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Thursday, April 17, 2014

**Abstract:** Give a smooth projective variety X over the function field K of a complex curve C, its algebraic de Rham cohomology forms a finite dimension K vector space. This vector space can be canonically realized as the generic fiber of a coherent sheaf on C, in two ways: one Hodge-theoretic and one geometric. I will explain an arithmetic version of this, where K is a number field. Here, p-adic Hodge theory takes the place of usual Hodge theory. This is joint work with Bhargav Bhatt.