{"title":"椭圆曲线的最小复盖模型","authors":"T. Fisher","doi":"10.1112/S1461157014000217","DOIUrl":null,"url":null,"abstract":"In this paper we give a new formula for adding $2$ -coverings and $3$ -coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$ -coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"112-127"},"PeriodicalIF":0.0000,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000217","citationCount":"0","resultStr":"{\"title\":\"Minimal models for -coverings of elliptic curves\",\"authors\":\"T. Fisher\",\"doi\":\"10.1112/S1461157014000217\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we give a new formula for adding $2$ -coverings and $3$ -coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$ -coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.\",\"PeriodicalId\":54381,\"journal\":{\"name\":\"Lms Journal of Computation and Mathematics\",\"volume\":\"17 1\",\"pages\":\"112-127\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2014-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1112/S1461157014000217\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Lms Journal of Computation and Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1112/S1461157014000217\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lms Journal of Computation and Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1112/S1461157014000217","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
In this paper we give a new formula for adding $2$ -coverings and $3$ -coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$ -coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.
期刊介绍：
LMS Journal of Computation and Mathematics has ceased publication. Its final volume is Volume 20 (2017). LMS Journal of Computation and Mathematics is an electronic-only resource that comprises papers on the computational aspects of mathematics, mathematical aspects of computation, and papers in mathematics which benefit from having been published electronically. The journal is refereed to the same high standard as the established LMS journals, and carries a commitment from the LMS to keep it archived into the indefinite future. Access is free until further notice.
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