Analytic Number Theory on Groups
群的解析数论
基本信息
- 批准号:9800048
- 负责人:
- 金额:$ 18.45万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:1998
- 资助国家:美国
- 起止时间:1998-07-01 至 2001-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
9800048GoldfeldThis project involves the study of discrete groups with applications to analytic number theory. The focus is on two specific applications. Modular symbols for congruence subgroups are integrals over closed cycles of a differential one-form, which are, by a theorem of Shimura, integral linear combinations of periods of the Jacobian variety associated to the congruence subgroup. The PI has previously conjectured that such integral coefficients have a polynomial growth in the level and that this implies the abc-conjecure. New types of automorphic forms (twisted by modular symbols) are introduced to study the moments of modular symbols. In particular, the investigation of the second moment may lead to progress in the direction of the abc-conjecture. The second application involves the celebrated Gross-Zagier formula for critical values of L-functions associated to modular forms. Recently, the PI and S. Zhang have found a much simpler and very direct proof of the L-value computation of Gross-Zagier. The central idea is to compute the base change of a holomorphic Poincare series. It is proposed to generalize this proof to the case of Hilbert modular L-series over totally real fields. The ultimate goal is to obtain new results in the direction of the Birch-Swinnerton-Dyer conjecture.This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.
这个项目涉及离散群的研究及其在解析数论中的应用。重点放在两个具体的应用上。同余子群的模符号是微分单形闭圈上的积分,根据Shimura定理,它是与同余子群相关的Jacobian簇的周期的整线性组合。PI以前曾猜想,这样的积分系数在水平上有多项式增长,这意味着ABC猜想。引入了一种新的自同构形式(由模符号扭曲)来研究模符号的矩。特别地,对二阶矩的研究可能导致ABC猜想的进展。第二个应用涉及著名的L临界值的格罗斯-扎吉尔公式--与模形式相关的函数。最近,Pi和Zhang找到了L的一个更简单、更直接的证明--Gross-Zagier的值计算。其中心思想是计算全纯Poincare级数的基变化。将这一证明推广到全实域上的Hilbert模L级数的情形。最终的目标是在Birch-Swinnerton-Dyer猜想的方向上获得新的结果。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究,它解决了一些问题,比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一,出于纯粹的美学原因,人们追寻了许多个世纪。然而,在过去的半个世纪里,它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Dorian Goldfeld其他文献
The functional equations of Langlands Eisenstein series for SL(n, ℤ)
- DOI:
10.1007/s11425-023-2213-y - 发表时间:
2023-10-16 - 期刊:
- 影响因子:1.500
- 作者:
Dorian Goldfeld;Eric Stade;Michael Woodbury - 通讯作者:
Michael Woodbury
Dorian Goldfeld的其他文献
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{{ truncateString('Dorian Goldfeld', 18)}}的其他基金
Analytic Number Theory and its Applications, July 14-18, 2014
解析数论及其应用,2014 年 7 月 14-18 日
- 批准号:
1403383 - 财政年份:2014
- 资助金额:
$ 18.45万 - 项目类别:
Standard Grant
Collaborative Research: Automorphic forms, representations and L-functions
合作研究:自同构形式、表示和 L 函数
- 批准号:
1001036 - 财政年份:2010
- 资助金额:
$ 18.45万 - 项目类别:
Standard Grant
COLLABORATIVE RESEARCH: EMSW21-RTG: JOINT COLUMBIA-CUNY-NYU RESEARCH TRAINING GROUP IN NUMBER THEORY
合作研究:EMSW21-RTG:哥伦比亚大学-纽约市立大学-纽约大学联合数论研究培训小组
- 批准号:
0739400 - 财政年份:2008
- 资助金额:
$ 18.45万 - 项目类别:
Continuing Grant
FRG: Collaborative Research: Combinatorial representation theory, multiple Dirichlet series and moments of L-functions
FRG:协作研究:组合表示理论、多重狄利克雷级数和 L 函数矩
- 批准号:
0652554 - 财政年份:2007
- 资助金额:
$ 18.45万 - 项目类别:
Standard Grant
Collaborative Research: FRG: Applications of Multiple Dirichlet Series to Analytic Number Theory
合作研究:FRG:多重狄利克雷级数在解析数论中的应用
- 批准号:
0354582 - 财政年份:2004
- 资助金额:
$ 18.45万 - 项目类别:
Standard Grant
Joint COLUMBIA-CUNY-NYU Number Theory Seminar
哥伦比亚大学-纽约市立大学-纽约大学联合数论研讨会
- 批准号:
0312270 - 财政年份:2003
- 资助金额:
$ 18.45万 - 项目类别:
Standard Grant
Mathematical Sciences: Analytic Number Theory on Groups
数学科学:群的解析数论
- 批准号:
9505584 - 财政年份:1995
- 资助金额:
$ 18.45万 - 项目类别:
Standard Grant
Mathematical Sciences: Analytic Number Theory on Groups
数学科学:群的解析数论
- 批准号:
9200716 - 财政年份:1992
- 资助金额:
$ 18.45万 - 项目类别:
Continuing Grant
Mathematical Sciences: Analytic Number Theory on Elliptic Curves
数学科学:椭圆曲线的解析数论
- 批准号:
9003907 - 财政年份:1990
- 资助金额:
$ 18.45万 - 项目类别:
Continuing Grant
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职业:解析数论的研究和途径
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- 批准号:
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