A Note on Logarithmic Growth Newton Polygons of p-Adic Differential Equations

Ohkubo, Shun

doi:10.1093/imrn/rnu017

Cited by 3 publications

(4 citation statements)

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“…0 K{t} are trivial as ∇-modules over K{t} (see [Ohk15]). To the best of the author's knowledge, at this point, there is no alternative proof of Dwork's conjecture or h n (M ) = h n (M E ) exploiting Andre's theorem in an essential way.…”

Section: Then Condition (I) Implies Condition (Ii) Moreover If K Is Perfect Then Condition (Ii) Implies Condition (I)mentioning

confidence: 99%

“…Dwork's conjecture in the introduction is an analogue of the above conjecture for -modules over , which is equivalent to saying, under André's notation, that lies on or above with the same left endpoints, i.e. ; note that the equality does not always hold for -modules over such that are trivial as -modules over (see [Ohk15]). To the best of the author's knowledge, at this point, there is no alternative proof of Dwork's conjecture or exploiting Andre's theorem in an essential way.…”

Section: Proof Of Theorem 04mentioning

confidence: 99%

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Logarithmic growth filtrations for -modules over the bounded Robba ring

Ohkubo¹

2021

Compositio Math.

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In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$ -adic differential equations $Dx=0$ on the $p$ -adic open unit disc $|t|<1$ , which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$ . Then, Dwork calculated the log-growth filtration for $p$ -adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$ -modules over $K[\![t]\!]_0$ , which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$ -modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

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Section: Then Condition (I) Implies Condition (Ii) Moreover If K Is Perfect Then Condition (Ii) Implies Condition (I)mentioning

confidence: 99%

Section: Proof Of Theorem 04mentioning

confidence: 99%

Logarithmic growth filtrations for -modules over the bounded Robba ring

Ohkubo¹

2021

Compositio Math.

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“…André defines his log-growth Newton polygons N P(M ) and N P(M E ) by fixing the right endpoints at (n, 0) ([And08, 3.3]), then proves Dwork's conjecture without the coincidence of the left endpoints ([And08, Theorem 4.1.1]). In [Ohk15], the author constructs M of rank two such that the left endpoints of N P(M ) and N P(M E ) do not coincide.…”

Section: Reverse Filtrationmentioning

confidence: 99%

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

Ohkubo¹

2018

Preprint

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In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for (ϕ, ∇)-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.

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“…• Transfer principles and effective convergence bounds: [34], [38], [64], [66], [93,Chapters 9,13,18]. • Logarithmic growth of p-adic solutions: [3], [29], [30], [33], [59], [60], [93,Chapter 18], [111], [125], [127]. • Stokes phenomena for complex connections: [105], [106], [108], [121], [149], [150],…”

Section: Appendix B Thematic Bibliographymentioning

confidence: 99%

Convergence Polygons for Connections on Nonarchimedean Curves

Kedlaya

2016

Simons Symposia

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In classical analysis, one builds the catalog of special functions by repeatedly adjoining solutions of differential equations whose coefficients are previously known functions. Consequently, the properties of special functions depend crucially on the basic properties of ordinary differential equations. This naturally led to the study of formal differential equations, as in the seminal work of Turrittin [167]; this may be viewed retroactively as a theory of differential equations over a trivially valued field. After the introduction of p-adic analysis in the early 20th century, there began to be corresponding interest in solutions of p-adic differential equations; however, aside from some isolated instances (e.g., the proof of the Nagell-Lutz theorem; see Theorem 3.4), a unified theory of p-adic ordinary differential equations did not emerge until the pioneering work of Dwork on the relationship between p-adic special functions and the zeta functions of algebraic varieties over finite fields (e.g., see [57,58]). At that point, serious attention began to be devoted to a serious discrepancy between the p-adic and complex-analytic theories: on an open p-adic disc, a nonsingular differential equation can have a formal solution which does not converge in the entire disc (e.g., the exponential series). One is thus led to quantify the convergence of power series solutions of differential equations involving rational functions over a nonarchimedean field; this was originally done by Dwork in terms of the generic radius of convergence [59]. This and more refined invariants were studied by numerous authors during the half-century following Dwork's initial work, as documented in the author's book [93].At around the time that [93] was published, a new perspective was introduced by Baldassarri [13] (and partly anticipated in prior unpublished work of Baldassarri and Di Vizio [15]) which makes full use of Berkovich's theory of nonarchimedean analytic spaces. Given a differential equation as above, or more generally a connection on a curve over a nonarchimedean field, one can define an invariant called the convergence polygon; this is a function from the underlying Berkovich topological space of the curve into a space of Newton polygons, which measures the convergence of formal horizontal sections and is well-behaved with respect to both the topology and the piecewise linear structure on the Berkovich space. One can translate much of the prior theory of p-adic differential equations into (deceptively) simple statements about the behavior of the convergence polygon; this process was carried out in a series of papers by Poineau and Pulita [141,133,135,136], as supplemented by work of this author [98] and upcoming joint work with Baldassarri [16].

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A Note on Logarithmic Growth Newton Polygons of p-Adic Differential Equations

Cited by 3 publications

References 5 publications

Logarithmic growth filtrations for -modules over the bounded Robba ring

Logarithmic growth filtrations for -modules over the bounded Robba ring

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

Convergence Polygons for Connections on Nonarchimedean Curves

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