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Splitting of almost ordinary abelian surfaces in families and the 𝑆-integrality conjectures

  • Ruofan Jiang EMAIL logo
Published/Copyright: May 6, 2025

Abstract

Let 𝐮 be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic 𝑝. Suppose Δ is an infinite set of positive integers such that ( m p ) = 1 for all m ∈ Δ . If 𝐮 does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of 𝐮 has endomorphism ring containing Z ⁱ [ x ] / ( x 2 − m ) for some m ∈ Δ . This implies that there are infinitely many places modulo which 𝐮 is not simple, generalizing the main result of [D. Maulik, A. N. Shankar and Y. Tang, Reductions of abelian surfaces over global function fields, Compos. Math. 158 (2022), 4, 893–950] to the non-ordinary case. As another application, we also generalize the 𝑆-integrality theorem for elliptic curves over number fields, as proved in [M. Baker, S.-I. Ih and R. Rumely, A finiteness property of torsion points, Algebra Number Theory 2 (2008), 2, 217–248], to the setting of abelian surfaces over global function fields.

Award Identifier / Grant number: DMS-2100436

Funding statement: The author is partially supported by the NSF grant DMS-2100436.

Acknowledgements

The author thanks Ananth Shankar for suggesting this problem and his help, thanks Asvin G, Qiao He, Jiaqi Hou, and Salim Tayou for valuable discussions, and thanks Jordan Ellenberg for pointing out some imprecision in the introduction. The author also thanks Keerthi Madapusi Pera for answering a question on toroidal compactifications and Martin Olsson for answering a question on log geometry.

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Received: 2024-05-10
Revised: 2025-04-04
Published Online: 2025-05-06
Published in Print: 2025-08-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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