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

从毕达哥拉斯学派所考虑的三角形边长到直到二十世纪末才被证明的费马最后定理，多项式方程的整数和有理数解几千年来一直吸引着人类。多项式方程的有理解和满足方程的复数所形成的几何形状之间有很深的联系，对于一类被广泛研究的方程来说，它看起来就像一个多孔的甜甜圈。这种联系也许最好地用Faltings定理来说明，这是一个开创性的现代结果，它表明由复数解定义的形状中的洞的数量对方程的有理解的数量施加了强烈的约束。这项拨款调查了这种关系的另一个方面，探索几何如何与通过增加超过有理数的潜在解来找到方程的新解的速度有关。此外，国际学生联合会将举办职业小组、研讨会和座谈会，这些活动将有助于招募和留住我们的学生，因为他们中的大多数人在科学领域的代表性不足。国际学生联合会还将举办一项艺术奖，以促进将艺术纳入数学教育，以及提高未被充分代表的数学学生的可见度、代表性和包容性。这笔赠款中的项目建立在先前关于伽罗瓦逆问题的几何面的工作基础上。在他们关于丢番图稳定性的程序中，Mazur和Rubin建议通过理解C的单个点在有理数上生成的数域的集合来研究曲线C；特别是，他们问这样的域扩张的集合在多大程度上决定了曲线。这笔赠款中提出的项目解决了与此相反的方向，即曲线的几何形状如何影响前述的一组现场扩展。拟议的项目从算术统计的角度根据场的程度和判别来处理这一集合的大小，作为判别界限的函数来衡量这一集合的渐近增长。将探索的其他方向包括将该计数限制为具有指定伽罗瓦群的域扩展，以及增加雅可比的等级的域扩展。这些项目扩展了PI和她的合作者正在进行的工作，使用了数字几何、希尔伯特不可约性、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优势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.