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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $P \in C(K)$\end{document}, then the K-rational geometric fundamental group of C/K is the Galois group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pi_1(C,P)=\rm{Gal}(F_{nr,P}/F)$\end{document} of the maximal unramified extension Fnr,P of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $F=\kappa(C)$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\tilde{\rho}_V:\pi_1(C,P)\rightarrow {\rm PGL}(V)$\end{document} with infinite image.¶In this paper we first derive some necessary and sufficient conditions that the projective Galois representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\tilde{\rho}_V:G_F\rightarrow {\rm PGL}(V)$\end{document} attached to a p-adic submodule \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $V\subset T_p(A)$\end{document} of the Tate-module of an abelian variety A/F factors over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pi_1(C,P)$\end{document}. 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.