On the Hasse principle for complete intersection varieties

在1961年，伯奇使用了哈迪-利特尔伍德圆方法的一般版本来验证所有阶形式系统的哈塞原理，只要这些形式有足够多的变量（这取决于阶和形式的数量）。从那时起，这一领域出现了许多发展，导致在特定情况下验证Hasse原理所需的变量数量显着减少，但这些结果都不适用于两个立方形式的系统。我们的目标是改进Birch关于两个三次型的结果，即Hasse原理在两个三次型至少有50+phi个变量时成立，其中phi是两个三次型的交簇的奇异轨迹的维数.我们的目标首先是发展一个二维形式的平均Van-der Corput差分方法，然后通过利用对两个积分求平均值所获得的额外节省，使用该值来找到较小弧的更好边界。然后，我们将执行Weyl差分，以获得出现在次弧中的指数和的显式界。这将使我们能够在Birch方法上节省6或7个变量。在此之后，我们将进一步调整Van-der Corput差分步骤，以获得a和中的部分Kloosterman细化。为了利用这一点，我们将使用泊松求和而不是Weyl差分。我们将需要适应和改进现有技术，以证明平方根抵消发生在指数和出现在小弧。这将有望使我们能够节省额外的3-5个变量。最后，我们的目标是纳入一个版本的圆方法，使用更大的间隔作为构建块的小弧，以获得进一步节省超过一个总和，并可能节省另一个变量。

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.