Let \(X\) be a smooth elliptic fibration over a smooth base \(B\). Under mild assumptions, the authors establish a Fourier-Mukai 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 Strominger-Yau-Zaslow version of mirror symmetry, to twisted sheaves, and to non-commutative geometry.