Tate modules of isocrystals and good reduction of Drinfeld modules
M. Mornev
TL;DR
This work builds a unified, isocrystal-based framework for studying Drinfeld modules and their reductions by introducing the ∞-adic Tate module and establishing a parallel with finite-place Tate theories. Central to the approach is the Tate-module construction for pure isocrystals, the Hartl–Pink classification of σ-bundles on the Fargues–Fontaine curve, and the base-change and reduction analyses that connect motives to Weil-group representations, including J.-K. Yu’s ∞-adic representation. The main contributions are a new reduction criterion at the infinite place, a full-faithfulness result for the ∞-adic Tate module functor under density hypotheses, and a robust bridge between Drinfeld-module motives, pure A-motives, and τ-sheaves via isocrystals. The findings provide a powerful, characteristic-free method to study good reduction and its relation to ramification, with potential extensions to A-motives and τ-sheaves that could impact p-adic and function-field arithmetic geometry.
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.