Luis Dieulefait is Associate Professor in the Department of Algebra and Geometry at the University of Barcelona. 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. Gerd Faltings is Managing Director of the Max Planck Institute for Mathematics in Bonn. Yuri I. Manin is Professor Emeritus at the Max Planck Institute for Mathematics in Bonn. B. Z. Moroz is Associate Professor at the Max Planck Institute for Mathematics in Bonn. Jean-Pierre Wintenberger is Professor at the Department of Mathematics at the University of Strasbourg.

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• 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 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 Kunyavskiĭ; 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.