Ramification Groups of Local Fields: with Geometric Applications (New Mathematical Monographs)

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.

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).