LEAPS-MPS: Number Fields Generated by Points of Curves and their Galois Groups

从毕达哥拉斯学派考虑的三角形边长到费马大定理（直到世纪末才被证明），多项式方程的有理解吸引了人类数千年。多项式方程的有理解与满足该方程的复数所形成的几何形状之间有着深刻的联系，对于广泛研究的一类方程来说，它看起来像一个多孔的甜甜圈。这种联系也许是最好的说明了法尔胜定理，开创性的现代结果表明，孔的数量所定义的形状复杂的解决方案强加了强大的限制的数量合理的解决方案的方程。该补助金调查这种关系的另一个方面，探索几何形状如何与通过增加超越有理数的潜在解来发现方程的新解的速率相关。此外，PI将举办职业小组，研讨会和座谈会，这将有助于我们的学生，其中大多数是代表性不足的科学招聘和保留。PI还将举办一个艺术奖，以促进数学教育中的艺术，以及知名度，代表性和代表性不足的数学学生的包容性。 该项目在此基础上建立在以前的工作逆伽罗瓦问题的几何方面。在他们关于丢番图稳定性的程序中，马祖尔和鲁宾建议通过理解由C的一个点在有理数上生成的数域的集合来研究曲线C;特别地，他们问这样的域扩张的集合在多大程度上决定了曲线。在这个补助金中提出的项目解决了一个相反的方向，曲线的几何形状如何影响上述一组字段扩展。拟议的项目解决这个集合的大小的程度和判别的领域从算术统计的角度来看，测量的渐近增长的判别界限的函数。将探讨的其他方向包括限制此计数到指定伽罗瓦群的字段扩展，以及增加雅可比矩阵秩的字段扩展。 该项目扩展了PI及其合作者正在进行的工作，使用了数字几何，希尔伯特不可约化，牛顿多边形和线性优化等工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Integer and rational solutions to polynomial equations have intrigued humans for thousands of years, from the triangle side lengths considered by the Pythagoreans to Fermat’s Last Theorem, which was only proven at the end of the twentieth century. There is a deep link between the rational solutions to a polynomial equation and geometry of the shape formed by the complex numbers that satisfy the equation, which for a widely-studied class of equations looks like a many-holed donut. This link is perhaps best illustrated by Faltings’ Theorem, a seminal modern result which shows that the number of holes in the shape defined by the complex number solutions imposes strong constraints on the number of rational solutions to the equation. This grant investigates another aspect of this relationship, exploring how the geometry relates to the rate at which new solutions to the equation are found by augmenting the potential solutions beyond the rational numbers. In addition, the PI will host career panels, workshops, and colloquia which will aid in recruitment and retention in the sciences of our students, the majority of whom are underrepresented. The PI will also run an art prize to foster inclusion of the arts in mathematics education, as well as visibility, representation, and inclusion of underrepresented math students. The projects in this grant build on previous work on a geometric facet of the inverse Galois problem. In their program on Diophantine stability, Mazur and Rubin suggest studying a curve C by understanding the set of number fields generated over the rationals by a single point of C; in particular, they ask to what extent the set of such field extensions determines the curve. The projects proposed in this grant address an inverse direction of this, how the geometry of the curve influences the aforementioned set of field extensions. The proposed projects address the size of this set in terms of the degree and discriminant of the fields from an arithmetic statistics perspective, measuring the asymptotic growth of this set as a function of the discriminant bound. Additional directions which will be explored include restricting this count to field extensions with specified Galois groups, and field extensions which increase the rank of the Jacobian. The projects expand on ongoing work by the PI and her collaborators, using tools such as geometry of numbers, Hilbert irreducibility, Newton polygons, and linear optimization.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.