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.
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. Gouvea, 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.