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Abstract
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Let
be a prime
number, let
be the ring of integers of a finite field extension
of
and let
be a complete valuation
ring of rank
and
mixed characteristic
.
We introduce and study the
integral Hodge polygon, a new invariant of
-divisible
groups over
endowed with
an action
of
. If
is unramified,
this invariant recovers the classical Hodge polygon and only depends on the reduction of
to the residue
field of
.
This is not the case in general, whence the attribute “integral”. The new
polygon lies between Fargues’ Harder–Narasimhan polygons of the
-power torsion parts
of
and another
combinatorial invariant of
called the Pappas–Rapoport polygon. Furthermore, the
integral Hodge polygon behaves continuously in families over a
-adic
analytic space.
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Keywords
$p$-divisible groups, Newton polygon, ramified action
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Mathematical Subject Classification
Primary: 14G35, 14L05
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Milestones
Received: 5 April 2023
Revised: 22 December 2023
Accepted: 8 January 2024
Published: 29 June 2024
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© 2024 MSP (Mathematical Sciences
Publishers). |
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