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 software in mathematics Geometry within the 2009-2010 educational year.

The notes via Laurent Berger supply an creation to *p*-adic Galois representations and Fontaine earrings, that are particularly valuable for describing many neighborhood deformation jewelry at *p* that come up clearly in Galois deformation theory.

The notes by way of Gebhard Böckle provide a entire direction on Galois deformation conception, ranging from the foundational result of Mazur and discussing intimately the idea 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 earrings over neighborhood ones are discussed.

The notes through Mladen Dimitrov current the fundamentals of the mathematics conception of Hilbert modular varieties and types, with an emphasis at the examine of the pictures of the connected Galois representations, on modularity lifting theorems over completely genuine quantity fields, and at the cohomology of Hilbert modular kinds with indispensable coefficients.

The notes by way of Lassina Dembélé and John Voight describe tools for appearing specific computations in areas of Hilbert modular types. those equipment depend upon the Jacquet-Langlands correspondence and on computations in areas of quaternionic modular types, either for the case of yes and indefinite quaternion algebras. a number of examples are given, and purposes to modularity of Galois representations are discussed.

The notes through Tim Dokchitser describe the evidence, received via the writer in a joint undertaking with Vladimir Dokchitser, of the parity conjecture for elliptic curves over quantity fields below the belief of finiteness of the Tate-Shafarevich staff. The assertion of the Birch and Swinnerton-Dyer conjecture is integrated, in addition to a close 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

by John

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