Non-abelian Fundamental Groups and Iwasawa

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 ArtinTakagi 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.

List of contributors	p. vi
Preface	p. vii
Lectures on anabelian phenomena in geometry and arithmetic	p. 1
On Galois rigidity of fundamental groups of algebraic curves	p. 56
Around the Grothendieck anabelian section conjecture	p. 72
From the classical to the noncommutative Iwasawa theory (for totally real number fields)	p. 107
On the M_H(G)-conjecture	p. 132
Galois theory and Diophantine geometry	p. 162
Potential modularity - a survey	p. 188
Remarks on some locally Q_p-analytic representations of GL₂(F) in the crystalline case	p. 212
Completed cohomology - a survey	p. 239
Tensor and homotopy criteria for functional equations of l-adic and classical iterated integrals	p. 258
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Non-abelian Fundamental Groups and Iwasawa Theory

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