---

This introduction to Shimura varieties covers key topics including non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; elliptic and modular curves over rings and more.

---

This book contains a detailed account of the result of the author´s recent Annals paper and JAMS paper on arithmetic invariant, including mi -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.

---

1 Non-triviality of Arithmetic Invariants.- 2 Elliptic Curves and Modular Forms.- 3 Invariants, Shimura Variety and Hecke Algebra.- 4 Review of Scheme Theory.- 5 Geometry of Variety.- 6 Elliptic and Modular Curves over Rings.- 7 Modular Curves as Shimura Variety.- 8 Non-vanishing Modulo p of Hecke L -values.- 9 p -Adic Hecke L -functions and their mi -invariants.- 10 Toric Subschemes in a Split Formal Torus.- 11 Hecke Stable Subvariety is a Shimura Subvariety.- References.- Symbol Index.- Statement Index.- Subject Index.

---

"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 Dabrowski, zbMATH, Vol. 1284, 2014)

---

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

---