Local-global principles: arithmetic statistics and obstructions

求解整数和有理数多项式方程的方法已经被寻求和研究了4000多年。有时很容易看出，一个多项式方程不允许有‘整体’(即积分或有理)解。例如，如果方程没有实数解，那么它显然没有整数解。寻找实数解更容易的原因是因为实数具有一个称为完备性的理想性质，这与实数形成一个没有间隙的连续体的事实有关。正是整数的离散性质使它们难以处理。通过观察其他具有完备性的奇异数系中的整数(称为p-进制数)，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元解，那么它就没有整数解。但是，如果该方程在实数域和所有p进数域中都有解，会怎样呢？这是否意味着它有一个合理的解决方案？如果我们有一族方程，其中这个问题的答案是肯定的，那么我们说哈斯原理对这个族是成立的。例如，哈斯原理适用于二次型。这意味着确定二次型是否有整数解是很容易的。然而，哈斯原理并不适用于三阶或更高阶的方程。这就引出了一些自然而然的问题，比如：哈斯原则多久失败一次？为什么它会失败？这项研究解决了某些方程式家族的这两个问题。为了回答第一个问题，我们将确定一族方程式，这些方程式可以用有意义的方式列举出来。然后，我们将确定哈斯原理是否适用于该族中的任何方程。对于那些可能发生故障的方程，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精确局部条件。最困难的一步将是计算出这个家族中有多大比例的方程式会失败。这将告诉我们，失败是否如我们所希望的那样，在家庭中是罕见的。如果失败很少，那么在这个族中随机选择一个方程就会满足哈斯原理，并且确定它是否有全局解相当于检查它是否有实数解和p-进制解。后一种计算可以在有限时间内进行，而目前还没有确定多项式方程是否有整数解的通用算法。第二个问题涉及对地方-全球原则的阻碍，例如哈斯原则。已知的最重要的梗阻是布劳尔-马宁梗阻。为了了解Brauer-Manin障碍对一系列品种的影响，有几个挑战需要克服。我们必须计算布劳尔群，它是量化障碍物的代数对象。然后，必须计算布劳尔群中每个元素所给出的障碍物。最后，必须确定布劳尔-马宁障碍是否足以解释家庭中所有局部-全球原则的失败。这个项目的第二部分将推动我们目前对这一过程中每一步的理解。

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.