Elliptic Curves and Arithmetic Invariants

1. Front Matter

   Pages i-xviii

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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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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.

• Hecke algebra
• Shimura variety
• arithmetic invariants
• elliptic curves
• modular forms
• scheme theory

“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)

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

  Haruzo Hida

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

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