TITLE: What is the current state of the crystalline analogue of the Weil conjectures?
QUESTION [9 upvotes]: In "F-isocrystals on open varieties results and conjectures" Faltings says:

"Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil conjectures."

My questions are: 
What is the current state of the crystalline analogue of the Weil conjectures?
Which are the articles and books about these conjectures?

REPLY [8 votes]: As far as I can tell, Kedlaya solved it 12 years ago.