Logarithmic growth and Frobenius filtrations for solutions ofp-adic differential equations

Abstract

For a $\nabla$-module M over the ring $K[[x]}_0$ of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations oil the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a $\phi-\nabla$-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, $M^\star$, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork

Logarithmic growth filtrations for -modules over the bounded Robba ring

“…Chiarellotto and Tsuzuki formulated Dwork's conjecture as an analogue of the Grothendieck-Katz specialization theorem for (ϕ, ∇)modules M over K [[t]] 0 (see also Remark 10.5). Recall that a slope of Dwork's conjecture is previously known in the case where the rank is less than or equal to two [CT09,Corollary 7.3]. In § 10, we prove the following result.…”

On the rationality and continuity of logarithmic growth filtration of solutions of p-adic differential equations

“…for any µ < λ. The rationality of the log-growth of f is proved by Chiarellotto and Tsuzuki in [CT09] when n = 2, then by Nakagawa in [Nak13] when n is arbitrary under the assumption that the number of breaks of the Newton polygon of b n X n + b n−1 X n−1 + · · · + b 0 as a polynomial over the Amice ring E is equal to n. Nakagawa's assumption is too strong since it is equivalent to assuming that the number of breaks of the Frobenius filtration of M tensored with E is equal to n. Unfortunately, a naïve attempt to generalize Nakagawa's result without the assumption on the Newton polygon seems to fail. To overcome this difficulty, we carefully choose a cyclic vector e in § 5: by definition, the Newton polygon of b n X n + b n−1 X n−1 + · · · + b 0 is the boundary of the lower convex hull of some set of points associated to the b i 's.…”

“…The second part is about a comparison of the log-growth filtration and the Frobenius slope filtration under a certain technical assumption, which is based on Dwork's work on the hypergeometric differential equation. They prove the conjecture in the rank 2 case in [CT09]. They also provide a complete answer to a generic version of their conjecture in [CT11].…”

Effective 𝑝-adic cohomology for cyclic cubic threefolds

“…Example 6.3.4. We next reduce 7 −1 Φ 1 modulo 7 2 and multiply by the degree 218 polynomial ∆ 13 1 ∆ 2 ∆ 7 3 . In the resulting matrix, each entry is congruent modulo t 500 to a polynomial of degree at most 211 (see worksheet).…”

“…Example 6.3.5. Finally, we evaluate these polynomials at t = 1, then divide by (∆ 13 1 ∆ 2 ∆ 7 3 )(1) to get a 7-adic matrix which is congruent to 7 −1 Φ 1 modulo 7 2 . Let A be a lift of this matrix to Z.…”