Ramification Groups of Local Fields

with Geometric Applications

By: Takeshi Saito

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Hardcover | 18 June 2026

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Hardcover
474 Pages

Dimensions(cm)
22.86 x 15.24 x 2.69

Hardcover

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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 Hasseâ"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 Grothendieckâ"Oggâ"Shafarevich formula. A new proof of the Deligneâ"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.

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