Arithmetic D-modules over Laurent series fields

19 February 2019
Daniel Caro

Let k be a characteristic $p>0$ perfect field, V be a complete DVR whose residue field is $k$ and fraction field $K$ is of characteristic $0$. We denote by $\mathcal{E}  _K$ the Amice ring with coefficients in $K$, and by $\mathcal{E} ^\dagger _K$ the bounded Robba ring with coefficients in $K$. Berthelot's classical theory of Rigid Cohomology over varieties $X/k((t))$ gives $\mathcal{E}  _K$-valued objects.  Recently, Lazda and Pal developed a refinement of rigid cohomology,
i.e. a theory of $\mathcal{E} ^\dagger _K$-valued Rigid Cohomology over varieties $X/k((t))$. Using this refinement, they proved a semistable version of the variational Tate conjecture. 

The purpose of this talk is to introduce to a theory of arithmetic D-modules with $\mathcal{E} ^\dagger _K$-valued cohomology which satisfies a formalism of Grothendieck‚Äôs six operations.