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Elliptic Curves and Arithmetic Invariants

  • Book
  • © 2013

Overview

Authors:
  1. Haruzo Hida

    1. Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

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  • Contains top-notch research that will interest both experts and advanced graduate students
  • Written by an expert renowned for his discovery that modular forms fall into families, otherwise known as "Hida families"
  • Limits material to elliptic modular curves and the corresponding Shimura curves in order to make the book more accessible to graduate students?
  • Includes many exercises, examples, and applications that provide motivation for the reader
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xviii
    Download chapter PDF

  2. Nontriviality of Arithmetic Invariants

    • Haruzo Hida
    Pages 1-42

  3. Elliptic Curves and Modular Forms

    • Haruzo Hida
    Pages 43-82

  4. Invariants, Shimura Variety, and Hecke Algebra

    • Haruzo Hida
    Pages 83-144

  5. Review of Scheme Theory

    • Haruzo Hida
    Pages 145-216

  6. Geometry of Variety

    • Haruzo Hida
    Pages 217-224

  7. Elliptic and Modular Curves over Rings

    • Haruzo Hida
    Pages 225-279

  8. Modular Curves as Shimura Variety

    • Haruzo Hida
    Pages 281-334

  9. Nonvanishing Modulo p of Hecke L-Values

    • Haruzo Hida
    Pages 335-365

  10. p-Adic Hecke L-Functions and Their μ-Invariants

    • Haruzo Hida
    Pages 367-386

  11. Toric Subschemes in a Split Formal Torus

    • Haruzo Hida
    Pages 387-403

  12. Hecke Stable Subvariety Is a Shimura Subvariety

    • Haruzo Hida
    Pages 405-426

  13. Back Matter

    Pages 427-449
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Keywords

About this book

This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-invariant, L-invariant, and similar topics.   This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties.  Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader.  Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory.  Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

Reviews

“The main aim of the book is to give an account of Hida’s results on arithmetic invariants in an accessible way. … The book is intended for mathematicians with some background on modular forms and is worthwhile for both graduate students and experts. … There are numerous examples, exercises, and remarks, all aimed at carefully helping the reader. In conclusion, this book is a very welcome addition to the mathematical literature.” (Florian Sprung, Mathematical Reviews, April, 2015)

“The author gives in this book a detailed account of results concerning arithmetic invariants, including µ-invariant and L-invariant. … it contains a detailed account of the author’s recent results concerning arithmetic invariants. The book, addressed to advanced graduate students and experts working in number theory and arithmetic geometry, is a welcome addition to this beautiful and difficult area of research.” (Andrzej Dąbrowski, zbMATH, Vol. 1284, 2014)

Authors and Affiliations

  • Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

    Haruzo Hida

About the author

Haruzo Hida is currently a professor of mathematics at University of California, Los Angeles.

Bibliographic Information

  • Book Title: Elliptic Curves and Arithmetic Invariants

  • Authors: Haruzo Hida

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4614-6657-4

  • Publisher: Springer New York, NY

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

  • Copyright Information: Springer Science+Business Media New York 2013

  • Hardcover ISBN: 978-1-4614-6656-7Published: 09 June 2013

  • Softcover ISBN: 978-1-4899-9092-1Published: 08 February 2015

  • eBook ISBN: 978-1-4614-6657-4Published: 13 June 2013

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XVIII, 450

  • Topics: Number Theory, Algebraic Geometry

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