Ramification Groups of Local Fields

Takeshi Saito is a Professor of Mathematics at School of Mathematical Sciences, the University of Tokyo, specialising in arithmetic geometry. He is the recipient of the Algebra Prize of the Mathematical Society of Japan (1998) and Spring Prize of the Mathematical Society of Japan (2001) and is the Israel Gelfand Chair in Mathematics at IHES (2024–2026).

• Introduction; Part I. Ramification of Henselian Discrete Valuation Fields: 1. Finite extensions; 2. Cohomological filtration; Part II. Cyclic Extensions: 3. Cyclic extensions of degree; 4. Trace of differential forms; 5. The Hasse–Arf theorem; Part III. Conductor and Refinements: 6. Swan conductor; 7. Conductor and differential forms; Part IV. Geometric Applications: 8. Grothendieck–Ogg–Shafarevich formula; 9. Reduced fiber theorem; 10. Nearby cycles on curves; Part V. Upper Ramification Subgroups: 11. Stable integral models; 12. Upper ramification subgroups; 13. Logarithmic variant and Artin–Schreier–Witt extensions; Part VI. Graded Quotients and Character-Istic Forms: 14. Graded quotients; 15. Characteristic forms; 16. Logarithmic characteristic forms and the refined Swan con-ductor; Solutions to exercises; References; Index.