Abstract
Given a Laurent polynomial over a flat \(\mathbf {Z}\)-algebra, Vlasenko defines a formal group law. We identify this formal group law with a coordinate system of a formal group functor. When the “Hasse–Witt matrix” of the Laurent polynomial is invertible, Vlasenko defines a matrix by taking a certain \(p\)-adic limit. We show that this matrix is the Frobenius of the Dieudonné module of this formal group modulo \(p\).
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Notes
Vlasenko pointed out to me that her integrality proof does not require \(\Delta \) to be full dimensional as we assumed here. Thus our integrality proof is not as general as hers. Assuming her integrality theorem, the results in Sect. 3 can go through for an arbitrary \(\Delta \), except Remark 2.15, which requires the knowledge about the relation between \(F_{f}\) and algebraic geometry.
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Acknowledgements
Professor Masha Vlasenko sent me a list of suggestions and corrections, and clarified some of my misconceptions. I would like to thank her for her invaluable help.
I am also grateful to Tsung-Ju Lee, for his suggestions on Example 1.22; to Shizhang Li, for pointing out how to use \(\delta \)-rings in the proof of Theorem 2.14; to Qixiao Ma and Luochen Zhao, for discussions on formal groups; to Chenglong Yu, for answering my questions about his paper; and to Jie Zhou, for discussions on GKZ systems.
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Zhang, D. On Vlasenko’s formal group laws. manuscripta math. 170, 167–191 (2023). https://doi.org/10.1007/s00229-021-01353-z
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DOI: https://doi.org/10.1007/s00229-021-01353-z