
Ramification Groups of Local Fields
with Geometric Applications
Takeshi Saito(Author)
Cambridge University Press
Will be published approx. on 18. June 2026
Book
Hardback
474 pages
978-1-009-61753-6 (ISBN)
Description
Ramification groups of local fields are essential tools for studying boundary behaviour in geometric objects and the degeneration of Galois representations. This book presents a comprehensive development of the recently established theory of upper ramification groups of local fields with imperfect residue fields, starting from the foundations. It also revisits classical theory, including the Hasse-Arf theorem, and offers an optimal generalisation via log monogenic extensions. The conductor of Galois representations, defined through ramification groups, has numerous geometric applications, notably the celebrated Grothendieck-Ogg-Shafarevich formula. A new proof of the Deligne-Kato formula is also provided; this result plays a pivotal role in the theory of characteristic cycles. With a foundational understanding of commutative rings and Galois theory, graduate students and researchers will be well-equipped to engage with this rich area of arithmetic geometry.
More details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Target group
College/higher education
Illustrations
Worked examples or Exercises
Dimensions
Height: 235 mm
Width: 157 mm
Thickness: 30 mm
Weight
831 gr
ISBN-13
978-1-009-61753-6 (9781009617536)
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Schweitzer Classification
Person
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).
Content
Introduction; Part I. Ramification of Henselian Discrete Valuation Fields: 1. Finite extensions; 2. Cohomological ?ltration; 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 ?ber theorem; 10. Nearby cycles on curves; Part V. Upper Ramification Subgroups: 11. Stable integral models; 12. Upper rami?cation 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 re?ned Swan con-ductor; Solutions to exercises; References; Index.