The Analytic Theory of Division Fields and Spectral L-functions

除法域和谱L函数的解析理论

• 批准号: 0355564
• 负责人: William Duke
• 金额: $ 40万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355564&HistoricalAwards=false
• 关键词: Analytic; Theory; Division; Fields; Spectral

Abstract for award DMS-0355564 of DukeDuke and his students will work on two broad sets of problems:to develop the analytic theory of division fields of elliptic curves and abelian varietiesand to understand the number-theoretic significance of special valuesof spectral L-functions and their derivatives.The first part may be described as ``nonabelian" analytic numbertheory and is motivated by applications to the mod p reductionsof elliptic curves and abelian varieties. So far progress has beenpossible assuming the Generalized Riemann Hypothesis (GRH), andits removal it one of the many challenges.The second part is motivated by some analogies Duke has observedbetween the generation of number fields by units occurring asspecial values of derivatives of L-functions (e.g. the ``Starkconjectures") and the special values of derivatives of Lfunctions associated to Riemann surfaces. A general goal of the proposal is to extend the reach of analytic number theory,both within number theory at large and also within mathematicsmore generally. It has become evident over the lastfew years that the techniques of analytic number theory have amuch broader applicability than was earlier appreciated.Such techniques also have potential importance in practical applications,especially in cryptography and related areas.

摘要奖DMS-0355564杜克杜克和他的学生将致力于两个广泛的问题：发展的分析理论的划分领域的椭圆曲线和阿贝尔varietiesand了解数论意义的特殊valuesof谱L-functions和他们的derivaties.The第一部分可以被描述为"nonabelian”解析数论和动机的应用modp reductionsof椭圆曲线和阿贝尔varieties。到目前为止，假设广义黎曼假设（GRH）的进展是可能的，它的消除是许多挑战之一。第二部分是由杜克的一些类比激发的，这些类比发生在作为L函数导数的特殊值（例如"斯塔克图”）和与黎曼曲面相关的L函数导数的特殊值的单位之间。 该提案的一个总体目标是扩展解析数论的范围，无论是在数论中，还是在更普遍的代数学中。近几年来，人们发现解析数论的技术比以前认识到的有更广泛的应用，这些技术在实际应用中也有潜在的重要性，特别是在密码学和相关领域。