Mathematical Sciences: Number Theory, Arithmetic Geometry, and Transcendence

数学科学：数论、算术几何和超越性

• 批准号: 8805216
• 负责人: Hugh Montgomery
• 金额: $ 44.2万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8805216&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Number; Theory; Arithmetic

This research is in many areas of number theory. Montgomery will pursue a number of basic questions in analytic number theory and related areas of harmonic analysis, diophantine approximation and the geometry of numbers. Most of these questions involve sets of numbers or vectors which are almost independent. The object is to exploit the almost independence appropriately in order to achieve the desired aim. Masser will study some diophantine problems which lend themselves into the following groupings: (a) using transcendence techniques to examine isomorphism classes of elliptic curves; (b) to obtain upper bounds for the number of points of small height on a fixed abelian variety; (c) to establish precise conditions for the algebraic independence of values of certain Mahler-type power series at algebraic points; and (d) to study the multiplicities of linear recurrences for a finite field. Milne's research object is to produce a comprehensive set of reciprocity laws describing how automorphisms of the complex numbers act on automorphic functions automorphic forms, their special values, their "Fourier-Jacobi series, and the Eisenstein series attached to cusp forms on boundary components. Keating's research object is to understand Igusa curves better by studying their function fields including applications to coding theory and factorization theory. He proposes to use Drinfeld's theory of extensions of function fields to produce a more explicit characterization of the function fields. This research is in very broad areas of number theory, the study of the properties of the integers. Montgomery's focus is on that part of the subject where classical analysis is used to understand these deep questions. Masser's research concentrates on using transcendence techniques (again analytic but of a very different type than Montgomery's techniques) to study problems concerning solutions of equations in integers. Milne combines algebraic and modern analytic techniques to study special functions that arise in number theory. Keating studies special domains that are important in number theory.

这项研究涉及数论的许多领域。 Montgomery 我将探讨解析数论中的一些基本问题 以及调和分析、丢番图逼近等相关领域 和数字的几何学。 这些问题大多涉及 几乎独立的数或向量的集合。 的 目的是适当地利用几乎独立性， 为了达到预期的目的。 Masser将研究一些 丢番图的问题，这借给自己到以下 （a）使用超越技术， 椭圆曲线的同构类;（B）得到上 上小高度点的数目的界限 阿贝尔品种;（c）建立精确的条件， Mahler型幂值的代数无关性 级数在代数点;及（d）研究多重性 一个有限域上的线性递归 Milne的研究 目标是制定一套全面的互惠法 描述复数的自同构如何作用于 自守函数自守形式，它们的特殊值， 他们的"傅里叶-雅可比级数，和爱森斯坦级数， 在边界分量上形成尖点。 基廷的研究对象 是通过研究伊古萨曲线的功能 包括编码理论和因子分解的应用 理论 他建议使用Drinfeld的扩展理论， 函数字段以产生更明确的 功能字段。 这项研究是在非常广泛的领域数论， 研究整数的性质。 蒙哥马利的重点是 在这个问题上，经典分析被用来 理解这些深刻的问题。 Masser的研究集中在 关于使用超越技术（再次分析，但非常 不同于蒙哥马利的技术）来研究问题 关于整数方程的解。 米尔恩联合公司 代数和现代分析技术来研究特殊的 在数论中出现的函数 基廷研究特别 在数论中很重要的领域。