Irrationality of Periods and Arithmetic of Abelian Varieties

周期的无理性与阿贝尔簇的算术

• 批准号: 2231958
• 负责人: Yunqing Tang
• 金额: $ 20.1万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2231958&HistoricalAwards=false
• 关键词: Irrationality; Periods; Arithmetic; Abelian; Varieties

This research project concerns work in Diophantine geometry and arithmetic geometry, which are essentially ways to understand solutions of families of polynomial equations. The first part of the project studies irrationality in Diophantine geometry. A classical way to prove a number is irrational is by showing that there exists a sequence of rational numbers that approximate this number very well. There are many interesting (conjecturally) irrational numbers in the literature with approximations by rational numbers that are not good enough to apply the classical methods. The principal investigator and collaborators will develop a new framework to explore the properties of certain power series constructed from these approximations in order to prove the conjectured irrationality in some important cases. The second part of the project studies the arithmetic of abelian varieties, which are higher dimensional analogues of elliptic curves. These geometric objects can be defined by polynomial equations over the integers. The principal investigator and collaborators will study the behavior of certain abelian varieties modulo different prime numbers. The proposed work includes the training of undergraduate and graduate students. For the first part, the classical way of proving irrationality can be formulated as studying the convergence radii of the power series associated to the rational approximations and comparing them to the denominator type of the power series. In earlier studies of rationality and algebraicity criterion of power series, the convergence radii have been replaced by many variants, which are numerically larger; therefore, there are rational approximations whose convergence radii are too small compared to the denominator type while these variants are large enough. The PI and collaborators expect to explore these larger radii variants to solve some irrationality questions. For the second part, the PI and collaborators expect to generalize Elkies’s theorem on infinitude of supersingular reductions of elliptic curves to certain abelian varieties parametrized by genus 0 Shimura curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目涉及丢番图几何和算术几何，它们本质上是理解多项式方程族解的方法。该项目的第一部分研究丢番图几何中的无理数。证明数字无理数的经典方法是证明：存在一个可以很好地逼近这个数的有理数序列。文献中存在许多有趣的（推测性的）无理数，它们的有理数逼近还不足以应用该数。项目的第二部分研究了阿贝尔簇的算术，主要研究者和合作者将开发一个新的框架，从这些近似中探索某些幂级数的性质，以证明猜想的非理性。是椭圆曲线的高维类似物。这些几何对象可以通过整数上的多项式方程来定义。主要研究者和合作者将研究某些阿贝尔簇模的行为。拟议的工作包括对本科生和研究生的培训，证明无理数的经典方法可以表述为研究与有理近似相关的幂级数的收敛无线电并将它们与分母进行比较。在早期对幂级数的有理性和代数性准则的研究中，收敛射数已被许多数值上较大的变体所取代，因此存在有理近似。与分母类型相比，其收敛半径太小，而这些变体足够大。 PI 和合作者希望探索这些较大的半径变体来解决一些非理性问题。对于第二部分，PI 和合作者希望推广 Elkies 定理。椭圆曲线到某些阿贝尔簇的无限超奇异约简，由属 0 Shimura 曲线参数化。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。