Introduction to p-adic differential equations
Abstract
In the ultrametric setting, linear differential equations present phenomena that do not appear over the complex field. Indeed, the solutions of such equations may fail to converge everywhere, even without thepresence of poles. This leads to a non-trivial notion of the radius of convergence, and its knowledge permits to obtain several interesting information about the equation. Notably, it controls the finite dimensionality of the de Rham cohomology. In this course we introduce the framework of p-adic differential equations.
Prerequisite: General algebra, Linear algebra, Galois theory of fields, Metric spaces.
References
G. Christol. “Le théorème de turritin p-adique (version du 11/06/2011)”.
G. Christol. “Modules différentiels et équations différentielles p-adiques”. In: Queen’s Papers in Pure and Applied Math (1983).
G. Christol and P. Robba. Équations différentielles p-adiques - Applications aux sommes exponentielles. Actualités Mathématiques. Hermann, 1994.
K. S. Kedlaya. p-adic differential equations. English. Cambridge: Cam- bridge University Press, 2010, pp. xvii + 380.isbn: 978-0-521-76879- 5/hbk.