Tate modules of isocrystals and good reduction of Drinfeld modules

TL;DR

This paper constructs the $ abla$-adic Tate module of Drinfeld modules using isocrystals, establishing a criterion for good reduction based on unramified Tate modules, and extends the approach to pure motives and isocrystals.

Contribution

It introduces a new construction of the $ abla$-adic Tate module via isocrystals and proves a criterion linking good reduction to unramified Tate modules.

Findings

The $ abla$-adic Tate module is unramified if and only if the Drinfeld module has good reduction.

The construction applies to pure $A$-motives and $F$-isocrystals in $p$-adic cohomology.

The approach uses the classification of vector bundles on the Fargues-Fontaine curve.

Abstract

A Drinfeld module has a $\mathfrak{p}$-adic Tate module not only for every finite place $\mathfrak{p}$ of the coefficient ring but also for $\mathfrak{p} = \infty$. 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.