CAREER: Genus one curves: rational points and arithmetic statistics

职业：属一曲线：有理点和算术统计

• 批准号: 1352598
• 负责人: Mirela Ciperiani
• 金额: $ 40.76万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352598&HistoricalAwards=false
• 关键词: CAREER; Genus; curves; rational; points

A central topic in number theory is the study of diophantine equations, that is, the search for integer solutions to polynomial equations with integer coefficients. It has proved highly fruitful to view such equations not only from an algebraic perspective but also from a geometric one; the field of arithmetic geometry provides a language to do so. Elliptic curves (and the closely related genus one curves) are a fascinating class of diophantine equations. The proposed research bears on basic questions of solvability of algebraic equations that define genus one curves, as well as other arithmetic-geometric aspects of such equations. The PI plans to organize a collaborative workshop with specialists from three different areas of number theory that impact the proposed research; and a lecture series on recent developments in number theory which will expose faculty, graduate students, and undergraduates to new and exciting mathematics in this area. The basic objects of study of the proposed research are genus one curves and their rational points.The PI will investigate them from three distinct perspectives: proving, in joint work with A. Wiles, the existence of solvable points on genus one curves defined over the rationals; analyzing the structure of local points on supersingular elliptic curves over Z_p extensions; and studying the arithmetic statistics of the existence of points on quadratic twists of elliptic curves viewed as genus one curves.

数论的一个中心课题是丢番图方程的研究，即寻找整系数多项式方程的整数解。它已被证明是非常富有成效的查看这些方程不仅从代数的角度，而且从几何之一;算术几何领域提供了一种语言这样做。椭圆曲线（以及与之密切相关的亏格一曲线）是一类迷人的丢番图方程。拟议的研究承担的基本问题的代数方程的可解性，定义属一曲线，以及其他算术几何方面的此类方程。PI计划组织一个合作研讨会，与来自数论三个不同领域的专家一起影响拟议的研究;以及关于数论最新发展的系列讲座，这将使教师，研究生和本科生接触到这一领域新的令人兴奋的数学。研究的基本对象是亏格1曲线及其有理点，PI将从三个不同的角度进行研究：证明，与A。Wiles，定义在有理数上的亏格为1的曲线上可解点的存在性，分析了Z_p扩张上超奇异椭圆曲线上局部点的结构，研究了作为亏格为1的椭圆曲线的二次扭上点的存在性的算术统计。