In  a joint program with G-T. Lan and M. Sheng we have established a p-adic  Hitchin- Simpson correspondence : given a smooth proper scheme X over the Witt ring W of a finite field F_q of characteristic p, then the category of GL_r-crystalline representations of the etale fundamental group of X over the fraction field K of W is equivalent to  the category of  periodic graded Higgs bundles on X. In a recent joint  paper with R-R.Sun and J-B.Yang we have worked out a PGL-version of the p-adic  Hitchin- Simpson correspondence  between the category of PGL_r-crystalline representations of the  etale fundamental group of X/K and the category of periodic graded Higgs bundles after twisting line bundles.  As an  application one shows that the etale fundamental group of any smooth  p-adic curve C of genus >1 carries irreducible PGL_r-crystalline representations for some  1<r<p using the existence  of stable graded Higgs bundles  on C/F_q  of rank-r and of degree 1.

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