Tate modules of isocrystals and good reduction of Drinfeld modules

Mornev, Max (Maxim)

doi:10.2140/ant.2021.15.909

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Tate modules of isocrystals and good reduction of Drinfeld modules

Max (Maxim) Mornev

Vol. 15 (2021), No. 4, 909–970

DOI: 10.2140/ant.2021.15.909

Abstract

A Drinfeld module has a $𝔭$ -adic Tate module not only for every finite place $𝔭$ of the coefficient ring but also for $𝔭 = \infty$ . This was discovered by J.-K. Yu in the form of a representation of the Weil group.

Following an insight of Taelman we construct the $\infty$ -adic Tate module by means of the theory of isocrystals. This applies more generally to pure $A$ -motives and to pure $F$ -isocrystals of $p$ -adic cohomology theory.

We demonstrate that a Drinfeld module has good reduction if and only if its $\infty$ -adic Tate module is unramified. The key to the proof is the theory of Hartl and Pink which gives an analytic classification of vector bundles on the Fargues–Fontaine curve in equal characteristic.

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Keywords

Drinfeld modules, isocrystals, Galois representations

Mathematical Subject Classification 2010

Primary: 11G09

Milestones

Received: 14 November 2019

Revised: 9 August 2020

Accepted: 17 October 2020

Published: 29 May 2021

Authors

Max (Maxim) Mornev
Department of Mathematics ETH Zürich Switzerland

Vol. 15, No. 4, 2021