Rational Points on Varieties

品种的理性点

• 批准号: 2441565
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2441565
• 关键词: Rational; Points; Varieties

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.

我将在算术几何和数论的一般领域工作，特别是研究某些丢番图方程的Brauer-Manin障碍。丢番图方程是若干个变量的多项式方程，数学家通常对求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，我们尝试回答是否存在任何这样的点，如果存在，有多少个。我们将通过观察这些丢番图方程的几何性质来实现这一点，并寻找是否存在任何障碍，以求存在整数或有理解。此外，本项目的目的是研究某些类型的丢番图方程(如log K3曲面)。丢番图方程具有这样的性质：如果它们有理数解，则可以“局部”地查看，即检查它是否对某个素数p有p-进给解。这对于二次曲线非常有效，其中Hasse-Minkowski原理指出对所有素数都有p-进给解既是一个充要条件。然而，Hasse-Minkowski定理并不普遍成立，例如，如果我们观察通过在椭圆曲线上体面地执行而产生的齐次空间，这些方程给出了丢番图方程，这些方程对所有素数都有p元解，但有理解集是空的。为了研究这种失效，我们将使用Brauer-Manin障碍。我们也可以预测丢番图方程上有理点的渐近行为，例如Manin猜想预测有界高度有理点的数目的精确渐近行为。我们将致力于在这一猜想的某些情况下取得进展。我们的研究并不局限于有理领域的工作；我们将通过研究数域上的解决方案来扩展这一点。在数论和算术几何范围内工作对未受过训练的人来说似乎不适用，但这对密码学领域有直接的应用。在数论中开发的许多工具被应用于密码学，如椭圆曲线密码学、RSA等等。本研究进一步发展了数论领域的研究成果，为以后的研究提供了工具，例如后量子密码学。此外，在数据科学领域中使用拓扑和代数几何已经成为一个日益激烈的研究领域，通过使用代数几何中的各种技术，我们为我们所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，这对于数论的进步和我们开发的其他人的工作是必不可少的。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的继续工作，并有望为他们提供高质量的研究。