Local-global principles: arithmetic statistics and obstructions

局部全局原则：算术统计和障碍

• 批准号: EP/S004696/1
• 负责人: Rachel Newton
• 金额: $ 15.48万
• 依托单位: University of Reading
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FS004696%2F1
• 关键词: Local; global; principles; arithmetic; statistics

Methods for solving polynomial equations in integers and rationals have been sought and studied for more than 4000 years. Sometimes it is easy to see that a polynomial equation admits no 'global' (meaning integral or rational) solution. For example, if the equation has no solution in the real numbers, then it clearly has no integer solution. The reason that looking for real solutions is easier is because the real numbers have a desirable property called completeness, which relates to the fact that the real numbers form a continuum with no gaps. It is the discrete nature of the integers which makes them difficult to deal with. By viewing the integers within other exotic number systems (called the p-adic numbers) that also enjoy the property of completeness, we can avail ourselves of other ways to rule out existence of integer solutions to polynomial equations. If the equation has no p-adic solution then it has no integer solution. But what if the equation has solutions in the field of real numbers and in all the p-adic fields? Does this mean it has a rational solution? If we have a family of equations where the answer to this question is yes, then we say the Hasse principle holds for that family. For example, the Hasse principle holds for quadratic forms. This means that determining whether a quadratic form has an integer solution is easy. However, there are equations of degree 3 and higher for which the Hasse principle fails. This leads to some natural questions, such as: How often does the Hasse principle fail? Why does it fail? This research addresses both of these questions for certain families of equations. To answer the first question, we will fix a family of equations which can be enumerated in a meaningful way. We will then determine whether the Hasse principle can fail for any equation in the family. For those equations where failures can occur, we will calculate an algebraic object which measures the severity of the failure and determines the precise local conditions which are responsible for the failure. The most difficult step will be to calculate what proportion of the equations in the family give failures. This will tell us whether failure is, as we hope, a rare occurrence in the family. If failures are rare, then a randomly chosen equation in the family will satisfy the Hasse principle and determining whether it has a global solution is equivalent to checking whether it has real and p-adic solutions. The latter calculation can be performed in finite time, whereas no general algorithm exists for determining whether a polynomial equation has an integer solution. The second question concerns obstructions to local-global principles such as the Hasse principle. The most important known obstruction is the Brauer-Manin obstruction. There are several challenges to be overcome in order to understand the consequences of the Brauer-Manin obstruction for a family of varieties. One must calculate the Brauer group, which is the algebraic object quantifying the obstruction. Then one must calculate the obstruction given by each element of the Brauer group. Finally, one must determine whether the Brauer-Manin obstruction suffices to explain all failures of local-global principles in the family. The second part of this project will push the boundaries of our current understanding of each step in this process.

求解整数和有理数多项式方程的方法已经被探索和研究了4000多年。有时很容易看出，多项式方程不承认“整体”（即积分或有理）解。例如，如果方程在真实的数中没有解，那么它显然没有整数解。寻找真实的解更容易的原因是因为真实的数有一个理想的属性，称为完备性，这与真实的数形成一个没有间隙的连续统的事实有关。正是整数的离散性使它们难以处理。通过观察其他具有完备性的奇异数系统（称为p-adic数）中的整数，我们可以利用其他方法来排除多项式方程的整数解的存在性。如果方程没有p-adic解，那么它就没有整数解。但如果方程在真实的数域和所有p-adic域中都有解，那又会怎样呢？这是否意味着它有一个合理的解决方案？如果我们有一个方程族，这个问题的答案是肯定的，那么我们说哈塞原理对这个方程族成立。例如，哈塞原理适用于二次型。这意味着确定一个二次型是否有整数解是很容易的。然而，存在哈塞原理失效的3次和更高次的方程。这就引出了一些很自然的问题，比如：哈塞原理失败的频率是多少？为什么会失败？这项研究解决了这两个问题的某些家庭的方程。为了回答第一个问题，我们将确定一个可以以有意义的方式枚举的方程族。然后，我们将确定哈塞原理是否对族中的任何方程失效。对于那些可能发生故障的方程，我们将计算一个代数对象，它可以测量故障的严重程度，并确定导致故障的精确局部条件。最困难的一步将是计算族中有多少比例的方程会失败。这将告诉我们，失败是否如我们所希望的那样，在家庭中很少发生。如果故障是罕见的，那么族中随机选择的方程将满足Hasse原理，并且确定它是否具有全局解等价于检查它是否具有真实的和p-adic解。后者的计算可以在有限的时间内进行，而没有一般的算法存在，以确定一个多项式方程是否有一个整数解。第二个问题涉及对哈塞原则等地方-全球原则的阻碍。已知最重要的梗阻是Brauer-Manin梗阻。为了理解Brauer-Manin障碍对一个品种家族的影响，有几个挑战需要克服。必须计算Brauer群，它是量化障碍的代数对象。然后必须计算Brauer群的每个元素所给出的阻塞。最后，我们必须确定布劳尔-马宁阻碍是否足以解释族中局部-整体原理的所有失败。这个项目的第二部分将推动我们目前对这个过程中每一步的理解的界限。

项目成果

Explicit uniform bounds for Brauer groups of singular K3 surfaces

奇异 K3 曲面的布劳尔群的显式均匀边界

• DOI: 10.5802/aif.3526
• 发表时间: 2023
• 期刊: Annales de l'Institut Fourier
• 作者: Balestrieri, Francesca;Johnson, Alexis;Newton, Rachel
• 通讯作者: Newton, Rachel

A bound on the primes of bad reduction for CM curves of genus 3

属 3 的 CM 曲线的不良约简素数上界

• DOI: 10.1090/proc/14975
• 发表时间: 2020
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Kiliçer P

Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg)

具有规定范数的数字字段（附录由 Yonatan Harpaz 和 Olivier Wittenberg 编写）

• DOI: 10.4171/cmh/528
• 发表时间: 2022
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Frei C

Distribution of genus numbers of abelian number fields

• DOI: 10.1112/jlms.12737
• 发表时间: 2022-09
• 期刊: Journal of the London Mathematical Society
• 作者: C. Frei;D. Loughran;Rachel Newton

Explicit methods for the Hasse norm principle and applications to A n and S n extensions

哈斯范数原理的显式方法及其在 An 和 S n 扩展中的应用

• DOI: 10.1017/s0305004121000268
• 发表时间: 2021
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: MACEDO A