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.

已寻求和研究整数和理性中的多项式方程的方法已有4000多年的历史。有时，很容易看出多项式方程不接受“全局”（意味着积分或理性）解决方案。例如，如果方程在实际数字中没有解决方案，则显然没有整数解决方案。寻找真实解决方案更容易的原因是因为实际数字具有所需的属性，称为完整性，这与实际数字形成没有空白的连续体的事实有关。整数的离散性使它们难以处理。通过查看其他外来数字系统（称为P-Adic数字）中的整数，这些系统也享有完整性的属性，我们可以利用其他方法来排除多项式方程的整数解决方案。如果方程没有P-ADIC解，则没有整数解。但是，如果方程在实际数字和所有P-ADIC领域中具有解决方案，该怎么办？这是否意味着它具有合理的解决方案？如果我们有一个方程式的家庭，这个问题的答案是肯定的，那么我们说Hasse原则为该家庭提供了。例如，Hasse原理具有二次形式。这意味着确定二次形式是否具有整数解决方案很容易。但是，Hasse原理会失败的3度及更高程度。这导致了一些自然问题，例如：Hasse原则多久一次失败？为什么失败？这项研究针对某些方程式家庭解决了这两个问题。为了回答第一个问题，我们将修复可以以有意义的方式列举的方程式。然后，我们将确定HASSE原理是否会因家庭中的任何方程式失败。对于可能发生故障的方程式，我们将计算一个代数对象，该对象测量故障的严重性并确定导致故障的确切地方条件。最困难的步骤是计算家庭中的方程式的哪些比例会导致失败。正如我们希望的那样，这将告诉我们失败是否在家庭中很少发生。如果失败很少，那么家庭中随机选择的方程式将满足Hasse原理，并确定其是否具有全球解决方案等同于检查其是否具有真实和P-ADIC解决方案。后一个计算可以在有限的时间内进行，而没有一般算法来确定多项式方程是否具有整数解决方案。第二个问题是涉及诸如Hasse原则之类的本地全球原则的障碍。最重要的障碍物是Brauer-Manin障碍物。为了了解brauer-manin障碍物对品种家庭的后果，需要克服一些挑战。必须计算Brauer组，该组是量化障碍物的代数对象。然后，必须计算Brauer组的每个元素给出的障碍物。最后，必须确定Brauer-Manin阻塞是否足以解释家庭中局部全球原则的所有失败。该项目的第二部分将突破我们当前对此过程中每个步骤的理解的界限。

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