IAS/Park City Mathematics Series 2001; 569 pp; softcover Volume: 9 ISBN10: 0821844482 ISBN13: 9780821844489 List Price: US$88 Member Price: US$70.40 Order Code: PCMS/9.S
 The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry. Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price. Readership Graduate students and research mathematicians interested in arithmetic algebraic geometry. Reviews "The book ... gives a good overview of the subject and proceeds naturally to more technical aspects of the theory. An attractive feature of the book is the presence of many exercises for students."  European Mathematical Society Newsletter Table of Contents Joe P. Buhler, Elliptic curves, modular forms, and applications  Preface
 Elliptic curves
 Points on elliptic curves
 Elliptic curves over \(\mathbf C\)
 Modular forms of level 1
 Lseries; Modular forms of higher level
 \(l\)adic representations
 The rank of elliptic curves over \(\mathbf Q\)
 Applications of elliptic curves
 Bibliography
Alice Silverberg, Open questions in arithmetic algebraic geometry  Overview
 Torsion subgroups
 Ranks
 Conjectures of Birch and SwinnertonDyer
 ABC and related conjectures
 Some other conjectures
 Bibliography
Kenneth A. Ribet and William A. Stein, Lectures on Serre's conjectures  Preface
 Introduction to Serre's conjecture
 Optimizing the weight
 Optimizing the level
 Exercises
 Appendix by Brian Conrad: The Shimura construction in weight 2
 Appendix by Kevin Buzzard: A mod \(\ell\) multiplicity one result
 Bibliography
Fernando Q. Gouvêa, Deformations of Galois representations  Introduction
 Galois groups and their representations
 Deformations of representations
 The universal deformation: existence
 The universal deformation: properties
 Explicit deformations
 Deformations with prescribed properties
 Modular deformations
 \(p\)adic families and infinite ferns
 Appendix 1 by Mark Dickinson: A criterion for existence of a universal deformation ring
 Appendix 2 by Tom Weston: An overview of a theorem of Flach
 Appendix 3 by Matthew Emerton: An introduction to the \(p\)adic geometry of modular curves
 Bibliography
Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves  Preface
 MordellWeil groups
 Selmer groups
 \(\Lambda\)modules
 Mazur's control theorem
 Bibliography
John Tate, Galois cohomology  Galois cohomology
 Bibliography
WenChing Winnie Li, The arithmetic of modular forms  Introduction
 Introduction to elliptic curves, modular forms, and CalabiYau varieties
 The arithmetic of modular forms
 Connections among modular forms, elliptic curves, and representations of Galois groups
 Bibliography
Noriko Yui, Arithmetic of certain CalabiYau varieties and mirror symmetry  Introduction
 The modularity conjecture for rigid CalabiYau threefolds over the field of rational numbers
 Arithmetic of orbifold CalabiYau varieties over number fields
 \(K3\) surfaces, mirror moonshine phenomenon
 Bibliography
