Delta theory of Anderson Modules II: Hodge-Pink structure

Abstract

In this article, using the theory of $\delta$-geometry, we construct a canonical $z$-isocrystal $({\mathbf{H}}_{\delta }(E),{\mathfrak{f}}^{\ast })$ admitting a Hodge-Pink structure for any abelian Anderson module $E$. The Hodge-Pink structure on ${\mathbf{H}}_{\delta }(E)$ induces a natural filtration $({\mathbf{H}}_{\delta }(E)\supset {\mathbf{X}}_{\mathrm{prim}}(E)\supset \{0\})$. The elements of ${\mathbf{X}}_{\mathrm{prim}}(E)$ are represented by primitive delta characters associated to $E$. We establish a natural morphism from ${\mathbf{H}}_{\delta }(E)$ to the associated de Rham cohomology module ${\mathbf{H}}_{\mathrm{dR}}^{\ast }(E)$, which is strictly compatible with the aforementioned filtration and the classical Hodge filtration $({\mathbf{H}}_{\mathrm{dR}}^{\ast }(E)\supset \mathrm{Lie}(E{)}^{\ast }\supset \{0\})$ on ${\mathbf{H}}_{\mathrm{dR}}^{\ast }(E)$. Moreover, we show that the map induces an isomorphism between ${\mathbf{X}}_{\mathrm{prim}}(E)$ and $\mathrm{Lie}(E{)}^{\ast }$. Hence our isomorphism provides an interesting interpretation of the invariant differentials of $E$ as primitive delta characters of $E$. Furthermore, when $E$ is a Drinfeld module, we show that the constructed $z$-isocrystal ${\mathbf{H}}_{\delta }(E)$ is weakly admissible. Consequently, the positive equal characteristic analogue of the Fontaine functor associates a crystalline $z$-adic Galois representation to the $\delta$-geometric object ${\mathbf{H}}_{\delta }(E)$. In the case, when $E$ is the Carlitz module, we show that the Galois representation associated to ${\mathbf{H}}_{\delta }(E)$ is indeed the usual one coming from the Tate module.

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