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Number theory currently has at least three different perspectives on non-abelian phenomena: the Langlands programme, non-commutative Iwasawa theory and anabelian geometry. In the second half of 2009, experts from each of these three areas gathered at the Isaac Newton Institute in Cambridge to explain the latest advances in their research and to investigate possible avenues of future investigation and collaboration. For those in attendance, the overwhelming impression was that number theory is going through a tumultuous period of theory-building and experimentation analogous to the late 19th century, when many different special reciprocity laws of abelian class field theory were formulated before knowledge of the Artin-Takagi theory. Non-abelian Fundamental Groups and Iwasawa Theory presents the state of the art in theorems, conjectures and speculations that point the way towards a new synthesis, an as-yet-undiscovered unified theory of non-abelian arithmetic geometry.

Iwasawa theory. --- Non-Abelian groups. --- Groups, Non-Abelian --- Groups, Nonabelian --- Nonabelian groups --- Group theory --- Iwasawa's theorem --- Algebraic fields

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Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the deepest and most beautiful theorem known about them. It is also the simplest example of a vast array of subsequent, unproven "main conjectures'' in modern arithmetic geometry involving the arithmetic behaviour of motives over p-adic Lie extensions of number fields. These main conjectures are concerned with what one might loosely call the exact formulae of number theory which conjecturally link the special values of zeta and L-functions to purely arithmetic expressions (the most celebrated example being the conjecture of Birch and Swinnerton-Dyer for elliptic curves). Written by two leading workers in the field, this short and elegant book presents in full detail the simplest proof of the "main conjecture'' for cyclotomic fields . Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. The masterly exposition is intended to be accessible to both graduate students and non-experts in Iwasawa theory.

Algebraic fields. --- Cyclotomy. --- Functions, Zeta. --- Equations, Cyclotomic --- Number theory --- Equations, Abelian --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Zeta functions --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra

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English literature --- Philosophers --- Biography: 1900-1999

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