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.

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Last updated: 09 Sep 2024