The theory of $\delta$
-geometry was developed by A. Buium based on the analogy with differential algebra where the analogue of a differential operator is played by a $\pi$
-derivation $\delta$
. A $\pi$
-derivation $\delta$
arises from the $\pi$
-typical Witt vectors and naturally associates with a lift of Frobenius $\phi$
. In this talk, we will discuss the theory of $\delta$
-geometry for Anderson modules. Anderson modules are higher dimensional generalizations of Drinfeld modules.
As an application of the above, we will construct a canonical $z$
-isocrystal $\mathbb{H}(E)$
with a Hodge- Pink structure associated to an Anderson module $E$
defined over a $\pi$
-adically complete ring $R$
with a fixed $\pi$
-derivation $\delta$
on it. Depending on a $\delta$
-modular parameter, we show that the $z$
-isocrystal $\mathbb{H}(E)$
is weakly admissible in the case of Drinfeld modules of rank $2$
. Hence, by the analogue of Fontaine’s mysterious functor in the positive characteristic case (as constructed by Hartl), one associates a Galois representation to such an $\mathbb{H}(E)$
. The relation of our construction with the usual Galois representation arising from the Tate module of $E$
is currently not clear. This is a joint work with Sudip Pandit.