ARITHMETIC AND GEOMETRY

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 Painlevé 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 arithmétique Jean-Louis Colliot-Thélène
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.

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.

•Modern survey ideal for graduate students entering the field
•Includes exciting new developments making it essential reading for established researchers in the area
•Written by the world's leading authorities in number theory

Authors
•Luis Dieulefait, Universitat de Barcelona
Luis Dieulefait is Associate Professor in the Department of Algebra and Geometry at the University of Barcelona.
•Gerd Faltings, Max-Planck-Institut für Mathematik, Bonn
D. R. Heath-Brown FRS is Professor of Pure Mathematics at the University of Oxford and has twice been an invited speaker at the International Congress of Mathematicians (ICM). He is one of a growing band of number theorists exploring the interface between analytic number theory and Diophantine geometry, which led him to co-organise the trimester of which this volume is the proceedings.
•D. R. Heath-Brown, University of Oxford
Gerd Faltings is Managing Director of the Max Planck Institute for Mathematics in Bonn.
•Y. V. Manin, Max-Planck-Institut für Mathematik, Bonn
Yuri I. Manin is Professor Emeritus at the Max Planck Institute for Mathematics in Bonn.