Arithmetic of Elliptic Curves

椭圆曲线的算术

基本信息

批准号：
9970382
负责人：
Joseph Silverman
金额：
$ 9万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1999
资助国家：
美国
起止时间：
1999-07-01 至 2003-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970382&HistoricalAwards=false
关键词：
Arithmetic Elliptic Curves

项目摘要

9970382Dr. Silverman will conduct research on the arithmetic of elliptic curves. He will do a theoretical analysis of a new algorithm for solving the discrete logarithm problem on elliptic curves and give optimizations for the various parameters required by the algorithm. He will study lower bounds for the canonical height on elliptic curves and prove a Lehmer-type bound for non-reciprocal points. He will investigate the relationship between the Frobenius trace values of an elliptic curve and the rank of the curve so as the study the variation of the rank in families of curves and give a probabilistic quantitative justification for Mestre's method of producing elliptic curves of high rank. All of these projects will significantly advance our knowledge of the arithmetic theory of elliptic curves, an area of mathematics which is of widespread interest in its own right and because of its applicability to number theory and other areas of mathematics.The proposed research project fits into the general framework of understanding the number theoretic properties of elliptic curves. This theory has been extensively studied for a long time, both for its intrinsic beauty and its applicability to other areas of number theory and mathematics. In recent years, the full power of modern arithmetic and algebraic geometry have been brought into play, with excellent results. All three main projects will answer important theoretical and practical questions concerning the arithmetic of elliptic curves and will spur further research in this field.

9970382dr。 Silverman将对椭圆曲线的算术进行研究。他将对新算法进行理论分析，以解决椭圆曲线上的离散对数问题，并对算法所需的各种参数进行优化。他将研究椭圆曲线上规范高度的下边界，并证明对非重点点结合的Lehmer型。他将研究椭圆曲线的Frobenius痕量值与曲线等级之间的关系，以便研究曲线家族中等级的差异，并为Mestre产生高级椭圆形曲线的方法提供了概率的定量理由。所有这些项目都将大大提高我们对椭圆曲线算术理论的了解，椭圆曲线的算术理论是数学领域，它自身具有广泛关注的兴趣，并且由于其适用于数字理论和其他数学领域的适用性。拟议的研究项目适合理解椭圆曲线的数量理论的一般框架。该理论已经进行了很长时间的广泛研究，既是其内在的美丽及其对其他数字理论和数学领域的适用性。近年来，现代算术和代数几何形状的全部力量已发挥作用，结果出色。这三个主要项目将回答有关椭圆曲线算术的重要理论和实用问题，并将刺激该领域的进一步研究。