Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T04:10:50.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations

Part of: Differential and difference algebra

Published online by Cambridge University Press:  18 February 2009

Bruno Chiarellotto  and
Nobuo Tsuzuki
Bruno Chiarellotto
Affiliation:
Dipartimento Matematica Pura e Applicata, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy, (chiarbru@math.unipd.it).
Nobuo Tsuzuki
Affiliation:
Mathematical Institute, Tohoku University, 6-3 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8578, Japan, (tsuzuki@math.tohoku.ac.jp).
Get access Rights & Permissions [Opens in a new window]

Abstract

For a ∇-module M over the ring K[[x]]0 of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a ϕ–∇-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

Keywords

p-adic differential equationslogarithmic growthNewton polygons

MSC classification

Secondary: 12H25: $p$-adic differential equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 3 , July 2009 , pp. 465 - 505
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.André, Y., Dwork's conjecture on the logarithmic growth of solutions of p-adic differential équations, Compositio Math. 144 (2008), 484–494.CrossRefGoogle Scholar
2.Baldassarri, F. and Chiarellotto, B., Algebraic versus rigid cohomology with lagarithmic coefficients, in Barsotti Symposium in Algebraic Geometry (ed. Messing, W. and Cristante, V.), Perspectives in Mathematics, Volume 15, pp. 11–50 (Academic Press, 1994).CrossRefGoogle Scholar
3.Berthelot, P., Géomètrie rigide et cohomologie des variétès algébriques de caractéristique p, Bull. Soc. Math. France 23 (1986), 7–32.Google Scholar
4.Chiarellotto, B., Espaces de Berkovich et équations différentielles p-adiques: une note, Rend. Sem. Mat. Padova 103 (2000), 193–209.Google Scholar
5.Chiarellotto, B. and Stum, B. Le, F-isocristaux unipotents, Compositio Math. 116 (1999), 81–110.CrossRefGoogle Scholar
6.Chiarellotto, B. and Tsuzuki, N., ϕ-modules over a p-adically complete discrete valuation ring, preprint.Google Scholar
7.Christol, G., Modules différentiels et équations différentielles p-adiques, Queen's Papers in Pure and Applied Mathematics, Volume 66 (Queen's University, Kingston, ON, 1983).Google Scholar
8.Christol, G. and Mebkhout, Z., Sur le théorème de l'indice des équations différentielles p-adiques, III, Annals Math. 151 (2000), 385–457.CrossRefGoogle Scholar
9.Deligne, P., Equations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Volume 163 (Springer, 1970).CrossRefGoogle Scholar
10.Dwork, B. M., On the zeta function of a hypersurface, II, Annals Math. 80 (1964), 227–299.CrossRefGoogle Scholar
11.Dwork, B. M., p-adic cycles, Publ. Math. IHES 37 (1969), 27–116.CrossRefGoogle Scholar
12.Dwork, B. M., On p-adic differential equations, II, Annals Math. 98 (1973), 366–376.CrossRefGoogle Scholar
13.Dwork, B. M., On p-adic differential equations, III, Invent. Math. 20 (1973), 35–45.CrossRefGoogle Scholar
14.Dwork, B. M., Lectures on p-adic differential equations, Grundlehren der mathematischen Wissenschaften, Volume 253 (Springer, 1982).CrossRefGoogle Scholar
15.Dwork, B. M. and Robba, Ph., On ordinary p-adic linear differential equations, Trans. Am. Math. Soc. 231 (1977), 1–46.Google Scholar
16.Grothendieck, A., Groupes de Barsotti–Tate et cristaux de Dieudonné, Séminaire de Mathématiques Supérieures (É;té, 1970), No. 45 (Les Presses de l'Université de Montréal, 1974).Google Scholar
17.Katz, N., Slope filtration of F-crystals, Astérisque 63 (1979), 113–164.Google Scholar
18.Kedlaya, K., A p-adic monodromy theorem, Annals Math. 160 (2004), 93–184.CrossRefGoogle Scholar
19.Kedlaya, K., Slope filtrations revisited, Documenta Math. 10 (2005), 447–525.CrossRefGoogle Scholar
20.Kedlaya, K., Slope filtrations for relative Frobenius, Astérisque 319 (2008), 259–301.Google Scholar
21.Matsuda, S., Katz's correspondence for quasi-unipotent overconvergent isocrystals, Compositio Math. 134 (2002), 1–34.CrossRefGoogle Scholar
22.Robba, Ph., On the index of p-adic differential operators, I, Annals Math. 101 (1975), 280–316.CrossRefGoogle Scholar
23.Robba, Ph., Sur les équations différentielles lineaires p-adiques, II, Pac. J. Math. 98 (1982), 393–418.CrossRefGoogle Scholar
24.Robba, Ph. and Christol, G., Équation différentielles p-adiques: application aux sommes exponentielles, Actualité Mathematiques (Hermann, Paris, 1994).Google Scholar
25.Tsuzuki, N., Slope filtration of quasi-unipotent overconvergent F-isocrystals, Annales Inst. Fourier 48 (1998), 379–412.CrossRefGoogle Scholar
26.Tsuzuki, N., On base change theorem and coherence in rigid cohomology, Documenta Math., Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 891–918.Google Scholar