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If C/K is a curve over a finitely generated field K with a K-rational point \(P \in C(K)\), then the K-rational geometric fundamental group of C/K is the Galois group \(\pi_1(C,P)=\rm{Gal}(F_{nr,P}/F)\) of the maximal unramified 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 examples of curves for which this group is infinite, and this is implied by the existence of projective p-adic representations \(\tilde{\rho}_V:\pi_1(C,P)\rightarrow {\rm PGL}(V)\) with infinite image.¶In this paper we first derive some necessary and sufficient conditions that the projective Galois representation \(\tilde{\rho}_V:G_F\rightarrow {\rm PGL}(V)\) attached to a p-adic submodule \(V\subset T_p(A)\) 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 representations: one by properties of moduli spaces and the other by cyclic coverings.

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