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  • © 2021

The Eigenbook

Eigenvarieties, families of Galois representations, p-adic L-functions

Birkhäuser

Authors:

  • Offers a comprehensive and solid introduction to the theory
  • Includes exercises and solutions
  • Presents a unique collection of knowledge on this theory not available elsewhere

Part of the book series: Pathways in Mathematics (PATHMATH)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xi
  2. Introduction

    • Joël Bellaïche
    Pages 1-3
  3. The ‘Eigen’ Construction

    1. Front Matter

      Pages 5-5
    2. Eigenalgebras

      • Joël Bellaïche
      Pages 7-46
    3. Eigenvarieties

      • Joël Bellaïche
      Pages 47-94
  4. Modular Symbols and L-Functions

    1. Front Matter

      Pages 95-95
    2. Abstract Modular Symbols

      • Joël Bellaïche
      Pages 97-118
  5. The Eigencurve and its p-Adic L-Functions

    1. Front Matter

      Pages 201-201
    2. The Eigencurve of Modular Symbols

      • Joël Bellaïche
      Pages 203-247
    3. p-Adic L-Functions on the Eigencurve

      • Joël Bellaïche
      Pages 249-261
    4. Solutions and Hints to Exercises

      • Joël Bellaïche
      Pages 287-309
  6. Back Matter

    Pages 311-316

About this book

​This book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental importance in number theory.

For graduate students and newcomers to this field, the book provides a solid introduction to this highly active area of research. For experts, it will offer the convenience of collecting into one place foundational definitions and theorems with complete and self-contained proofs.

Written in an engaging and educational style, the book also includes exercises and provides their solution.

Reviews

“This book represented hope. If I read it carefully, maybe I would finally get to know what they were all talking about, and gain some real insight into what are obviously very important and influential ideas. While I cannot claim to be an expert by now, my first skim through, skipping all the exercises, has provided me with a satisfying foundation, and I found that revisited passages responded well to a second reading to consolidate what I had learned.” (Neil P. Dummigan, Mathematical Reviews, May, 2023)

“Complete proofs (or detailed references) of all statements are given and many exercises (with their solutions or hints) are included, hence the book may be addressed to graduate students working in this beautiful area of number theory and arithmetic algebraic geometry. This is a welcome addition to the literature in a field.” (Andrzej Dąbrowski, zbMATH 1493.11002, 2022)

Authors and Affiliations

  • Department of Mathematics, Brandeis University, Waltham, USA

    Joël Bellaïche

About the author

Joël Bellaïche studied at the École Normale Supérieure of Paris and obtained his PhD at the university of Orsay. He is now professor at Brandeis University. His research interest are in Number Theory, Automorphic Forms, Galois Representations, L-functions, and Algebraic Geometry.

Bibliographic Information

  • Book Title: The Eigenbook

  • Book Subtitle: Eigenvarieties, families of Galois representations, p-adic L-functions

  • Authors: Joël Bellaïche

  • Series Title: Pathways in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-77263-5

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Hardcover ISBN: 978-3-030-77262-8Published: 13 August 2021

  • Softcover ISBN: 978-3-030-77265-9Published: 14 August 2022

  • eBook ISBN: 978-3-030-77263-5Published: 11 August 2021

  • Series ISSN: 2367-3451

  • Series E-ISSN: 2367-346X

  • Edition Number: 1

  • Number of Pages: XI, 316

  • Number of Illustrations: 15 b/w illustrations, 1 illustrations in colour

  • Topics: Number Theory

Buy it now

Buying options

eBook USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 69.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access