Explicit Moduli Spaces and Arithmetic Applications

显式模空间和算术应用

• 批准号: 1406066
• 负责人: Wei Ho
• 金额: $ 13.46万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406066&HistoricalAwards=false
• 关键词: Explicit; Moduli; Spaces; Arithmetic; Applications

Studying polynomial equations and their solutions dates back thousands of years. For certain types of polynomials, understanding these solutions has important applications, ranging from engineering to biology to elliptic curve cryptography. It is still a very difficult question to determine how many solutions an arbitrary polynomial equation might have, especially if one looks specifically for solutions lying in the natural numbers or the rational numbers. A simple example of this phenomenon is Fermat's Last Theorem, which states that there are no positive integer solutions to an equation of the form an n-th power is equal to the sum of two n-th powers if n is an integer larger than 2, so for example a cube of a positive integer cannot be equal to the sum of two cubes of positive integers. While the theorem is easy to state, the method of proof involves many deep ideas in number theory and arithmetic geometry, including the idea of studying many such polynomial equations all at once (the idea of "moduli spaces"). The PI intends to study such spaces and use them to understand properties of certain types of polynomials and associated geometric objects, such as elliptic curves.The PI works in the intersection of three fields: number theory, algebraic geometry, and representation theory. The main theme of this proposal involves the use of representation theory to explicitly construct moduli spaces in algebraic geometry; these constructions also may be applied to solve counting problems in number theory by using geometry-of-numbers techniques. The PI intends to use methods and tools developed in all of these subjects over the last hundred years, including constructions from classical algebraic geometry, ideas from Lie theory, and sieve techniques from analytic number theory.

研究多项式方程及其解可以追溯到数千年前。 对于某些类型的多项式，理解这些解决方案具有重要的应用，从工程到生物学再到椭圆曲线密码学。 确定任意多项式方程可能有多少个解仍然是一个非常困难的问题，特别是如果专门寻找自然数或有理数中的解。这种现象的一个简单例子是费马大定理，该定理指出，如果 n 是大于 2 的整数，则 n 次方等于两个 n 次方之和的方程不存在正整数解，因此例如一个正整数的立方不能等于两个正整数的立方之和。 虽然该定理很容易陈述，但证明方法涉及数论和算术几何中的许多深刻思想，包括同时研究许多此类多项式方程的思想（“模空间”的思想）。 PI 打算研究此类空间，并利用它们来理解某些类型的多项式和相关几何对象（例如椭圆曲线）的属性。PI 的工作涉及三个领域的交叉领域：数论、代数几何和表示论。 该提案的主题涉及使用表示论来显式构造代数几何中的模空间；这些结构还可以通过使用数几何技术来解决数论中的计数问题。 PI 打算使用过去一百年来在所有这些学科中开发的方法和工具，包括经典代数几何的构造、李理论的思想以及解析数论的筛选技术。