Diophantine problems

数论是数学中最古老的领域之一，但即使在今天，它仍然在该学科内外提供意想不到的应用。它还有一点臭名昭著，因为它有一些经典问题，它们的特点是很容易陈述，但显然很难解决。我们提议的研究集中在这一性质的一些结果上；我们称我们的方法为“丢番图问题的显式方法”。我们用来证明我们的结果的机制有些不同。我们使用的一个基本领域是丢番图近似，它经典地寻求衡量有理数逼近无理数的程度。我们的建议具有一定的新颖性，在于它将这些技术与Wiles用来证明Fermat最后定理的那些著名技术的修改相结合(以及来自数论、解析和组合的其他领域的结果)。*我们建议的工作集中在两个主题相关的问题上--椭圆曲线的理论和计算方面，以及丢番图方程的经典问题的显式解。贯穿这两个问题的一个共同线索是它们与求解一类被称为S单位方程的问题的联系。在三次数域上求出这类方程的解使我们能够列出所有具有有理系数和“小”导体的椭圆曲线(自Shafarevich的工作以来一直被认为是一个有限问题)。将其扩展到更高次的域允许将这种分析进行到数域上的椭圆曲线。我们建议的研究将提供这类曲线表，极大地扩展了当前的文献，并提供了求解S单位方程的计算工具，这些工具应该可以在其他各种情况下使用。*我们的方法还将允许我们在一些其他经典问题上取得进展，包括求递归序列中的移位幂、各种多项式指数方程和一般的n项S单位方程。为了实现这一点，我们必须首先加强和推广最近关于三元方程的一些结果，这些结果源于相关Galois表示的模性以及Thue-Siegel的超几何方法。在实现后一个目标的过程中，我们被引导到一个与Martin，O‘Bryant和Rechnitzer合作的解析和计算数论的项目中，在这个项目中，我们寻求获得完全显式的界，其中误差项至少为算术级数中计数素数的每个标准函数节省一个对数。

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.