Irrationality of Periods and Arithmetic of Abelian Varieties

本研究计画涉及丢番图几何与算术几何的研究，这是了解多项式方程族解的基本方法。该项目的第一部分研究丢番图几何中的非理性。证明一个数是无理数的经典方法是证明存在一个很好地近似这个数的有理数序列。在文献中有许多有趣的（从理论上讲）无理数，它们用有理数近似，但不足以应用经典方法。主要研究者和合作者将开发一个新的框架来探索从这些近似构造的某些幂级数的性质，以证明在一些重要情况下的非理性。第二部分研究了阿贝尔簇的算法，阿贝尔簇是椭圆曲线的高维类似物。这些几何对象可以由整数上的多项式方程定义。主要研究者和合作者将研究某些阿贝尔变种模不同素数的行为。拟议的工作包括培训本科生和研究生。 对于第一部分，证明非理性的经典方法可以表述为研究与有理逼近相关联的幂级数的收敛半径，并将它们与幂级数的分母类型进行比较。在幂级数的有理性和代数性判据的早期研究中，收敛半径被许多变量所取代，这些变量在数值上更大;因此，存在收敛半径与分母类型相比太小的有理逼近，而这些变量足够大。PI和合作者希望探索这些更大的半径变体，以解决一些不合理的问题。对于第二部分，PI和合作者希望将Elkies关于椭圆曲线的超奇异约化的无穷大定理推广到由亏格0 Shimura曲线参数化的某些阿贝尔簇。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

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 In earlier studies of rationality and algebraicity criterion of power series, the convergence radii have been replaced by many variants, which are numerically larger; therefore, there are rational approximations whose convergence radii are too small compared to the denominator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.