By Gebhard Böckle, Lassina Dembélé, Mladen Dimitrov, Tim Dokchitser
The notes during this quantity correspond to complicated classes held on the Centre de Recerca Matemàtica as a part of the learn software in mathematics Geometry within the 2009-2010 educational year.
The notes via Laurent Berger offer an creation to p-adic Galois representations and Fontaine jewelry, that are particularly valuable for describing many neighborhood deformation earrings at p that come up obviously in Galois deformation theory.
The notes through Gebhard Böckle provide a entire direction on Galois deformation conception, ranging from the foundational result of Mazur and discussing intimately the speculation of pseudo-representations and their deformations, neighborhood deformations at locations 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 means of Mladen Dimitrov current the fundamentals of the mathematics conception of Hilbert modular varieties and kinds, with an emphasis at the learn of the photographs of the connected Galois representations, on modularity lifting theorems over absolutely genuine quantity fields, and at the cohomology of Hilbert modular forms with necessary coefficients.
The notes via Lassina Dembélé and John Voight describe equipment for acting specific computations in areas of Hilbert modular kinds. those equipment depend upon the Jacquet-Langlands correspondence and on computations in areas of quaternionic modular varieties, either for the case of sure and indefinite quaternion algebras. numerous examples are given, and functions to modularity of Galois representations are discussed.
The notes by means of Tim Dokchitser describe the facts, bought by means of the writer in a joint undertaking with Vladimir Dokchitser, of the parity conjecture for elliptic curves over quantity fields less than the belief of finiteness of the Tate-Shafarevich staff. The assertion of the Birch and Swinnerton-Dyer conjecture is incorporated, in addition to an in depth research of neighborhood and worldwide root numbers of elliptic curves and their classification.
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