Opens in a new window]
Nuno Hultberg
Affiliation:
Institute for Theoretical Sciences, Westlake University, 600 Dunyu Rd, Xihu District, Hangzhou, Zhejiang, 310030, P. R. China e-mail: [email protected]
Andreas Mihatsch*
Affiliation:
School of Mathematical Sciences, Zhejiang University, 866 Yuhangtang Rd, Hangzhou, 310058, P. R. China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjecture and the arithmetic transfer conjecture from Li and Mihatsch (2023, Preprint, arXiv:2307.11716) in cases of Hasse invariant $1/2$.

Keywords

Arithmetic fundamental lemmaquaternion algebraRapoport–Zink spaceorbital integralintersection number

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 14G35: Modular and Shimura varieties
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 23
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Drinfeld, V. G., Coverings of $p$ -adic symmetric domains . Funkcional. Anal. i Priložen. 10(1976), no. 2, 29–40.Google Scholar
Guo, J., On a generalization of a result of Waldspurger . Canad. J. Math. 48(1996), no. 1, 105–142.CrossRefGoogle Scholar
Howard, B. and Li, Q., Intersections in Lubin–Tate space and biquadratic Fundamental Lemmas. Preprint, 2020. arXiv:2010.07365.Google Scholar
Jacquet, H. and Rallis, S., Uniqueness of linear periods . Compositio Math. 102(1996), no. 1, 65–123.Google Scholar
Kottwitz, R., Base change for unit elements of Hecke algebras . Compositio Math. 60(1986), no.2, 237–250.Google Scholar
Leslie, S., Xiao, J. and Zhang, W., Periods and heights for unitary symmetric pairs, in preparation.Google Scholar
Li, Q., An intersection number formula for CM cycles in Lubin–Tate towers. Mathematical Journal 171(2022), no.9, 1923–2011.Google Scholar
Li, Q., A computational proof of the linear Arithmetic Fundamental Lemma for $G{L}_4$ . Journal of Mathematics 74(2022), no.2, 381–427.Google Scholar
Li, Q., A computational proof for the biquadratic Linear AFL for $G{L}_4$ , in preparation.Google Scholar
Li, Q. and Mihatsch, A., On the linear AFL: The non-basic case. Preprint, 2022. arXiv:2208.10144.Google Scholar
Li, Q. and Mihatsch, A., Arithmetic transfer for inner forms of $G{L}_{2n}$ . Preprint, 2023. arXiv:2307.11716.Google Scholar
Mihatsch, A., Local constancy of intersection numbers . Algebra Number Theory 16(2022), no. 2, 505–519.CrossRefGoogle Scholar
Mihatsch, A. and Zhang, W., On the arithmetic fundamental lemma conjecture over a general $p$ -adic field. J. Eur. Math. Soc. (JEMS) 26(2024), no. 12, 4831–4901.Google Scholar
Zhang, W., On arithmetic fundamental lemmas , Invent. Math. 188(2012), no. 1, 197–252.CrossRefGoogle Scholar
Zhang, W., Periods, cycles, and L-functions: a relative trace formula approach . In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 487–521.Google Scholar
Zhang, W., Weil representation and arithmetic fundamental lemma . Ann. Math. (2) 193(2021), no. 3, 863–978.CrossRefGoogle Scholar
Zhang, Z., Maximal parahoric arithmetic transfers, resolutions and modularity. Math. J. 174(2025), no. 1, 1–129.Google Scholar