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Differential modules on p-adic polyannuli

Part of: Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  19 May 2009

Kiran S. Kedlaya  and
Liang Xiao
Kiran S. Kedlaya
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA ([email protected])
Liang Xiao
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA ([email protected])
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Abstract

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 0. This extends prior work in the one-dimensional 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.

Keywords

differential modulespolyannulinonarchimedean fieldgeneric radius of convergence

MSC classification

Secondary: 14G22: Rigid analytic geometry
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 1 , January 2010 , pp. 155 - 201
Copyright
Copyright © Cambridge University Press 2010

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References

1.Baldassarri, F. and di Vizio, L., Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces, preprint arXiv 0709.2008v3 (2008).Google Scholar
2.Berkovich, V. G., Spectral theory and analytic geometry over non-archimedean fields, Mathematical Surveys and Monographs, Volume 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
3.Christol, G. and Dwork, B., Modules différentielles sur les couronnes, Annales Inst. Fourier 44 (1994), 663–701.CrossRefGoogle Scholar
4.Dwork, B., Gerotto, G. and Sullivan, F., An introduction to G-functions, Annals of Mathematics Studies, Volume 133 (Princeton University Press, 1994).Google Scholar
5.Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, Volume 150 (Springer, 1995).CrossRefGoogle Scholar
6.Grothendieck, A., Éléments de géométrie algébrique, IV, Étude locale des schémas et des morphismes de schémas, I, Publ. Math. IHES 20 (1964).CrossRefGoogle Scholar
7.Kedlaya, K. S., Semistable reduction for overconvergent F-isocrystals, I, Unipotence and logarithmic extensions, Compositio Math. 143 (2007), 1164–1212.CrossRefGoogle Scholar
8.Kedlaya, K. S., Swan conductors for p-adic differential modules, I, A local construction, Alg. Num. Theory 1 (2007), 269–300.CrossRefGoogle Scholar
9.Kedlaya, K. S., The p-adic local monodromy theorem for fake annuli, Rend. Sem. Mat. Univ. Padova 118 (2007), 101–146.Google Scholar
10.Kedlaya, K. S., Semistable reduction for overconvergent F-isocrystals, III, Local semistable reduction at monomial valuations, Compositio Math. 145 (2009), 143–172.CrossRefGoogle Scholar
11.Kedlaya, K. S., p-adic differential equations (version of 19 01 2009), course notes (available at http://math.mit.edu/~kedlaya/papers/).CrossRefGoogle Scholar
12.Kedlaya, K. S., Swan conductors for p-adic differential modules, II, Global variation, preprint arXiv 0705.0031v3 (2009).Google Scholar
13.Kedlaya, K. S., Good formal structures for flat meromorphic connections, I, Surfaces, preprint arXiv 0811.0190v3 (2009).Google Scholar
14.Kedlaya, K. S. and Tynan, P., Detecting integral polyhedral functions, Confluentes Math. 1 (2009), 1–23.CrossRefGoogle Scholar
15.Matsuda, S., Conjecture on Abbes–Saito filtration and Christol?–Mebkhout filtration, in Geometric aspects of Dwork theory, Volume II, pp. 845–856 (de Gruyter, Berlin, 2004).CrossRefGoogle Scholar
16.Ore, O., Theory of non-commutative polynomials, Annals Math. 34 (1933), 480–508.CrossRefGoogle Scholar
17.Ribenboim, P., The theory of classical valuations (Springer, 1999).CrossRefGoogle Scholar
18.Sabbah, C., Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque 263 (2000).Google Scholar
19.Schneider, P., Nonarchimedean functional analysis (Springer, 2002).CrossRefGoogle Scholar
20.Young, P. T., Radii of convergence and index for p-adic differential operators, Trans. Am. Math. Soc. 333 (1992), 769–785.Google Scholar