By Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight

The notes during this quantity correspond to complicated classes held on the Centre de Recerca Matemàtica as a part of the study application 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 in particular worthy for describing many neighborhood deformation jewelry at p that come up evidently in Galois deformation theory.

The notes via Gebhard Böckle provide a complete path 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 displays of world deformation jewelry over neighborhood ones are discussed.

 The notes via Mladen Dimitrov current the fundamentals of the mathematics conception of Hilbert modular varieties and forms, with an emphasis at the research of the pictures of the hooked up Galois representations, on modularity lifting theorems over completely actual quantity fields, and at the cohomology of Hilbert modular forms with essential coefficients.

 The notes via Lassina Dembélé and John Voight describe tools for acting particular computations in areas of Hilbert modular varieties. those equipment depend upon the Jacquet-Langlands correspondence and on computations in areas of quaternionic modular kinds, either for the case of convinced and indefinite quaternion algebras. a number of examples are given, and functions to modularity of Galois representations are discussed.

 The notes by way of Tim Dokchitser describe the facts, got by way of the writer in a joint venture with Vladimir Dokchitser, of the parity conjecture for elliptic curves over quantity fields lower 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 international root numbers of elliptic curves and their classification.

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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

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