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 HasseArf 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 GrothendieckOggShafarevich formula. A new proof of the DeligneKato 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.
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Fully develops the theory of ramification groups from the foundations, spanning classical theory to recent developments.
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 HasseArf theorem; Part III. Conductor and Refinements:
6. Swan
conductor;
7. Conductor and differential forms; Part IV. Geometric
Applications:
8. GrothendieckOggShafarevich formula;
9. Reduced ber
theorem;
10. Nearby cycles on curves; Part V. Upper Ramification Subgroups:
11. Stable integral models;
12. Upper ramication subgroups;
13. Logarithmic
variant and ArtinSchreierWitt extensions; Part VI. Graded Quotients and
Character-Istic Forms:
14. Graded quotients;
15. Characteristic forms;
16.
Logarithmic characteristic forms and the rened Swan con-ductor; Solutions to
exercises; References; Index.
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 (20242026).
