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Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves

Authors: Kiran S. Kedlaya and Michael Temkin
Journal: Proc. Amer. Math. Soc. 146 (2018), 489-495
MSC (2010): Primary 12J25
Published electronically: August 7, 2017
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Abstract: We show that for $ k$ a perfect field of characteristic $ p$, there exist endomorphisms of the completed algebraic closure of $ k((t))$ which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for $ p$ a prime and $ \mathbb{C}_p$ a completed algebraic closure of $ \mathbb{Q}_p$, there exist closed points of the Fargues-Fontaine curve associated to $ \mathbb{C}_p$ whose residue fields are not (even abstractly) isomorphic to $ \mathbb{C}_p$ as topological fields.

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Additional Information

Kiran S. Kedlaya
Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California, 92093

Michael Temkin
Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California, 92093

Received by editor(s): July 3, 2016
Received by editor(s) in revised form: March 19, 2017
Published electronically: August 7, 2017
Additional Notes: The first author received additional support from NSF grants DMS-1101343 and DMS-1501214 and from UC San Diego (Stefan E. Warschawski Professorship)
The second author was supported by the Israel Science Foundation (grant No. 1159/15)
Some of this work was carried out during the MSRI fall 2014 semester program “New geometric methods in number theory and automorphic forms” supported by NSF grant DMS-0932078
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2017 Kiran S. Kedlaya and Michael Temkin

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