We prove that Frobenius eigenvalues of & ell;-adic cohomology and & ell;-adic intersection cohomology of rigid spaces overpadic local fields are algebraic integers and we give bounds for their padic valuations. As an application, we deduce bounds for their weights, proving conjectures of Bhatt, Hansen, and Zavyalov. We also give examples of monodromy-pure perverse sheaves on projective curves with non monodromy-pure cohomology, answering a question of Hansen and Zavyalov. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Publication:

ADVANCES IN MATHEMATICS

http://dx.doi.org/10.1016/j.aim.2026.111149

Author:

Qing Lu

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

mail addresses: qlu@bnu.edu.cn

Weizhe Zheng

Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

University of the Chinese Academy of Sciences, Beijing 100049, China

Corresponding author

E-mail addresses: wzheng@math.ac.cn