Number Theory Problems Over Local Fields and Function Fields

局部域和函数域的数论问题

• 批准号: 0600919
• 负责人: Brian Conrad
• 金额: $ 44.07万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600919&HistoricalAwards=false
• 关键词: Number; Theory; Problems; Over; Local

DMS-0600919Brian ConradThe PI proposes to work on some geometric questions that arise in number-theoretic questions over global function fields and p-adic fields. In the theory of overconvergent classical modular forms, the canonical subgroup of Lubin and Katz (for a p-adic analytic family of elliptic curves) has been an important tool. The recent interest in p-adic modular forms beyond the classical case has motivated several different constructions of canonical subgroups for abelian varieties,and recent work of the PI has led to another higher-dimensional theory that, at least in its geometric aspects, has a wider range of applicability than the other approaches. The PI proposes to use deformation-theoretic methods to make explicit certain abstract estimates in the theory, and to draw arithmetic consequences for p-adic families of automorphic forms.In another direction, the PI and co-workers have used methods from deformation theory and rigid-analytic geometry to develop a theory of a new global parity obstruction to randomness properties of prime specialization of inseparable irreducible polynomials overcertain coordinate rings of curves over finite fields. This has given rise to families of elliptic curves with unexpected rank behaviorover global function fields (under some standard conjectures) and seems likely to have further Diophantine applicationsto Mordell--Weil ranks for families of abelian varieties over global function fields. The PI proposes to develop a betterunderstanding of this basic arithmetic phenomenon and to work out some further consequences for ranks in families.Number theory is among the oldest subjects in mathematics, since it is ultimately concerned with the relatively concreteproblem of studying properties of whole numbers. Such problems can take on the form of finding whole number solutionsto systems of equations in many variables, or problems relating to properties of primes, and so on. Typically one has to bring in some deeper structural information (provided by geometric, algebraic, or analytic ideas) to make progress on such questions, and in the last several decades a very wide range of sophisticated geometric techniques have been developed to attack these problems. Moreover, though the initial motivation comes from within pure mathematics, security ofelectronic telecommunications has come to be related in an essential way to many of these number-theoretic problems (such asprime factorization and studying points on curves over finite fields) and the geometric concepts used to attack them. This proposal focuseson both geometric and prime factorization questions that arise in several number-theoretic settings, aiming to develop somenew theoretical methods and to apply them to specific problems. In addition, the PI proposes to continue his long-standing tradition of supervising high-level number theory research by talented high school students and giving talks to larger groups of students at the high school, undergraduate, and graduate levels. He will also complete two books that, together with assorted freely available notes already posted on his web site, will provide useful references for graduate students who wish to learn about some fundamental topics in arithmetic geometry.

DMS-0600919 Brian Conradthe Pi建议研究在整体函数域和p-ady域上的数论问题中出现的一些几何问题。在超收敛经典模型理论中，Lubin和Katz的正则子群(对于p元解析椭圆曲线族)一直是一个重要的工具。最近对经典情形以外的p-进模形式的兴趣促使了关于交换簇的几种不同的典范子群的构造，并且PI最近的工作导致了另一种高维理论，至少在它的几何方面，比其他方法具有更广泛的适用性。PI建议使用形变理论方法对理论中的某些抽象估计进行显式的确定，并给出p元自同构形式族的算术推论。在另一个方向上，PI和他的同事利用形变理论和刚性解析几何的方法发展了一种新的全局奇偶障碍理论，该理论对有限域上不可分不可约多项式的素数特化的随机性性质进行了研究。这产生了在全局函数域上具有意想不到的秩行为的椭圆曲线族(在一些标准猜想下)，并且似乎有可能进一步应用于全局函数域上的阿贝尔变异族的Mordell-Weil秩.PI建议对这一基本算术现象有一个更好的理解，并得出一些关于家庭等级的进一步结果。数论是数学中最古老的学科之一，因为它最终涉及到研究整数性质这一相对具体的问题。这类问题可以采取寻找多变量方程组的整数解的形式，或与素数的性质有关的问题，等等。通常，人们必须引入一些更深层次的结构信息(由几何、代数或分析思想提供)才能在这些问题上取得进展，在过去的几十年里，已经发展了非常广泛的复杂几何技术来解决这些问题。此外，尽管最初的动机来自于纯数学，但电子通信的安全性已经以一种本质的方式与许多这些数论问题(如素数分解和研究有限域上的曲线上的点)以及用来攻击它们的几何概念有关。这项建议集中在几个数论背景下出现的几何和素因式分解问题，旨在发展一些新的理论方法，并将它们应用于具体问题。此外，PI建议继续他的长期传统，即监督有天赋的高中生进行高水平的数论研究，并为更多的高中、本科生和研究生水平的学生进行演讲。他还将完成两本书，连同他网站上已经张贴的各种免费提供的笔记，将为希望学习算术几何基础主题的研究生提供有用的参考。