Arithmetic Algebraic Geometry

Elliptic curves, modular forms, and applications, Joe P. Buhler: elliptic curves; points on elliptic curves; elliptic curves over $\mathbf$; modular forms of level 1; L-series - modular forms of higher level; $l$-adic representations; the rank of elliptic curves over $\mathbfQ$; applications of elliptic curves. Open questions in arithmetic algebraic geometry, Alice Silverberg: overview; torsion subgroups; ranks; conjectures of Birch and Swinnerton-Dyer; ABC and related conjectures; some other conjectures. Lectures on Serre's conjectures, Kenneth A. Ribet and William A. Stein: 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. Deformations of Galois representations, Fernando Q. Gouvea: 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; a criterion for existence of a universal deformation ring; an overview of a theorem of Flach; an introduction to the $p$-adic geometry of modular curves. Introduction to Iwasawa theory for elliptic curves, Ralph Greenberg: Mordell-Weil groups; Selmer groups; $\Lambda$-modules; Mazur's control theorem. Galois cohomology, John Tate: Galois cohomology. The arithmetic of modular forms, Wen-Ching Winnie Li: 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. Arithmetic of certain Calabi-Yau varieties and mirror symmetry, Noriko Yui: 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.