Continuity and finiteness of the radius of convergence of a <i>p</i>-adic differential equation <i>via</i> potential theory

Poineau, Jérôme; Pulita, Andrea

doi:10.1515/crelle-2013-0086

Cited by 4 publications

(11 citation statements)

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“…4. This strengthens the continuity theorem of [38,39,42]. • We show that every curve admits a triangulation such that locally at any interior point, the connection decomposes into a particularly simple form; see §5.…”

Section: Introductionsupporting

confidence: 66%

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Local and global structure of connections on nonarchimedean curves

Kedlaya¹

2015

Compositio Math.

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Abstract. Consider a vector bundle with connection on a p-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author's 2010 book and on subsequent improvements by Baldassarri and Poineau-Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

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Section: Introductionsupporting

confidence: 66%

“…The proof in [42] is a bit different, making use of a combinatorial criterion for piecewise affinity. A proof in terms of p-adic potential theory is given in [39]. Another proof, essentially a streamlined version of the above arguments, is given in [7].…”

Section: 3mentioning

confidence: 99%

Local and global structure of connections on nonarchimedean curves

Kedlaya¹

2015

Compositio Math.

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“…It is worth mentioning that the result stated above when applied to smooth Berkovich analytic curves over a field of characteristic zero bears some similarity with theorems proved in [2,11]. In [2], the author, Baldassarri studies a system of differential equations defined over an analytic domain of the affine line over a non-Archimedean real-valued field of characteristic zero.…”

Section: Introductionmentioning

confidence: 67%

“…If one preserves the restrictions on the field k and considers the case of a system of differential equations defined instead over a smooth Berkovich curve, then a similar result holds true. In [11], the authors, Poineau and Pulita prove that associated to a system of differential equations over a smooth Berkovich curve, there exists a locally finite graph contained in the curve and a retraction of the curve onto it such that the radius of convergence function is constant along the fibres of the retraction. The result we prove in this paper and the results of Poineau-Pulita and Baldassari show that the behaviour of certain functions of interest are controlled by finite simplicial complexes associated to them.…”

Section: Introductionmentioning

confidence: 99%

Finite morphisms between projective varieties and Skeleta

Welliaveetil

2015

Proceedings of the London Mathematical Society

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Abstract. In this paper we study finite surjective morphisms between irreducible projective varieties in terms of the morphisms they induce between the respective analytifications. The background for the principal result is as follows. Let V ′ and V be irreducible, projective varieties over an algebraically closed, non-Archimedean real valued field k with V normal and φ : V ′ → V be a finite surjective morphism . Let x ∈ V an (L) (Remark 1.1), where L/k is an algebraically closed complete non-Archimedean real valued field extension. We associate canonically to x an L-point of the space (V × k L) an which lies on the fibre over x and denote this point x L . The embedding of V into some n-dimensional projective space defines in a natural way a family of open neighbourhoods of x L contained in (V × k L) an . Each element of this family is parametrized by an (n + 1) 2 -tuple which quantifies its size. Of particular interest to us will be those neighbourhoods O whose preimage for the morphism (φ × id L ) an decomposes into the disjoint union of homeomorphic copies of O via (φ × id L ) an . Let Gx L denote the sub collection of elements of this form. Theorem 1.7 shows that there exists a deformation retraction of the space V an onto a finite simplicial complex such that along the fibres of the retraction the size of the largest element belonging to Gx L is constant.

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“…Proof. For (a), see [134,Proposition 6.2.11] or [16] (or reduce to the case where N (x) has all slopes greater than − log R(x) and then apply [98, Theorem 5.3.6]). A similar argument implies (b) because in this case, R(x) = 1 and the zero slopes are forced to make a nonpositive contribution to the index.…”

Section: Subharmonicity and Indexmentioning

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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Continuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory

Cited by 4 publications

References 3 publications

Local and global structure of connections on nonarchimedean curves

Local and global structure of connections on nonarchimedean curves

Finite morphisms between projective varieties and Skeleta

Convergence Polygons for Connections on Nonarchimedean Curves

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