Memoirs of the American Mathematical Society 2008; 90 pp; softcover Volume: 193 ISBN10: 0821840924 ISBN13: 9780821840924 List Price: US$64 Individual Members: US$38.40 Institutional Members: US$51.20 Order Code: MEMO/193/901
 Let \(X\) be a smooth elliptic fibration over a smooth base \(B\). Under mild assumptions, the authors establish a FourierMukai equivalence between the derived categories of two objects, each of which is an \(\mathcal{O}^{\times}\) gerbe over a genus one fibration which is a twisted form of \(X\). The roles of the gerbe and the twist are interchanged by the authors' duality. The authors state a general conjecture extending this to allow singular fibers, and they prove the conjecture when \(X\) is a surface. The duality extends to an action of the full modular group. This duality is related to the StromingerYauZaslow version of mirror symmetry, to twisted sheaves, and to noncommutative geometry. Table of Contents  Introduction
 The Brauer group and the TateShafarevich group
 Smooth genus one fibrations
 Surfaces
 Modified \(T\)duality and the SYZ conjecture
 Appendix A. Duality for representations of \(1\)motives, by Dmitry Arinkin
 Bibliography
