{"title":"-因子的椭圆曲面与交点","authors":"Laura Demarco, Niki Myrto Mavraki","doi":"10.4171/jems/1354","DOIUrl":null,"url":null,"abstract":"Suppose $\\mathcal{E} \\to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\\overline{\\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\\mathbb{R}$-divisors $\\overline{D}_X$ on the base curve $B$ over a number field, for each $X \\in E(k)\\otimes \\mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\\overline{D}_X\\cdot \\overline{D}_Y$, as a biquadratic form on $E(k)\\otimes \\mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N\\'eron-Tate height functions on the fibers $E_t(\\overline{\\mathbb{Q}})$ of $\\mathcal{E}$ over $t \\in B(\\overline{\\mathbb{Q}})$: given points $P_1, \\ldots, P_m \\in E(k)$ with $m\\geq 2$, there exist an infinite sequence $t_n\\in B(\\overline{\\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' \\in E_{t_n}(\\overline{\\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\\{P_{1, t_n}', \\ldots, P_{m,t_n}'\\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, \\ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $\\mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Elliptic surfaces and intersections of adelic $\\\\mathbb{R}$-divisors\",\"authors\":\"Laura Demarco, Niki Myrto Mavraki\",\"doi\":\"10.4171/jems/1354\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Suppose $\\\\mathcal{E} \\\\to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\\\\overline{\\\\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\\\\mathbb{R}$-divisors $\\\\overline{D}_X$ on the base curve $B$ over a number field, for each $X \\\\in E(k)\\\\otimes \\\\mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\\\\overline{D}_X\\\\cdot \\\\overline{D}_Y$, as a biquadratic form on $E(k)\\\\otimes \\\\mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N\\\\'eron-Tate height functions on the fibers $E_t(\\\\overline{\\\\mathbb{Q}})$ of $\\\\mathcal{E}$ over $t \\\\in B(\\\\overline{\\\\mathbb{Q}})$: given points $P_1, \\\\ldots, P_m \\\\in E(k)$ with $m\\\\geq 2$, there exist an infinite sequence $t_n\\\\in B(\\\\overline{\\\\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' \\\\in E_{t_n}(\\\\overline{\\\\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\\\\{P_{1, t_n}', \\\\ldots, P_{m,t_n}'\\\\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, \\\\ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $\\\\mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2020-12-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/jems/1354\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jems/1354","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors
Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors $\overline{D}_X$ on the base curve $B$ over a number field, for each $X \in E(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadratic form on $E(k)\otimes \mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N\'eron-Tate height functions on the fibers $E_t(\overline{\mathbb{Q}})$ of $\mathcal{E}$ over $t \in B(\overline{\mathbb{Q}})$: given points $P_1, \ldots, P_m \in E(k)$ with $m\geq 2$, there exist an infinite sequence $t_n\in B(\overline{\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\{P_{1, t_n}', \ldots, P_{m,t_n}'\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, \ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $\mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.