English

Tate modules of isocrystals and good reduction of Drinfeld modules

Number Theory 2025-04-23 v5

Abstract

A Drinfeld module has a p\mathfrak{p}-adic Tate module not only for every finite place p\mathfrak{p} of the coefficient ring but also for p=\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 AA-motives and to pure FF-isocrystals of pp-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.

Keywords

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