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The integral Hodge polygon for $p$-divisible groups with endomorphism structure

Stéphane Bijakowski and Andrea Marrama

Vol. 6 (2024), No. 2, 189–224

Let p be a prime number, let 𝒪F be the ring of integers of a finite field extension F of p and let 𝒪K be a complete valuation ring of rank 1 and mixed characteristic (0,p). We introduce and study the integral Hodge polygon, a new invariant of p-divisible groups H over 𝒪K endowed with an action ι of 𝒪F. If F|p is unramified, this invariant recovers the classical Hodge polygon and only depends on the reduction of (H,ι) to the residue field of 𝒪K. This is not the case in general, whence the attribute “integral”. The new polygon lies between Fargues’ Harder–Narasimhan polygons of the p-power torsion parts of H and another combinatorial invariant of (H,ι) called the Pappas–Rapoport polygon. Furthermore, the integral Hodge polygon behaves continuously in families over a p-adic analytic space.

$p$-divisible groups, Newton polygon, ramified action
Mathematical Subject Classification
Primary: 14G35, 14L05
Received: 5 April 2023
Revised: 22 December 2023
Accepted: 8 January 2024
Published: 29 June 2024
Stéphane Bijakowski
Centre de Mathématiques Laurent Schwartz (CMLS), CNRS,
Ecole polytechnique
Andrea Marrama
Centre de Mathématiques Laurent Schwartz (CMLS), CNRS,
Ecole polytechnique
Dipartimento di Matematica “Tullio Levi-Civita”
Università degli Studi di Padova