Kedlaya, Kiran S.; Tuitman, Jan Effective convergence bounds for Frobenius structures on connections. (English) Zbl 1276.14032 Rend. Semin. Mat. Univ. Padova 128, 7-16 (2012). 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. Reviewer: Nikolaj M. Glazunov (Kyïv) Cited in 8 Documents 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.) Keywords:Picard-Fuchs equation; Gauss-Manin connection; Frobenius lift; Frobenius structure; effective convergence bounds PDFBibTeX XMLCite \textit{K. S. Kedlaya} and \textit{J. Tuitman}, Rend. Semin. Mat. Univ. Padova 128, 7--16 (2012; Zbl 1276.14032) Full Text: DOI arXiv Link References: [1] B. DWORK - P. ROBBA, Effective p-adic bounds for solutions of homo- geneous linear differential equations. Tran s. Amer. Math. Soc., 259 (2) (1980), pp. 559-577. · Zbl 0439.12016 [2] K. S. KEDLAYA, p-adic Differential Equations. Cambridge University Press, 2010. · Zbl 1213.12009 [3] K. S. KEDLAYA, Effective p-adic cohomologyfor cyclic cubic threefolds. In Computational Algebraic and Analytic Geometry of Low-dimensional Varieties. Amer. Math. Soc., 2012. Available at http://math.mit.edu/\(kedlaya/papers/.\) [4] A. LAUDER, Rigid cohomologyand p-adic point counting. J. TheÂor. Nombres Bordeaux, 17 (2005), pp. 169-180. · Zbl 1087.14020 [5] A. LAUDER, A recursive method for computing zeta functions of varieties. LMS J. Comput. Math., 9 (2006), pp. 222-269. · Zbl 1108.14018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.