Fundamental groups and good reduction criteria for curves over positive characteristic local fields

Abstract

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian (ϕ,∇)-module over the bounded Robba ring E † K , whose underlying unipotent group (after base changing to the Amice ring E K ) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, p-adic analogue of Oda's theorem that a semistable curve over a p-adic field has good reduction iff the Galois action on its ℓ-adic unipotent fundamental group is unramified.

Around -independence

“…The point provides a fibre functor from the subcategory of unipotent isocrystals, and again we may define to be the corresponding pro-unipotent group scheme over . It was also shown in [Laz16, § 5] how to put a canonical ‘ -module structure’ on , that is, a -module structure on its Hopf algebra, which simply as a -module (i.e. forgetting the Hopf algebra structure) is a direct limit of its finite-dimensional sub- -modules.…”

“…In other words, we have finite dimensionality, vanishing in the expected degrees, versions with compact support or support in a closed subscheme, Poincaré duality, Künneth formula, etc. We also constructed a category of overconvergent isocrystals on relative to , and in [Laz16, Corollary 2.2] it was proved that this category is Tannakian. The point provides a fibre functor from the subcategory of unipotent isocrystals, and again we may define to be the corresponding pro-unipotent group scheme over .…”

“…This is a Tannakian category by [Shi00, Proposition 4.14], and defines a fibre functor We let denote the corresponding pro-unipotent group. To put a Frobenius and monodromy operator on this pro-unipotent group, that is, to turn this into a non-abelian -module, we follow the approach of [Laz16]. So let denote considered with the log structure of the punctured point, and the structure morphism.…”

“…By [Laz16, Proposition 9.9] the affine group scheme over underlying , that is, the affine group scheme obtained by forgetting -module structures, is simply . Hence we may view this latter group scheme canonically as having the structure of a non-abelian -module.…”

“…For and finite one again needs to take a different approach to define the pro-unipotent fundamental group, using Tannakian duality and the theory of isocrystals to produce a pro-unipotent group scheme on which Frobenius acts. When is equicharacteristic local, one needs to appeal to the categories of overconvergent isocrystals introduced in [LP16] and studied in [Laz16] in order to define the pro-unipotent fundamental group as a ‘non-abelian -module’ over the bounded Robba ring . Again, appealing to the -adic local monodromy theorem allows us to produce a non-abelian -adic Weil–Deligne representations in this case.…”