Logarithmic growth filtrations for $(\varphi,\nabla)$-modules over the bounded Robba ring

TL;DR

This paper proves a conjecture relating the log-growth filtration and Frobenius slope filtration for $(, abla)$-modules over the bounded Robba ring, with applications to Newton polygons and Dwork's conjecture.

Contribution

It establishes a comparison theorem between log-growth and Frobenius slope filtrations, introducing a new criterion for pure bounded quotient and applying it to Newton polygon conjectures.

Findings

Proved the conjecture on the comparison of filtrations.

Developed a new criterion for pure of bounded quotient.

Extended results to log-growth Newton polygons and Frobenius Newton polygons.

Abstract

In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for $(\varphi,\nabla)$-modules over the bounded Robba ring. The main result is a proof of a conjecture proposed by B. Chiarellotto and N. Tsuzuki on a comparison between the log-growth filtration and Frobenius slope filtration. One of the ingredients of the proof is a new criterion for pure of bounded quotient, which is a notion introduced by Chiarellotto and Tsuzuki to formulate their conjecture. We also give several applications to log-growth Newton polygons, including a conjecture of Dwork on the semicontinuity, and an analogue of a theorem due to V. Drinfeld and K. Kedlaya on Frobenius Newton polygons for indecomposable convergent $F$-isocrystals.