<i>p</i>-adic Differential Equations

Kedlaya, Kiran S.

doi:10.1017/cbo9780511750922

Cited by 108 publications

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“…We then make a series of calculations on a one-dimensional annulus over a nonarchimedean field which itself carries one or more commuting derivations. We consider modules equipped with commuting actions of both the base derivations and the derivation in the geometric direction, and obtain results in the spirit of those in [11]. We finally extend these results to higher-dimensional spaces (which may still carry derivations on the base field) by using some careful analysis of convex functions on polyhedral subsets of R n .…”

Section: Introductionmentioning

confidence: 84%

“…As discovered originally by Christol-Dwork [3], and amplified by the first author [11], in the situation of Definition 1. …”

Section: Moving Along Frobeniusmentioning

confidence: 90%

“…The latter computes Yoneda extensions; see, e.g., [11,Lemma 5 Hypothesis 1.2.5. For the rest of this subsection, we assume that K is a field of characteristic zero complete for a nonarchimedean norm | · | and equipped with a derivation ∂ with operator norm |∂| K < ∞, and that V is a nonzero ∂-differential module over K. Definition 1.2.6.…”

Section: Differential Fields and Differential Modulesmentioning

confidence: 99%

“…Let F be the completion of Frac R under |·| 1 . By [11,Lemma 1.3.7], we can construct a basis of M ⊗ F defining a supremum norm | · | [11,Lemma 8.6.1] gives a basis of K 1/t, t defining |·| ′ M . However, we can approximate that basis arbitrarily closely with a basis of M itself, because R is dense in K 1/t, t under | · | 1 , and any element of R with an inverse in K 1/t, t also has an inverse in R. Any sufficiently good approximation will define the same supremum norm.…”

Section: Then There Exists a Basis Of M Defining A Supremum Normmentioning

confidence: 99%

“…(In fact, one also obtains these data for each point of the Berkovich1 analytic space; this is the point of view adopted in ongoing work of Baldassarri and di Vizio, starting with [1].) By pulling together techniques from the literature and adding one or two new ideas, one can make fairly definitive statements about the nature of this variation; this was done by the first author in a course given in fall 2007, whose compiled notes constitute the volume [11].…”

Section: Introductionmentioning

confidence: 99%

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Differential modules onp-adic polyannuli

Kedlaya

Xiao

2009

J. Inst. Math. Jussieu

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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.

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Section: Introductionmentioning

confidence: 84%

“…As discovered originally by Christol-Dwork [3], and amplified by the first author [11], in the situation of Definition 1. …”

Section: Moving Along Frobeniusmentioning

confidence: 90%

Section: Differential Fields and Differential Modulesmentioning

confidence: 99%

Section: Then There Exists a Basis Of M Defining A Supremum Normmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Differential modules onp-adic polyannuli

Kedlaya

Xiao

2009

J. Inst. Math. Jussieu

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Frobenius Action on a Hypergeometric Curve and an Algorithm for Computing Values of Dwork’s p-adic Hypergeometric Functions

Asakura

2021

Transcendence in Algebra, Combinatorics, Geometry and Number Theory

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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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p-adic Differential Equations

Cited by 108 publications

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Differential modules onp-adic polyannuli

Differential modules onp-adic polyannuli

Frobenius Action on a Hypergeometric Curve and an Algorithm for Computing Values of Dwork’s p-adic Hypergeometric Functions

Convergence Polygons for Connections on Nonarchimedean Curves

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