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Effective convergence bounds for Frobenius structures on connections. (English) Zbl 1276.14032

An improvement of the previous bound in question of the paper by K. S. Kedlaya [“Effective \(p\)-adic cohomology for cyclic threefolds”, in: Computational Algebraic and Analytic Geometry of Low dimensional Varieties. Amer. Math. Soc. Vol. 572 (2012)] is given.
Let \(p\) be a prime, \(n\) a positive integer, and \(\mathbb{F}_q\) the finite field with \(q = p^n\) elements. Let \({\mathbb{Q}}_q\) denote the unique unramified extension of degree \(n\) of the field of \(p\)-adic numbers. Let \(U\) be an open dense subscheme of the projective space \({\mathbb{P}}^{1}_{{\mathbb{Q}}_q}\) with nonempty complement \(Z\). Let \(V\) be the rigid analytic subspace of \({\mathbb{P}}^{1}_{{\mathbb{Q}}_q}\) which is the complement of the union of the open disks of radius \(1\) around the points of \(Z\). A Frobenius structure on \( {\mathcal E}\) with respect to \(\sigma\) is an isomorphism \({\mathcal F}: \sigma^{*}{\mathcal E} \simeq {\mathcal E} \) of vector bundles with connection defined on some strict neighborhood of \(V\).
A meromorphic connection on \({\mathbb{P}}^{1}\) over a \(p\)-adic field admits a Frobenius structure defined over a suitable rigid analytic subspace. Authors of the paper under review give an effective convergence bound for this Frobenius structure by studying the effect of changing the Frobenius lift. They also give an example indicating that their bound is optimal.
The techniques used are computational. This is a good place to see the interplay between matrix representation of a Frobenius structure and a Gauss-Manin connection.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
11G25 Varieties over finite and local fields
11M38 Zeta and \(L\)-functions in characteristic \(p\)
14G22 Rigid analytic geometry
12H25 \(p\)-adic differential equations
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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References:

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