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.