CIRM - Videos & books Library - An overview on some recent results about $p$-adic differential equations over Berkovich curves

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Authors : Pulita, Andrea (Author of the conference)
CIRM (Publisher )

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Berkovich curves p-adic differential equations radius of convergence convergence Newton polygon controlling graph decomposition by the radii index theorem de Rham cohomology questions of the audience

Abstract : I will give an introductory talk on my recent results about $p$-adic differential equations on Berkovich curves, most of them in collaboration with J. Poineau. This includes the continuity of the radii of convergence of the equation, the finiteness of their controlling graphs, the global decomposition by the radii, a bound on the size of the controlling graph, and finally the finite dimensionality of their de Rham cohomology groups, together with some local and global index theorems relating the de Rham index to the behavior of the radii of the curve. If time permits I will say a word about some recent applications to the Riemann-Hurwitz formula.

MSC Codes :
12H25 - $p$-adic differential equations
14G22 - Rigid analytic geometry

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Film maker :

Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-03-28_Pulita.mp4

Information on the Event

Event Title : $p$-adic analytic geometry and differential equations / Géométrie analytique et équations différentielles $p$-adiques
Event Organizers : Lebacque, Philippe ; Nicaise, Johannes ; Poineau, Jérôme
Dates : 27/03/17 - 31/03/17
Event Year : 2017
Event URL : http://conferences.cirm-math.fr/1609.html

Citation Data

DOI : 10.24350/CIRM.V.19153403
Cite this video as: Pulita, Andrea (2017). An overview on some recent results about $p$-adic differential equations over Berkovich curves. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19153403
URI : http://dx.doi.org/10.24350/CIRM.V.19153403

[Multi angle] Explosion of Lyapunov exponents using non-Archimedean geometry / Author of the conference Favre, Charles.
[Multi angle] Arithmetic $D$-modules and existence of crystalline companion / Author of the conference Abe, Tomoyuki.
[Multi angle] $D$-modules and $p$-curvatures / Author of the conference Esnault, Hélène.
[Multi angle] de Rham theorem in non-Archimedean analytic geometry / Author of the conference Berkovich, Vladimir.

Bibliography

Baldassarri, F. (2010). Continuity of the radius of convergence of differential equations on $p$-adic analytic curves. Inventiones Mathematicae, 182(3), 513-584 - http://dx.doi.org/10.1007/s00222-010-0266-7

Kedlaya, Kiran S. (2016). Convergence polygons for connections on Nonarchimedean curves. In Baker, Matthew (ed.) et al., Nonarchimedean and tropical geometry (pp. 51-97). Cham: Springer - http://dx.doi.org/10.1007/978-3-319-30945-3_3

Kedlaya, Kiran S. (2015). Local and global structure of connections on nonarchimedean curves. Compositio Mathematica, 151, No. 6, 1096-1156 - http://dx.doi.org/10.1112/S0010437X14007830

Kedlaya, Kiran S. (2010). $p$-adic differential equations. Cambridge: Cambridge University Press (ISBN 978-0-521-76879-5/hbk). xvii, 380 p. (2010). - http://www.cambridge.org/fr/academic/subjects/mathematics/number-theory/p-adic-differential-equations?format=HB&isbn=9780521768795

Poineau, J., & Pulita, A. (2015). Continuity and finiteness of the radius of convergence of a $p$-adic differential equation via potential theory. Journal für die Reine und Angewandte Mathematik, 707, 125-147 - http://dx.doi.org/10.1515/crelle-2013-0086

Poineau, J., & Pulita, A. (2015). The convergence Newton polygon of a $p$-adic differential equation. II: Continuity and finiteness on Berkovich curves. Acta Mathematica, 214(2), 357-393 - http://dx.doi.org/10.1007/s11511-015-0127-8

Poineau, J., & Pulita, A. (2014). The convergence Newton polygon of a $p$-adic differential equation IV : local and global index theorems - https://arxiv.org/abs/1309.3940

Poineau, J., & Pulita, A. (2013). The convergence Newton polygon of a $p$-adic differential equation III : global decomposition and controlling graphs - https://arxiv.org/abs/1308.0859

Pulita, A. (2015). The convergence Newton polygon of a $p$-adic differential equation. I: Affinoid domains of the Berkovich affine line. Acta Mathematica, 214(2), 307-355 - http://dx.doi.org/10.1007/s11511-015-0126-9

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