Tate modules of isocrystals and good reduction of Drinfeld modules

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.