Weil representations associated to isocrystals over function fields

TL;DR

This paper constructs Weil group representations linked to isocrystals over function fields, extending Tate module concepts to a broader class of objects and revealing new $ ext{p}$-adic Weil representations for Drinfeld modules.

Contribution

It generalizes Tate module constructions to non-pure isocrystals and extends the compatible system of Galois representations to Weil group representations over function fields.

Findings

Constructed Weil group representations for all places of the function field.

Extended Tate module functor to non-pure isocrystals.

Discovered new $ ext{p}$-adic Weil representations for Drinfeld modules.

Abstract

Every Anderson $A$-motive $M$ over a field determines a compatible system of Galois representations on its Tate modules at almost all primes of $A$. This adapts easily to $F$-isocrystals, which are rational analogues of $A$-motives for the global function field $F:=\operatorname{Quot}(A)$. We extend this compatible system by constructing a Weil group representation associated to $M$ for every place of $F$. To this end we generalize the Tate module construction to a tensor functor on $F_{\mathfrak{p}}$-isocrystals that are not necessarily pure. To prove that this yields a compatible system, we work out how that construction behaves under reduction of $M$. As an offshoot we obtain a new kind of $\wp$-adic Weil representations associated to Drinfeld modules of special characteristic $\wp$.