Algebraic Geometry

Download E-books Elliptic Curves, Hilbert Modular Forms and Galois Deformations (Advanced Courses in Mathematics - CRM Barcelona) PDF

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 study application in mathematics Geometry within the 2009-2010 educational year.

The notes through Laurent Berger offer an advent to p-adic Galois representations and Fontaine earrings, that are specially priceless for describing many neighborhood deformation earrings at p that come up certainly in Galois deformation theory.

The notes through Gebhard Böckle supply a complete path on Galois deformation concept, ranging from the foundational result of Mazur and discussing intimately the speculation 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 worldwide deformation earrings over neighborhood ones are discussed.

 The notes via Mladen Dimitrov current the fundamentals of the mathematics conception of Hilbert modular types and kinds, with an emphasis at the examine of the photographs of the connected Galois representations, on modularity lifting theorems over absolutely genuine quantity fields, and at the cohomology of Hilbert modular types with fundamental coefficients.

 The notes by means of Lassina Dembélé and John Voight describe equipment for appearing particular computations in areas of Hilbert modular kinds. those tools rely on the Jacquet-Langlands correspondence and on computations in areas of quaternionic modular types, either for the case of sure 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, received by means 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 workforce. The assertion of the Birch and Swinnerton-Dyer conjecture is incorporated, in addition to a close learn of neighborhood and worldwide root numbers of elliptic curves and their classification.

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