$p$-adic analytic geometry and differential equations / Géométrie analytique et équations différentielles $p$-adiques

Collection $p$-adic analytic geometry and differential equations / Géométrie analytique et équations différentielles $p$-adiques

Organizer(s) Lebacque, Philippe ; Nicaise, Johannes ; Poineau, Jérôme

Date(s) 27/03/2017 - 31/03/2017

linked URL http://conferences.cirm-math.fr/1609.html

00:00:00 / 00:00:00

4 5

Chapters

An overview on some recent results about $p$-adic differential equations over Berkovich curves

By Andrea Pulita

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.

Information about the video

Date of recording 28/03/2017
Date of publication 06/04/2017
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19153403
Cite this video Pulita, Andrea (28/03/2017). An overview on some recent results about $p$-adic differential equations over Berkovich curves. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19153403
URL https://dx.doi.org/10.24350/CIRM.V.19153403

Domain(s)

Number Theory

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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