Delta Theory of Anderson Modules I: Differential Characters

In this article we develop the theory of differential or delta characters (the arithmetic analogue of Manin characters) of Anderson modules. Here we generalize the construction by Borger and Saha of the canonical finite rank $R$-module $\mathbf{H}(E)$ with a semilinear operator on it to any Anderson module $E$, where $R$ is the base ring which is a $\pi$-adically complete discrete valuation ring with a fixed lift of Frobenius $\phi$ on it. Then we show that $\mathbf{H}(E)$ admits a functorial map to the de Rham cohomology $\mathbf{H}_{\mathrm{dR}}^*(E)$ of $E$ which also preserves the Hodge filtration.
We also prove that the module of delta characters $\mathbf{X}_{\infty}(E)$ is finite and free as an $R\{\phi^{*}\}$-module. This leads to a strengthened version of an analogous result by Buium on the generation of differential characters of abelian varieties. We also construct a family of differential modular functions that play the analogous role of $f_{\mathrm{jet}}$ constructed by Buium for elliptic curves.
In a subsequent article, the finite rank $R$-module $\mathbf{H}(E)$ will lead to the construction of a canonical $z$-isocrystal $\mathbf{H}_{\delta}(E)$ with a Hodge-Pink filtration on it and we will show that $\mathbf{H}_{\delta}(E)$ is an admissible $z$-isocrystal.