Diophantine problems

丢番图问题

• 批准号: RGPIN-2018-03734
• 负责人: Bennett, Michael
• 金额: $ 2.55万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=645167
• 关键词: Diophantine; problems

Number theory is one of the most ancient fields within mathematics and yet, even today, continues to provide unexpected applications within and without the discipline. It is also somewhat notorious for having classical problems that have the feature that they are easy to state, yet, apparently, hard to solve. Our proposed research focusses on a number of results of this nature; we term our approach "explicit methods for Diophantine problems". The machinery we employ to prove our results is somewhat diverse. One of the basic fields we utilize is that of Diophantine approximation, which, classically, seeks to measure how well rational numbers approximate irrational ones. Where our proposal has a certain amount of novelty is in its combining these techniques with modifications of those famously used by Wiles to prove Fermat's Last Theorem (together with results coming from other areas of number theory, analytic and combinatorial).******Our proposed work is centred upon two thematically-connected problems - theoretical and computational aspects of elliptic curves and explicit solution of classical problems from Diophantine equations. A common thread running through much of these two problems is their connection to solving a class of what are known as S-unit equations. Finding the solutions to such equations over cubic number fields enables us to tabulate all elliptic curves with rational coefficients and "small" conductor (known to be a finite problem since work of Shafarevich). Extending this to fields of higher degree allows one to carry this analysis to elliptic curves over number fields. Our proposed research will provide tables of such curves that greatly extend the current literature, as well as computational tools for solving S-unit equations that should find use in a wide variety of other settings.******Our methods will also allow us to make progress on a number of other classical problems, including that of finding shifted powers in recurrence sequences, various polynomial-exponential equations, and the general n-term S-unit equation. To carry this out, we must first sharpen and generalize a number of recent results on ternary equations arising from the modularity of associated Galois representations, as well as the hypergeometric method of Thue-Siegel. In the course of carrying out this latter goal, we are led to a project in analytic and computational number theory, joint with Martin, O'Bryant and Rechnitzer, where we seek to obtain completely explicit bounds with error terms saving at least a logarithm for each standard function counting primes in arithmetic progression.

数字理论是数学中最古老的领域之一，即使在今天，也继续提供和没有学科的意外应用。遇到具有易于说明的特征的经典问题也有些臭名昭著，但显然很难解决。我们提出的研究重点是这种性质的许多结果。我们称我们的方法为“对养生问题的明确方法”。我们用来证明我们的结果的机械有些多样。我们利用的基本字段之一是二磷酸近似值，从经典上讲，它试图衡量理性数字近似非理性的数字。我们的建议具有一定的新颖性，就是将这些技术与威尔斯（Willes）著名人物证明Fermat的最后定理的技术相结合（以及来自其他数字理论，分析和组合的结果）。***** ***** *我们提出的工作集中在两个主题连接的问题上 - 椭圆曲线的理论和计算方面，以及从毒液方程式对经典问题的明确解决方案。通过这两个问题中的大部分问题，它们与求解一类称为s单位方程式的类别的联系。在立方数字段中找到此类方程的解使我们能够用合理系数和“小”导体（自Shafarevich的工作以来已知是有限的问题）将所有椭圆曲线制成表格。将其扩展到更高程度的领域，使人们可以将此分析带到数字字段上的椭圆曲线。我们拟议的研究将提供如此曲线的表，这些曲线极大地扩展了当前文献，以及解决S-UNIT方程的计算工具，这些方程式应该在其他各种其他设置中找到使用。******我们的方法也将允许我们要在许多其他经典问题上取得进展，包括在复发序列，各种多项式指数方程和一般的N期限S单位方程中找到移动的功率。要执行此操作，我们必须首先对由相关Galois表示的模块化以及Thue-Siegel的超几何方法以及Thue-Siegel产生的三元方程进行锐化并概括一些最新结果。在实现后一个目标的过程中，我们被带到了一个分析和计算编号理论的项目，与马丁，奥布莱恩特和Rechnitzer联合，我们试图在其中获得完全明确的界限，并至少保存一个对数的错误术语每个标准函数都计算算术进展中的素数。