Projective p-adic representations of the K-rational geometric fundamental group

If C/K is a curve over a finitely generated field K with a K-rational point P is an element of C(K), then the K-rational geometric fundamental group o f Cl K is the Galois group, pi (1)(C, P) = Gal(F-nr,F-P/F) of the maximal u nramified extension F-nr, (P) of F = kappa (C) in which P splits completely . In view of the Fontaine-Mazur Conjecture, it is of interest to know example s of curves for which this group is infinite, and this is implied by the ex istence of projective p-adic representations (p) over tilde (V) : pi (1)(C, P) --> PGL(V) with infinite image. In this paper we first derive some necessary and sufficient conditions that the projective Galois representation <(<rho>)over tilde>(V) : G(F) --> PGL (V) attached to a p-adic submodule V subset of T-p(A) of the Tate-module of an abelian variety A/F factors over pi (1) (C, P). We then apply this (in positive characteristic) to present two constructions for such representati ons: one by properties of moduli spaces and the other by cyclic coverings.