Differential modules onp-adic polyannuli

Abstract

We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic zero. This extends prior work in the onedimensional case of Christol, Dwork, Robba, Young, et al. Our results do not require positive residue characteristic; thus besides their relevance to the study of Swan conductors for isocrystals, they are germane to the formal classification of flat meromorphic connections on complex manifolds.

Purity for overconvergence

“…So we have the notion of the intrinsic generic radius of convergence I R(E, ρ) for ρ ∈ [λ, 1] n × [0, 1] m which is defined by Kedlaya-Xiao [14]. By [17, 2.7], (E, ∇) is overconvergent if and only if I R(E, 1) = 1, where 1 := (1, .…”

Ramification of higher local fields approaches and questions

“…We label the exceptional divisors as follows: for each coprime pair (m, n) ∈ N, there is exactly one such exceptional divisor D n/m+n such that, for the valuation v corresponding to D n/m+n , we have v(x) = n and v(y) = m. Along this divisor, we have a Swan conductor Sw(F; D n/m+n ). This proposition is a special case of the result for higher dimensional X and for an intersection point of simple normal crossing divisors, as proved in [KeX,Ke11a,Ke10b]. (The essential part of the proof is in [KeX].…”

“…We explain the main result of [KeX,Ke11a,Ke10b] on variation of Swan conductors by an example. We take…”

“…When K is of mixed characteristic under some mild condition, it is proved in [X12a] that one can "fake" the Robba ring construction above and apply the recent results [Ke10a,KeX] on p-adic equation equations to deduce the Hasse-Arf theorem.…”

Swan conductors for p-adic differential modules. II Global variation

“…It is the more careful analysis of differential modules on p-adic polyannuli in the joint paper [22] with Xiao that makes the stronger result possible; consequently, we consider all results in this section to be joint work with Xiao. This was needed to ensure the existence of such that (3.2) holds, as we were unable to prove this otherwise.…”