On the Hasse principle for complete intersection varieties

完全交叉品种的哈斯原理

• 批准号: 1769648
• 金额: --
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1769648
• 关键词: Hasse; principle; complete; intersection; varieties

In 1961, Birch used a general version of the Hardy-Littlewood circle method to verify the Hasse principle for systems of forms of all degrees, provided that these forms had sufficiently many variables (this depends both on the degree, and the number of forms). There have been many developments in this area since then which have led to a significant reduction in the number of variables required to verify the Hasse principle in specific cases, however none of these results have been applicable to systems of two cubic forms. We aim to improve on Birch's result for two cubic forms which states that the Hasse principle is true provided that the forms are in at least 50+phi variables, where phi is the dimension of the singular locus of the intersection variety of the two forms.We firstly aim to develop a two-dimensional version of the averaged Van-der Corput differencing method, and then use this to find a better bound for the minor arcs by taking advantage of the extra saving gained by averaging over both integrals. We will then perform Weyl differencing to get an explicit bound for the exponential sums which appear in the minor arcs. This will enable us to save 6 or 7 variables over Birch's method.After this, we will adapt the Van-der Corput differencing step further to get partial Kloosterman refinement in the a sum. In order to take advantage of this, we will then use Poisson summation instead of Weyl differencing. We will need to adapt and improve upon current state of the art techniques in order to prove that square-root cancellation occurs in the exponential sums which arise in the minor arcs. This will hopefully enable us to save an additional 3-5 variables.Finally, we aim to incorporate a version of the circle method which uses larger intervals as building blocks for the minor arcs, in order to get further saving over the a sum and potentially save another variable.

1961年，Birch使用Hardy-Littlewood圆方法的通用版本来验证所有阶形式系统的哈斯原理，前提是这些形式具有足够多的变量（这取决于阶数和形式的数量）。此后该领域取得了许多进展，导致在特定情况下验证哈斯原理所需的变量数量显着减少，但这些结果均不适用于两种三次形式的系统。我们的目标是改进两种三次形式的 Birch 结果，该结果指出，只要形式至少有 50+phi 变量，哈斯原理就成立，其中 phi 是两种形式的交集奇异轨迹的维数。我们首先旨在开发平均 Van-der Corput 差分方法的二维版本，然后利用它所获得的额外节省，使用它来找到更好的短弧界限。 两个积分的平均值。然后，我们将执行 Weyl 差分，以获得小弧中出现的指数和的显式界限。这将使我们能够比 Birch 的方法节省 6 或 7 个变量。此后，我们将进一步调整 Van-der Corput 差分步骤，以在总和中获得部分 Kloosterman 细化。为了利用这一点，我们将使用泊松求和而不是韦尔差分。我们需要适应和改进当前最先进的技术，以证明平方根抵消发生在小弧中出现的指数和中。这有望使我们能够节省额外的 3-5 个变量。最后，我们的目标是合并一个圆方法的版本，该方法使用较大的间隔作为小弧的构建块，以便进一步节省总和并可能节省另一个变量。