By Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight
The notes during this quantity correspond to complex classes held on the Centre de Recerca Matemàtica as a part of the examine application in mathematics Geometry within the 2009-2010 educational year.
The notes via Laurent Berger supply an advent to p-adic Galois representations and Fontaine earrings, that are specifically worthy for describing many neighborhood deformation earrings at p that come up obviously in Galois deformation theory.
The notes by way of Gebhard Böckle provide a entire direction on Galois deformation concept, ranging from the foundational result of Mazur and discussing intimately the idea of pseudo-representations and their deformations, neighborhood deformations at areas l ≠ p and native deformations at p that are flat. within the final section,the result of Böckle and Kisin on shows of world deformation earrings over neighborhood ones are discussed.
The notes by way of Mladen Dimitrov current the fundamentals of the mathematics idea of Hilbert modular kinds and kinds, with an emphasis at the examine of the photographs of the hooked up Galois representations, on modularity lifting theorems over completely actual quantity fields, and at the cohomology of Hilbert modular kinds with imperative coefficients.
The notes by means of Lassina Dembélé and John Voight describe tools for appearing particular computations in areas of Hilbert modular varieties. those equipment rely on the Jacquet-Langlands correspondence and on computations in areas of quaternionic modular varieties, either for the case of certain and indefinite quaternion algebras. a number of examples are given, and functions to modularity of Galois representations are discussed.
The notes by means of Tim Dokchitser describe the evidence, received via the writer in a joint venture with Vladimir Dokchitser, of the parity conjecture for elliptic curves over quantity fields less than the idea of finiteness of the Tate-Shafarevich staff. The assertion of the Birch and Swinnerton-Dyer conjecture is incorporated, in addition to an in depth examine of neighborhood and worldwide root numbers of elliptic curves and their classification.
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Additional resources for Elliptic Curves, Hilbert Modular Forms and Galois Deformations (Advanced Courses in Mathematics - CRM Barcelona)
Elliptic Curves, Hilbert Modular Forms and Galois Deformations (Advanced Courses in Mathematics - CRM Barcelona) by Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight