A $1$-dimensional formal group over the prismatization of $\operatorname{Spf}\:\mathbb{Z}_p$

Pub Date : 2024-03-26 DOI:10.4310/pamq.2024.v20.n1.a7
Vladimir Drinfeld
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Abstract

Let $\Sigma$ denote the prismatization of $\operatorname{Spf}\:\mathbb{Z}_p$. The multiplicative group over $\Sigma$ maps to the prismatization of $\mathbb{G}_m \times \operatorname{Spf}\:\mathbb{Z}_p$. We prove that the kernel of this map is the Cartier dual of some $1$-dimensional formal group over $\Sigma$. We obtain some results about this formal group (e.g., we describe its Lie algebra). We give a very explicit description of the pullback of the formal group to the quotient stack $Q/\mathbb{Z}^\times_p$, where $Q$ is the $q$-de Rham prism.
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$operatorname{Spf}\:\mathbb{Z}_p$棱镜化上的1$维形式群
让 $\Sigma$ 表示 $\operatorname{Spf}\:\mathbb{Z}_p$ 的棱镜化。$\Sigma$ 上的乘法群映射到 $\mathbb{G}_m \times \operatorname{Spf}\:\mathbb{Z}_p$ 的棱柱化。我们证明这个映射的内核是某个超过 $\Sigma$ 的 1$ 维形式群的卡蒂埃对偶。我们得到了关于这个形式群的一些结果(例如,我们描述了它的李代数)。我们给出了形式群对商栈 $Q/\mathbb{Z}^\times_p$ 的拉回的非常明确的描述,其中 $Q$ 是 $q$-de Rham 棱镜。
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