Canonical Vector Heights on Algebraic $\text{K}3$ Surfaces with Picard Number Two
Canonical Vector Heights on Algebraic $\text{K}3$ Surfaces with Picard Number Two
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皮卡德二号代数 $ ext{K}3$ 曲面上的规范向量高度
DOI：
10.4153/cmb-2003-048-x
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发表时间：
2003
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影响因子：
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通讯作者：
A. Baragar
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文献类型：
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作者：
A. Baragar
设$V$是定义在数域$K$上的代数$\Text{K}3$曲面。设$V$有Picard数2和一个无限的自同构群$\mathcal{A}\，=\，\Text{Aut(}V/K\Text{)}$。本文引入了向量高度$\mathbf{h}：\，V\，\to\，\Text{Pic(}V\Text{)}\，\oTimes\，\mathbb{R}的概念，并证明了具有如下性质的标准向量高度$\mathbf{\hat{h}}$的存在：$$\widehat{\mathbf{h}}\，\Left(\sigma P\right)\，=\，{{\sigma}_{*}}\宽帽子{\mathbf{h}}\左侧(P\右侧)${{h}_{D}}(P)\，=\，\mathbf{\hat{h}}(P)\，\cot\，D\，+\，O(1)，$$其中$\sigma\，\in\，\mathcal{A}，{{\sigma}_{*}}$是$\sigma$的推进(${{\sigma}^{-1}}$)，${{h}_{D}}$是与除数$D$相关的Weil高度。$O(1)$所隐含的有界函数不依赖于$P$。这使我们可以解决一些算术问题。例如，证明了数学{A}-轨道上对数高度有界的有理点的个数满足${N}_{数学{A}(P)}}(t，\，D)\，=\，\#\{Q\，\in\，\数学{A}(P)\，：\，{{h}_{D}}(Q)，<；，t，=，Frc{\Mu(P)}{S\，\log\，\omega}\，\log t\，+\，O\Left(\log\Left(\mathbf{\hat{h}}(P)\，\CDot\，D\，+\，2\Right)。$$这里，$\Mu(P)$是非负整数，$S$是正整数，$\omega$是实二次基本单位。
Abstract Let $V$ be an algebraic $\text{K}3$ surface defined over a number field $K$ . Suppose $V$ has Picard number two and an infinite group of automorphisms $\mathcal{A}\,=\,\text{Aut(}V/K\text{)}$ . In this paper, we introduce the notion of a vector height $\mathbf{h}:\,V\,\to \,\text{Pic(}V\text{)}\,\otimes \,\mathbb{R}$ and show the existence of a canonical vector height $\mathbf{\hat{h}}$ with the following properties: $$\widehat{\mathbf{h}}\,\left( \sigma P \right)\,=\,{{\sigma }_{*}}\widehat{\mathbf{h}}\left( P \right)$$ $${{h}_{D}}(P)\,=\,\mathbf{\hat{h}}(P)\,\cdot \,D\,+\,O(1),$$ where $\sigma \,\in \,\mathcal{A},\,{{\sigma }_{*}}$ is the pushforward of $\sigma $ (the pullback of ${{\sigma }^{-1}}$ ), and ${{h}_{D}}$ is a Weil height associated to the divisor $D$ . The bounded function implied by the $O(1)$ does not depend on $P$ . This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an $\mathcal{A}$ -orbit satisfies $${{N}_{\mathcal{A}(P)}}(t,\,D)\,=\,\#\{Q\,\in \,\mathcal{A}(P)\,:\,{{h}_{D}}(Q)\,<\,t\}\,=\,\frac{\mu (P)}{s\,\log \,\omega }\,\log t\,+\,O\left( \log \left( \mathbf{\hat{h}}(P)\,\cdot \,D\,+\,2 \right) \right).$$ Here, $\mu (P)$ is a nonnegative integer, $s$ is a positive integer, and $\omega $ is a real quadratic fundamental unit.