Tate modules of isocrystals and good reduction of Drinfeld modules
Number Theory
2025-04-23 v5
Abstract
A Drinfeld module has a -adic Tate module not only for every finite place of the coefficient ring but also for . 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 -adic Tate module by means of the theory of isocrystals. This applies more generally to pure -motives and to pure -isocrystals of -adic cohomology theory. We demonstrate that a Drinfeld module has good reduction if and only if its -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.
Cite
@article{arxiv.1910.11057,
title = {Tate modules of isocrystals and good reduction of Drinfeld modules},
author = {M. Mornev},
journal= {arXiv preprint arXiv:1910.11057},
year = {2025}
}
Comments
52 pages; final version as submitted to Algebra & Number Theory