Number Theory Problems Over Local Fields and Function Fields

brian conradpi提出在全局函数域和p进域上研究数论问题中出现的一些几何问题。在经典模形式的超收敛理论中，Lubin和Katz的正则子群（关于椭圆曲线的p进解析族）是一个重要的工具。最近对超越经典情况的p进模形式的兴趣激发了阿贝尔变体正则子群的几种不同构造，PI的最近工作导致了另一种高维理论，至少在其几何方面，比其他方法具有更广泛的适用性。PI提出用变形理论的方法来明确理论中的某些抽象估计，并给出自同构形式的p进族的算术结果。在另一个方向上，PI及其同事利用变形理论和刚性解析几何的方法，发展了一种新的全局奇偶阻碍理论，以阻止有限域上不可分割的不可约多项式在曲线坐标环上的素专化的随机性。这导致了椭圆曲线族在全局函数域上（在一些标准猜想下）具有意想不到的秩行为，并且似乎可能有进一步的抛世应用到全局函数域上的阿贝尔变种族的莫德尔-韦尔秩。PI建议更好地理解这一基本的算术现象，并进一步计算出家庭等级的一些结果。数论是数学中最古老的学科之一，因为它最终关注的是研究整数性质的相对具体的问题。这类问题可以采取寻找多变量方程组的整数解的形式，或者与素数性质有关的问题，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）才能在这类问题上取得进展，在过去的几十年里，已经开发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纯数学，但电子通信的安全性已经以一种基本的方式与许多这些数论问题（如质因数分解和研究有限域上曲线上的点）以及用于攻击它们的几何概念相关。本提案主要针对在数论环境中出现的几何和质因数分解问题，旨在发展一些新的理论方法并将其应用于具体问题。此外，PI还提议继续他长期以来的传统，即指导有才华的高中生进行高水平的数论研究，并向更多的高中生、本科生和研究生讲课。他还将完成两本书，连同各种免费提供的笔记已经张贴在他的网站上，将为希望学习算术几何一些基本主题的研究生提供有用的参考。

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.