Number Theory Problems Over Local Fields and Function Fields

DMS-0600919Brian Conrad PI 建议研究全局函数域和 p 进数域的数论问题中出现的一些几何问题。 在过收敛经典模形式的理论中，Lubin 和 Katz 的规范子群（对于椭圆曲线的 p 进解析族）一直是一个重要的工具。最近对经典情况之外的 p-adic 模形式的兴趣激发了阿贝尔簇的规范子群的几种不同构造，并且 PI 最近的工作导致了另一种更高维的理论，至少在其几何方面，它比其他方法具有更广泛的适用性。 PI建议使用变形理论方法在理论中做出明确的某些抽象估计，并为自守形式的p进数族得出算术结果。在另一个方向上，PI和他的同事使用变形理论和刚性解析几何的方法开发了一种新的全局宇称阻碍理论，该理论阻碍了有限域上曲线的某些坐标环上不可分的不可约多项式的素数特化的随机性特性。 这导致了椭圆曲线族在全局函数域上具有意想不到的排序行为（在某些标准猜想下），并且似乎可能对莫德尔（Mordell）有进一步的丢番图应用——全局函数域上阿贝尔簇的魏尔排序。 PI 提议更好地理解这一基本算术现象，并找出对家庭等级的进一步影响。数论是数学中最古老的学科之一，因为它最终涉及研究整数性质的相对具体问题。 此类问题可以表现为寻找多变量方程组的整数解，或与素数性质相关的问题，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）才能在此类问题上取得进展，并且在过去几十年中，已经开发了一系列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，但电子电信的安全性已经与许多这些数论问题（例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在本质上相关。 该提案重点关注几种数论环境中出现的几何和素因数分解问题，旨在开发一些新的理论方法并将其应用于特定问题。 此外，PI 建议继续其长期以来的传统，即监督有才华的高中生的高水平数论研究，并为更多的高中、本科生和研究生群体进行演讲。 他还将完成两本书，连同已经发布在他的网站上的各种免费笔记，将为希望学习算术几何中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.