Diophantine problems

丢番图问题

• 批准号: RGPIN-2018-03734
• 负责人: Bennett, Michael
• 金额: $ 5.1万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758481
• 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.

数论是数学中最古老的领域之一，但即使在今天，数论仍然在学科内外提供意想不到的应用。它也因经典问题而臭名昭著，这些问题具有易于陈述但显然难以解决的特点。我们提出的研究重点是这种性质的一些结果；我们将我们的方法称为“丢番图问题的显式方法”。我们用来证明结果的机制有些不同。我们使用的基本领域之一是丢番图近似，传统上它试图测量有理数逼近无理数的程度。我们的提案具有一定新颖性的地方在于，它将这些技术与怀尔斯证明费马大定理所使用的著名技术的修改相结合（以及来自数论、分析和组合其他领域的结果）。我们提出的工作集中在解决两个主题相关的问题 - 椭圆曲线的理论和计算方面以及丢番图方程经典问题的显式解。贯穿这两个问题的一个共同线索是它们与求解一类所谓的 S 单元方程的联系。在三次数域上找到此类方程的解使我们能够将所有具有有理系数和“小”导体的椭圆曲线制成表格（自 Shafarevich 的工作以来已知这是一个有限问题）。将其扩展到更高阶的领域允许人们将此分析进行到数域上的椭圆曲线。我们提出的研究将提供此类曲线的表格，极大地扩展了当前的文献，以及用于求解 S 单位方程的计算工具，这些工具应该在各种其他设置中得到使用。我们的方法还将使我们能够在以下方面取得进展：许多其他经典问题，包括求递归序列中的移位幂、各种多项式指数方程和一般 n 项 S 单元方程。为了实现这一目标，我们必须首先锐化和概括由相关伽罗瓦表示的模性以及 Thue-Siegel 的超几何方法产生的三元方程的一些最新结果。在实现后一个目标的过程中，我们与 Martin、O'Bryant 和 Rechnitzer 联合开展了一个解析和计算数论项目，在该项目中，我们寻求获得完全明确的误差项边界，从而至少节省一个对数每个标准函数都以算术级数计算素数。