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

2013/09/16 by Poineau, Jérôme, Pulita, Andrea · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1309.3940

Abstract

In this paper and its sequel we consider locally-free \mathscrO_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.

Cited by

• Comparison between formal slopes and p-adic slopes

Related

• The convergence Newton polygon of a p-adic differential equation III : global decompositions
• The convergence Newton polygon of a p-adic differential equation IV : controlling graphs
• The convergence Newton polygon of a p-adic differential equation I : Affinoid domains of the Berkovich affine line
• Convergence polygons for connections on nonarchimedean curves
• Continuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory