Arithmetic and Geometry [Paperback]

The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.In this proceedings volume leading authorities describe the state of the art in number theory, particularly Diophantine and arithmetic geometry. The book gives an excellent overview of the subject for graduate students and it is essential reading for researchers wishing to keep abreast of recent developments.In this proceedings volume leading authorities describe the state of the art in number theory, particularly Diophantine and arithmetic geometry. The book gives an excellent overview of the subject for graduate students and it is essential reading for researchers wishing to keep abreast of recent developments.The 'Arithmetic and Geometry' trimester, held at the Hausdorff Research Institute for Mathematics in Bonn, focussed on recent work on Serre's conjecture and on rational points on algebraic varieties. The resulting proceedings volume provides a modern overview of the subject for graduate students in arithmetic geometry and Diophantine geometry. It is also essential reading for any researcher wishing to keep abreast of the latest developments in the field. Highlights include Tim Browning's survey on applications of the circle method to rational points on algebraic varieties and Per Salberger's chapter on rational points on cubic hypersurfaces.Preface Luis Dieulefait, Gerd Faltings, D. R. Heath-Brown, Yuri I. Manin, B. Z. Moroz and Jean-Pierre Wintenberger; Introduction; List of participants; Trimester seminar; Workshop on the Serre conjecture; The research conference; 1. Galois groups of local fields, Lie algebras, and ramification Victor Abrashkin; 2. A characterisation of ordinary modular eigenforms with CM Rajender Adibhatla and Panagiotis Tsaknias; 3. Selmer complexes and p-adic Hodge theory Denis Benois; 4. A survey of applications of the circle method to rational points T. D. Browning; 5. Arithmetic differential equations of PainlÃ(