Arithmetic and Geometry

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 Painleve VI type Alexandru Buium and Yuri I. Manin; 6. Differential calculus with integers Alexandru Buium; 7. Un calcul de groupe de Brauer et une application arithmetique Jean-Louis Colliot-Thelene; 8. Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity Luis Dieulefait and Ariel Pacetti; 9. Big image of Galois representations and congruence ideals Haruzo Hida and Jacques Tilouine; 10. The skew-symmetric pairing on the Lubin-Tate formal module M. A. Ivanov and S. V. Vostokov; 11. Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry Boris Kunyavskii; 12. On the l-adic regulator as an ingredient of Iwasawa theory L. V. Kuz'min; 13. On a counting problem for G-shtukas Ngo Dac Tuan; 14. Modular forms and Calabi-Yau varieties Kapil Paranjape and Dinakar Ramakrishnan; 15. Derivative of symmetric square p-adic L-functions via pull-back formula Giovanni Rosso; 16. Uniform bounds for rational points on cubic hypersurfaces Per Salberger; 17. Descent on toric fibrations Alexei N. Skorobogatov; 18. On filtrations of vector bundles over P1Z A. Smirnov; 19. On the dihedral Euler characteristics of Selmer groups of Abelian varieties Jeanine Van Order; 20. CM values of higher Green's functions and regularized Petersson products Maryna Viazovska.