Rational Points on Varieties

我将在算术几何和数论的一般领域工作，特别是研究Brauer-Manin对某些丢番图方程的障碍。丢番图方程是一些变量的多项式方程，通常数论家对整数或有理数的解感兴趣。显式计算这些点可能很困难，相反，我们试图回答是否存在任何这样的点，如果存在，有多少。我们将通过观察这些丢番图方程的几何性质来做到这一点，并寻找是否存在任何障碍，以获得整数或有理解。此外，这个项目的目的是看看某些类别的丢番图方程丢番图方程具有这样的性质：如果它们具有有理解，则可以“局部地”查看，即检查它是否具有针对某个素数p的p-adic解。这对于二次曲线非常有效，其中Hasse-Minkowski原理指出所有素数都有p-adic解是一个充分必要条件。然而，哈塞-闵可夫斯基定理一般不成立，例如，如果我们看看通过执行体面的椭圆曲线上出现的齐次空间，这些目前丢番图方程有p-adic解决方案的所有素数，但一套合理的解决方案是空的。为了研究这种失败，我们将使用Brauer-Manin阻塞。人们也可以预测的渐近行为的合理点的丢番图方程，例如Manin猜想预测精确的渐近行为的数量合理点的有界高度。我们将致力于在这个猜想的某些情况下取得进展。我们的研究不局限于有理数领域，我们将通过研究数域的解来扩展这一点。数论和算术几何中的工作似乎不适用于未经训练的眼睛，但这对密码学领域有直接的应用。数论中开发的许多工具都应用于密码学，如椭圆曲线密码学，RSA等等。这项研究进一步深化了数论领域的成果，可以为未来的研究（例如后量子密码学）提供工具。此外，在数据科学领域使用拓扑和代数几何已经成为越来越激烈的研究领域，通过使用代数几何中的各种技术，我们为我们使用的类似技术的应用奠定了基础。此外，我们正在为更广泛的纯数学研究框架做出贡献，这对数论的发展至关重要，并允许其他人开展我们已经开发的工作。我的资助机构（EPSRC）的目标是在其广泛的投资组合中进一步扩大研究领域。我的研究将有助于EPSRC的持续工作，并希望为他们提供高质量的研究。

I will be working in the general area of arithmetic geometry and number theory, specifically looking at Brauer-Manin obstructions to certain Diophantine equations. A Diophantine equation is a polynomial equation in some number of variables and generally a number theorist is interested in seeking solutions in the integers or the rationals. Explicitly computing these points can be difficult, instead we attempt to answer if there exists any such points and if so how many. We will do this by looking at the geometric properties of these Diophantine equations and seek if any obstructions exist to there being a integer or rational solution. Moreover the aim this project is to look at certain classes of Diophantine equations (such as log K3 surfaces).Diophantine equations have the property of if they have a rational solution the one can look "locally" i.e. check if it has p-adic solutions for some prime p. This works perfectly well for conics, where the Hasse-Minkowski principle states that having p-adic solutions for all primes is both a necessary and sufficient condition. However the Hasse-Minkowski theorem does not hold in general for example if we look at homogenous spaces that occur through performing decent on elliptic curves, these present diophantine equations which have p-adic solutions for all primes but the set of rational solution is empty. To study such failures we will use the Brauer-Manin obstruction.One can also predict the asymptotic behaviour of a rational points on diophantine equations, for example the Manin conjecture predicts the precise asymptotic behaviour of the number of rational points of bounded height. We will aim to make progress in certain cases of this conjecture. Our research is not bounded by working over the field of rationals; we will extend this by studying solutions over number fields.Working within number theory and arithmetic geometry may seem inapplicable to the untrained eye however this has direct applications to the field of cryptography. Many of the tools developed within number theory are applied in cryptography such as elliptic curve cryptography, RSA and many more. This research furthers results within number theory which can provide tools for future research for example in post quantum cryptography. Further to this using topology and algebraic geometry in the area of data science has been an increasingly intense area of research, by using various techniques in algebraic geometry we lay foundations for applications of similar techniques used by us. Furthermore we are contributing to the wider framework of pure mathematics research, this is essential for the progression of number theory and to allow others to work we have developed.My funding body (EPSRC) has aims of further expanding research areas within their wide portfolio. My research would contribute to the continued work of EPSRC and will hopefully supply them with high quality research.