Ramification Groups of Local Fields : with Geometric Applications

This book develops the modern theory of upper ramification groups for local fields with imperfect residue fields, alongside classical results like the Hasse–Arf theorem. It introduces log monogenic extensions and provides new proofs of key formulas, offering essential tools for researchers in arithmetic geometry and Galois representation theory.

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.