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Arithmetic Algebraic Geometry
Edited by: Brian Conrad, University of Michigan, Ann Arbor, MI, and Karl Rubin, Stanford University, CA
A co-publication of the AMS and IAS/Park City Mathematics Institute.
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IAS/Park City Mathematics Series
2001; 569 pp; softcover
Volume: 9
ISBN-10: 0-8218-4448-2
ISBN-13: 978-0-8218-4448-9
List Price: US$83
Member Price: US$66.40
Order Code: PCMS/9.S
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Elliptic Curves, Modular Forms, and Their L-functions - Alvaro Lozano-Robledo

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 co-published 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

  • Introduction
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
  • L-series; 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 Swinnerton-Dyer
  • 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
  • Mordell-Weil groups
  • Selmer groups
  • \(\Lambda\)-modules
  • Mazur's control theorem
  • Bibliography
John Tate, Galois cohomology
  • Galois cohomology
  • Bibliography
Wen-Ching Winnie Li, The arithmetic of modular forms
  • Introduction
  • Introduction to elliptic curves, modular forms, and Calabi-Yau varieties
  • The arithmetic of modular forms
  • Connections among modular forms, elliptic curves, and representations of Galois groups
  • Bibliography
Noriko Yui, Arithmetic of certain Calabi-Yau varieties and mirror symmetry
  • Introduction
  • The modularity conjecture for rigid Calabi-Yau threefolds over the field of rational numbers
  • Arithmetic of orbifold Calabi-Yau varieties over number fields
  • \(K3\) surfaces, mirror moonshine phenomenon
  • Bibliography
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