Diophantine equations and local-global principles: into the wild

丢番图方程和局部全局原理：深入实践

• 批准号: MR/T041609/2
• 负责人: Rachel Newton
• 金额: $ 113.35万
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=MR%2FT041609%2F2
• 关键词: Diophantine; equations; local; global; principles

Studying integer (whole number) solutions to polynomial equations is the oldest field in mathematics, containing problems that have remained unsolved for millennia. Furthermore, its applications to cryptography and security make it one of the most high-impact areas of pure mathematics. Cryptosystems rely on the computational hardness of mathematical problems to protect our data. The realm of integer solutions to polynomial equations is a natural source of hard problems to underpin modern cryptosystems. For example, it can claim credit for the development of elliptic curve cryptography (ECC). This is a public key cryptographic system that has been widely used for over a decade by big players such as the USA National Security Agency and Microsoft. For instance, ECC is used to protect our credit card details when we make purchases over the internet. Cybersecurity is of crucial national importance in protecting data at the individual, corporate and state level and its role in daily life is increasing as more of our economic, administrative and social interactions take place online.The deep knowledge of elliptic curves needed for the development of ECC was gained by pursuing blue sky research in mathematics, of which the most famous recent example is Andrew Wiles' 1995 proof of Fermat's Last Theorem. This concerns one particular family of polynomial equations, namely x^n+y^n = z^n. When n=2, this is Pythagoras' equation relating the side lengths of a right-angled triangle. There are infinitely many integer solutions to this equation (e.g. x = 3, y = 4, z = 5) and we even have a formula for them. However, when n is greater than 2, the behaviour is very different. Fermat conjectured in 1637 that there were no positive integer solutions to the equation x^n+y^n = z^n for n greater than 2. The proof of this fact took more than 350 years and required the development of very advanced mathematical techniques. In September 2019, Google announced that they had achieved 'quantum supremacy', having developed a quantum computer that performed a task in 200 seconds where a top-range supercomputer would take 10,000 years. This stunning achievement presents a looming crisis for the cryptosystems protecting our data. A quantum computer that can solve the mathematical problems underlying current cryptosystems in seconds rather than millennia would be able to decrypt encrypted data and compromise its security. Security agencies and technology companies are urgently seeking new, and harder, mathematical problems to underlie post-quantum cryptographic systems and they are keen to collaborate with mathematicians to achieve this.My proposal is to study integer solutions to a much larger and more complex class of polynomial equations than elliptic curves, using a wide variety of techniques from number theory, algebra, geometry and analysis. The modern approach looks first for so-called local solutions and then investigates whether a collection of them can be patched together to form a global (meaning integer) solution. However, this local-global method is not always successful. I will study the reasons for its failure and conduct a statistical analysis of the frequency of these failures within families of equations. I will break new ground by tackling cases that have so far been untouched due to their complexity: the 'wild' in my title is an adjective used by mathematicians to describe mathematical objects whose behaviour is particularly difficult to handle. Recent breakthroughs in number theory mean the time is ripe to grapple with these wild problems. I will collaborate with leading cryptographers to explore possibilities arising from my research for new hard mathematical problems that can be used to underpin cryptosystems that can resist attacks by quantum computers.

研究多项式方程的整数解决方案是数学中最古老的领域，其中包含数千年尚未解决的问题。此外，它在密码学和安全性上的应用使其成为纯数学最高影响的领域之一。密码系统依靠数学问题的计算硬度来保护我们的数据。多项式方程的整数解决方案的领域是基础现代密码系统的严重问题的自然来源。例如，它可以声称对椭圆曲线密码学（ECC）的发展的信用。这是一个公共密钥加密系统，已被美国国家安全局和微软等大型参与者广泛使用了十年。例如，当我们通过互联网购买时，ECC用于保护我们的信用卡详细信息。 Cyber​​security is of crucial national importance in protecting data at the individual, corporate and state level and its role in daily life is increasing as more of our economic, administrative and social interactions take place online.The deep knowledge of elliptic curves needed for the development of ECC was gained by pursuing blue sky research in mathematics, of which the most famous recent example is Andrew Wiles' 1995 proof of Fermat's Last Theorem.这涉及一个特定的多项式方程系列，即x^n+y^n = z^n。当n = 2时，这是pythagoras的方程式，与右角三角形的侧面长度有关。该方程式有许多整数解（例如x = 3，y = 4，z = 5），我们甚至为它们具有公式。但是，当n大于2时，行为大不相同。费马特（Fermat）在1637年猜想，对于n大于2的方程式x^n+y^n = z^n没有正整数解决方案。这一事实的证明花费了350多年，需要开发非常高级的数学技术。 2019年9月，Google宣布他们已经实现了“量子至上”，这是开发了一台量子计算机，该计算机在200秒内执行了一项任务，而顶级超级计算机将需要10，000年。这一令人惊叹的成就为保护我们数据的密码系统带来了迫在眉睫的危机。可以在几秒钟内而不是千年来解决当前密码系统基础的数学问题的量子计算机将能够解密加密数据并损害其安全性。安全机构和技术公司迫切地寻求新的，更困难的数学问题来构成后量化后加密系统的基础，并且他们热衷于与数学家合作以实现这一目标。我的建议是研究比椭圆形曲线更大，更复杂的多项式等级类别的整数解决方案，而不是椭圆形曲线，而不是椭圆形的曲线，使用多种技术，分析了各种各样的技术。现代方法首先要寻找所谓的本地解决方案，然后研究是否可以将它们的集合整合在一起以形成全球（含义整数）解决方案。但是，这种本地全球方法并不总是成功的。我将研究其失败的原因，并对方程家族中这些失败的频率进行统计分析。我将通过解决迄今为止由于其复杂性而无法触及的案例来打破新的立场：我的标题中的“野性”是数学家用来描述数学对象的形容词，其行为尤其难以处理。数量理论的最新突破意味着该时间已经成熟，可以解决这些狂野的问题。我将与领先的密码学家合作，探索我对新的硬数学问题的研究引起的可能性，这些问题可用于支持可以抵抗量子计算机攻击的密码系统。

项目成果

Explicit uniform bounds for Brauer groups of singular K3 surfaces

奇异 K3 曲面的布劳尔群的显式均匀边界

• DOI: 10.5802/aif.3526
• 发表时间: 2023
• 期刊: Annales de l'Institut Fourier
• 作者: Balestrieri, Francesca;Johnson, Alexis;Newton, Rachel
• 通讯作者: Newton, Rachel

Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg)

具有规定范数的数字字段（附录由 Yonatan Harpaz 和 Olivier Wittenberg 编写）

• DOI: 10.4171/cmh/528
• 发表时间: 2022
• 期刊: Commentarii Mathematici Helvetici
• 影响因子: 0.9
• 作者: Frei C

The Hasse norm principle for abelian extensions -- corrigendum

阿贝尔扩张的哈斯范数原理——勘误表

• DOI: 10.48550/arxiv.2308.11640
• 发表时间: 2023
• 作者: Frei C

Explicit methods for the Hasse norm principle and applications to A n and S n extensions

哈斯范数原理的显式方法及其在 An 和 S n 扩展中的应用

• DOI: 10.1017/s0305004121000268
• 发表时间: 2021
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: MACEDO A

Distribution of genus numbers of abelian number fields

• DOI: 10.1112/jlms.12737
• 发表时间: 2022-09
• 期刊: Journal of the London Mathematical Society
• 作者: C. Frei;D. Loughran;Rachel Newton