Nyhet

Ramification Groups of Local Fields

with Geometric Applications

AvTakeshi Saito

Inbunden, Engelska, 2026

Del 53 i serien New Mathematical Monographs

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Beskrivning

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.

Produktinformation

Utgivningsdatum:2026-06-18
Mått:157 x 235 x 30 mm
Vikt:872 g
Format:Inbunden
Språk:Engelska
Serie:New Mathematical Monographs
Antal sidor:474
Förlag:Cambridge University Press
ISBN:9781009617536

Utforska kategorier

Talteori inom Naturvetenskap och teknik
Matematik inom Naturvetenskap och teknik
Topologi inom Naturvetenskap och teknik

Mer om författaren

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).

Innehållsförteckning

Introduction; Part I. Ramification of Henselian Discrete Valuation Fields: 1. Finite extensions; 2. Cohomological ﬁltration; 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 ﬁber theorem; 10. Nearby cycles on curves; Part V. Upper Ramification Subgroups: 11. Stable integral models; 12. Upper ramiﬁcation 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 reﬁned Swan con-ductor; Solutions to exercises; References; Index.