Deformations for Function Fields

AbstractWe consider a question of describing the one-dimensionalP-adic representations that lift a given representation over a finite field of the absolute Galois group of a function field. In this case, the characterization of abelianp-power extensions of fields of characteristicpcan be extended to abelian pro-p-extensions, and refined to allow only restricted ramification at the places ofK, and can be a tool for analyzing one-dimensionP-adic representations. We then turn to the problem of classifying those representations which can be realized as the action of the Galois group on the division points of a rank one Drinfeld module, discussing both results and a conjecture about the form of the representations that arise in this manner