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I'm curious about Faltings' "A p-adic Simpson correspondence ". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?

Edit: Annette Werner's survey "Vector Bundles on Curves over C_p" seems to be related.

Edit: The first part of a "new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0". An other related article.

Edit: today new in arxiv - "Non-abelian Hodge theory for algebraic curves over characteristic p"

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This doesn't answer the question, but you might want to check out Martin Olsson's Towards non--abelian $P$--adic Hodge theory in the good reduction case.

In another (maybe too different to be useful - I'm totally ignorant here) direction there's the work of Ogus and Vologodsky Nonabelian Hodge Theory in Characteristic p.

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Thanks! After browsing both articles I could not see how they and Faltings' article relate to each other. –  Thomas Riepe Apr 13 '10 at 11:13
    
A comment from an expert: "The relationship between the three approaches is not so clear, and in fact the aim of each of the theories is quite different. For example the Ogus-Vologodsky theory is with positive characteristic coefficients, whereas Olsson and the work of Faltings is with char 0 coefficients. Also in Olsson the idea is to work with a big category of isocrystals and look at the action of Frobenius/Galois on the whole category, whereas Faltings approach is to study the category of representations of the fundamental group over the ground field." –  Thomas Riepe Apr 16 '10 at 23:34
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