Number Theory and Algebraic Geometry

数论与代数几何

• 批准号: 0200687
• 负责人: Noam Elkies
• 金额: $ 12.05万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200687&HistoricalAwards=false
• 关键词: Number; Theory; Algebraic; Geometry

Abstract Noam Elkies The investigator studies problems in number theory, notablythe arithmetic of varieties such as elliptic curves and surfacesand their relations and applications to other areas of mathematicssuch as error-correcting codes and Euclidean lattices. Specificprojects include: proof and computational exploitation of newrelations between p-adic modular forms of weight 3/2 and the arithmeticof quadratic twists of odd analytic rank; refinement of the Shioda-Usuitheory relating Mordell-Weil lattices with Weyl groups, andgeneralization to certain complex reflection groups; continued studyof optimal recursive towers and the investigator's modularity conjecturefor such towers; extended study of the connection between bilinear(Somos) recurrences, theta sequences, and explicit moduli spaces;and novel applications of lattice reduction to study and efficientlycompute solutions of Diophantine equations and inequalities.The investigator continues his study of problems in number theoryconcerning the mathematical structure of algebraic solutionsof equations. This should lead both to better understandingof those equations and their solutions (elliptic curves, etc.),and to further connections with other topics such as efficientcomputation, error-correcting codes, and sphere packing.

研究数论中的问题，特别是椭圆曲线和曲面等变量的算术，以及它们与纠错码和欧几里德格等几何学其他领域的关系和应用。 具体项目包括：权为3/2的p-adic模形式与奇解析秩的二次扭运算之间的新关系的证明和计算利用;将Mordell-Weil格与Weyl群联系起来的Shioda-Weil理论的改进，以及对某些复反射群的推广;最佳递归塔的继续研究和这种塔的研究者的模性定理;扩展研究双线性（Somos）递归，θ序列和显式模空间之间的联系;和新的应用格减少研究和有效地计算解决方案的丢番图方程和不等式。调查继续他的研究问题的数论有关的代数解的数学结构。 这将有助于更好地理解这些方程及其解（椭圆曲线等），并进一步与其他主题，如有效计算，纠错码，球包装的联系。