Arithmetic of Elliptic Curves

椭圆曲线的算术

• 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

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.

9970382博士西尔弗曼将进行椭圆曲线算法的研究。他将对解决椭圆曲线上离散对数问题的新算法进行理论分析，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界，并证明非互易点的莱默型界限。 他将研究椭圆曲线的 Frobenius 迹值与曲线的秩之间的关系，从而研究曲线族中的秩的变化，并为 Mestre 产生高秩椭圆曲线的方法提供概率定量论证。 所有这些项目都将极大地增进我们对椭圆曲线算术理论的了解，椭圆曲线算术理论是一个因其本身而受到广泛关注的数学领域，因为它适用于数论和其他数学领域。拟议的研究项目适合理解椭圆曲线数论性质的一般框架。 该理论因其内在美及其对数论和数学其他领域的适用性而被广泛研究了很长时间。近年来，现代算术和代数几何的力量得到充分发挥，取得了优异的成绩。 所有三个主要项目都将回答有关椭圆曲线运算的重要理论和实践问题，并将推动该领域的进一步研究。