The convergence Newton polygon of a $p$-adic differential equation V : local index theorems

Abstract: In this paper and its sequel we consider locally-free $\mathscr{O}_X$-modules together with a connection over a quasi-smooth Berkovich curve $X$. We obtain necessary and sufficient conditions for the finite dimensionality of their de Rham cohomology over local domains such as disks and annuli. We deal with both analytic and meromorphic connections and we derive index formulas relating the index to the behavior of the radii of convergence of their solutions at the boundary of the curve $X$. We introduce the notion of absolute local index. We prove that it is an intrinsic notion extending the previous notions of Robba's generalized index and that of $p$-adic exponents. This condition arises at the boundary of the curve $X$ and it is an exact condition for the finite dimensionality of the de Rham cohomology. We derive comparison results between formal, meromorphic and analytic de Rham cohomologies.