On the Hasse principle for complete intersection varieties

1961年，Birch使用Hardy-Littlewood圆法的一般版本验证了所有次数的形式系统的哈斯原理，只要这些形式有足够多的变量(这取决于次数和形式的数量)。自那时以来，这一领域有了许多发展，导致在特定情况下验证哈斯原理所需的变量数量显著减少，但这些结果都不适用于两个三次形式的系统。我们的目的是改进Birch关于两个三次形的结果，如果这两个三次形的变量至少为50+Phi，则Hasse原理成立，其中Phi是两个三次形的交集的奇异轨迹的维度。我们首先致力于发展一个二维版本的平均Van-der Corput差分法，然后利用它通过对两个积分平均而获得的额外节省来寻找更好的小圆弧的界。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。这将使我们能够比Birch方法节省6到7个变量。在此之后，我们将进一步修改Van-der Corput差分步骤，以在a和中得到部分Krousterman精化。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出现在小圆弧中的指数和中。这将有望使我们能够节省额外的3-5个变量。最后，我们的目标是加入一个版本的圆形方法，该方法使用更大的间隔作为次要圆弧的构建块，以获得比a和更进一步的节省，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.