The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. Andre proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. In this paper, we prove a certain generalization of their results, which is stated by the unramifiedness of l-adic etale cohomology groups for K3 surfaces over finitely generated fields of characteristic 0. As a corollary, we get the original Shafarevich conjecture for K3 surfaces without assuming the extendability of polarization, which is stronger than the results of Andre and She. Moreover, as an application, we get the finiteness of twists of K3 surfaces via a finite extension of characteristic 0 fields.

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关于 K3 曲面的 Shafarevich 猜想的上同调推广

K3 曲面的 Shafarevich 猜想断言了 K3 曲面在固定数量域上的同构类的有限性，允许从有限位置的固定有限集进行良好的归约。安德烈对固定度数的极化 K3 面证明了这个猜想，最近她又对未指定度数的极化 K3 面证明了这一猜想。在本文中，我们证明了他们的结果的某种概括，这由在特征为 0 的有限生成域上 K3 曲面的 l-adic etale 上同调群的非分支性表示。作为推论，我们得到了 K3 曲面的原始 Shafarevich 猜想不假设极化的延展性，这比 Andre 和 She 的结果要强。此外，作为应用程序，