Noncommutative Iwasawa Theory and Explicit Reciprocity Laws

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We propose to study the arithmetic of elliptic curves and modular forms, through two powerful modern techniques - algebraic number theory and algebraic geometry. We hope that, amongst other things, our work will shed light on the Birch-Swinnerton-Dyer conjecture, one of the most mysterious and challenging unsolved problems in number theory, which is related to one of the oldest problems in mathematics - that of determining the possible values of the area of a right angled triangle, the lengths of all of whose sides are rational numbers.