On the Hasse principle for complete intersection varieties

1961年，Birch使用了Hardy-Littlewood圆方法的一般版本来验证所有程度形式系统的Hasse原理，前提是这些形式有足够多的变量（这取决于程度和形式的数量）。从那时起，这一领域有了许多发展，使在具体情况下验证哈斯原理所需的变量数量大大减少，但是这些结果都不适用于两立方形式的系统。我们的目标是改进Birch的两种三次形式的结果，该结果表明Hasse原理是正确的，前提是形式至少有50+phi变量，其中phi是两种形式的相交变化的奇异轨迹的维数。首先，我们的目标是开发一个二维版本的平均范-德-科普特差分方法，然后使用它来找到一个更好的小弧的边界，利用额外节省的优势，对两个积分进行平均。然后，我们将执行Weyl差分，以得到出现在小弧中的指数和的显式边界。这将使我们在Birch的方法上节省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，以得到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。我们需要适应和改进目前的技术水平，以证明在小弧中出现的指数和中发生平方根消去。这将使我们能够保存额外的3-5个变量。最后，我们的目标是合并一个圆形方法的版本，它使用较大的间隔作为小弧的构建块，以便在求和上得到进一步的节省，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.