We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners, which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.

中文翻译：

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椭圆 K3 曲面和雅可比行列式的导出等价

我们对 K3 表面的 Fourier-Mukai 伙伴上的椭圆纤维振动进行了详细研究，我们将其称为派生椭圆结构。我们根据 Hodge 理论数据对导出的椭圆结构进行完全分类，类似于描述 Fourier-Mukai 伙伴的导出 Torelli 定理。在皮卡德二级中，导出的椭圆结构完全由判别群的拉格朗日子群决定。因此，我们证明对于一大类 Picard 2 阶椭圆 K3 曲面，所有 Fourier-Mukai 伙伴都是雅可比行列式，并且我们部分地将这个结果扩展到非闭域。我们还证明存在具有 Fourier-Mukai 伙伴的椭圆 K3 曲面，它们不是原始 K3 曲面的雅可比行列式。这对哈塞特和钦克尔提出的问题给出了否定的答案。