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

Abstract

In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for $(\phi ,\nabla )$-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.

60 pages

References in corpus (2)

• Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations
• A note on logarithmic growth of solutions of $p$-adic differential equations without solvability