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 *-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.

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

Autorentext

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

Inhalt
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 Lvalues.- 9 p-Adic Hecke L-functions and their **-invariants.- 10 Toric Subschemes in a Split Formal Torus.- 11 Hecke Stable Subvariety is a Shimura Subvariety.- References.- Symbol Index.- Statement Index.- Subject Index.