We construct a descent-of-scalars criterion for $K$-linear abelian categories. Using advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti–Tate groups on complete curves over a finite field. We conjecture that such Barsotti–Tate groups “come from” a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over ${\mathbb{𝔽}}_q$. Along the way we formulate a conjecture on the field-of-coefficients of certain compatible systems.

中文翻译：

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Rank 2 局部系统、Barsotti-Tate 群和 Shimura 曲线

我们构建了一个标量下降标准$ķ$- 线性阿贝尔范畴。使用由于 Abe 导致的 Langlands 对应关系的进步，我们在有限域上的完整曲线上建立了某些 2 级局部系统和某些 Barsotti-Tate 群之间的对应关系。我们推测这样的 Barsotti-Tate 群“来自”一个假椭圆曲线族。作为这些想法的应用，我们提供了一个标准${\mathbb{𝔽}}_q$. 在此过程中，我们对某些兼容系统的系数场进行了猜想。