The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.

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 Kunyavski¿; 12. On the ¿-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.