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The absolute Galois group of Q is an object that contains a wealth of number-theoretic information. The study of its representations has been an intense focus of attention of algebraic number theorists. There are well-known so-called reciprocity conjectures about these representations attached to the names of Artin, Langlands, Serre, Fontaine and Mazur. I will try to explain some of these conjectures, and recent progress on them.
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