Arithmetical Algebraic Geometry

算术代数几何

• 批准号: 0070839
• 负责人: Douglas Ulmer
• 金额: $ 6万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070839&HistoricalAwards=false
• 关键词: Arithmetical; Algebraic; Geometry

Dr. Ulmer proposes three projects in arithmetical algebraicgeometry,related to Galois representations, modular forms, andelliptic curves,both over number fields and over function fields. The firstprojectproposed is to study the reduction modulo a prime p ofcertainrepresentations of the Galois group of the p-adic numbers,using therecent work of Colmez and Fontaine on p-adic Hodge theory. In thesecond project, Dr. Ulmer has constructed a subgroup of thelocalpoints at suitable places on an elliptic curve over afunction field; this subgroup contains the global points. He proposes to use these localpoints to study the conjecture of Birch and Swinnerton-Dyerforelliptic curves over function fields. The third project Dr.Ulmerproposes is to study a new class of problems in thecohomology ofvarieties which are inspired by classical non-vanishingresults forL-series. The new questions come by reinterpretingvanishing resultsusing Grothendieck's analysis of L-functions and lead topurelygeometric questions. In some instances these questions canbe treatedusing monodromy results of Katz and Sarnak.This proposal falls into the general area of arithmeticalalgebraicgeometry - a subject that blends two of the oldest areas ofmathematics: number theory and geometry. This combinationhas provedextraordinarily fruitful, having recently solved problemsthatwithstood the efforts of generations. Among its manyconsequences arenew error correcting codes which are used in computerstorage deviceslike compact disks and hard drives and secure informationtransmissionschemes which are used for financial transactions on theinternet.

Ulmer博士提出了算术代数测定学的三个项目，这些项目与Galois表示，模块化形式和ellipriptic曲线有关，包括数字字段和过度功能字段。 第一个项目验证是使用科尔梅斯和丰丹对P-Adic Hodge理论的P-Adic数字的GALOIS组的最佳表现。在第三个项目中，乌尔默博士在腹部椭圆形曲线的合适位置建造了一个小组。该子组包含全球点。他建议使用这些局部点来研究功能场上桦木和Swinnerton-Dyerforelliptic曲线的猜想。 第三个项目杜尔默植物博士是研究宇宙学的新问题，这些问题受到经典的非狂欢派系的启发。 新问题是通过重新解释了Grothendieck对L功能和铅型几何问题的分析的结果。 在某些情况下，这些问题可以治疗Katz和Sarnak的单肌结果。该提案属于算术杂志的一般领域 - 该主题融合了两个最古老的掌握区域：数字理论和几何形状。 最近解决了几代人的努力，这一组合证明了富有成果的卓有成效。 在其许多后果中，在计算机设备般的紧凑型磁盘和硬盘驱动器中使用的校正校正代码以及安全的信息传输化学，这些磁盘用于TheNerternet上的财务交易。