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GROUP ACTION ON THE DEFORMATIONS OF A FORMAL GROUP OVER THE RING OF WITT VECTORS

Part of: Algebraic number theory: local and $p$-adic fields Algebraic groups

Published online by Cambridge University Press:  20 December 2017

OLEG DEMCHENKO  and
ALEXANDER GUREVICH
OLEG DEMCHENKO
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Universitetsky pr. 28, Stary Petergof, St. Petersburg, 198504, Russia email vasja@eu.spb.ru
ALEXANDER GUREVICH
Affiliation:
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel email gurevich@math.huji.ac.il
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Abstract

A recent result by the authors gives an explicit construction for a universal deformation of a formal group $\unicode[STIX]{x1D6F7}$ of finite height over a finite field $k$. This provides in particular a parametrization of the set of deformations of $\unicode[STIX]{x1D6F7}$ over the ring ${\mathcal{O}}$ of Witt vectors over $k$. Another parametrization of the same set can be obtained through the Dieudonné theory. We find an explicit relation between these parameterizations. As a consequence, we obtain an explicit expression for the action of $\text{Aut}_{k}(\unicode[STIX]{x1D6F7})$ on the set of ${\mathcal{O}}$-deformations of $\unicode[STIX]{x1D6F7}$ in the coordinate system defined by the universal deformation. This generalizes a formula of Gross and Hopkins and the authors’ result for one-dimensional formal groups.

MSC classification

Primary: 14L05: Formal groups, $p$-divisible groups 11S31: Class field theory; $p$-adic formal groups
Type
Article
Information
Nagoya Mathematical Journal , Volume 235 , September 2019 , pp. 42 - 57
Copyright
© 2017 Foundation Nagoya Mathematical Journal  

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Footnotes

The first author has been supported by the Russian Science Foundation grant no. 6.53.924.2016.

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