DEFORMATIONS FOR FUNCTION-FIELDS

We consider a question of describing the one-dimensional P-adic repres entations that lift a given representation over a finite field of the absolute Galois group of a function field. In this case, the character ization of abelian p-power extensions of fields of characteristic p ca n be extended to abelian pro-p-extensions, and refined to allow only r estricted ramification at the places of K, and can be a tool for analy zing one-dimension P-adic representations. We then turn to the problem of classifying those representations which call be realized as the ac tion of the Galois group on the division points of a rank one Drinfeld module, discussing both results and a conjecture about the form of th e representations that arise in this manner. (C) 1998 Academic Press.