Diophantine geometry via analytic number theory

多项式方程在自然界中非常常见，可以用来描述无数的物理和数学现象。例如，毕达哥拉斯的定理指出，一个边长为a&lt；=b&lt；=c的直角三角形具有这样的性质，即这些长度总是满足二次方程a^2+b^2=c^2。尝试并确定给定的多项式方程在什么情况下允许整数解是非常自然的步骤。对于毕达哥拉斯方程，这个问题在公元250年被丢番图完全回答了，我们从他那里继承了“丢番图方程”这个术语。丢番图实际上成功地将一般解写成了毕达哥拉斯方程的整数。对于一般的丢番图方程，有三种基本的可能性：要么我们可以证明有无穷多个解(如上所述)，或者我们可以证明只有有限多个解(就像Wiles对Fermat方程a^k+b^k=c^k，当k&gt；2时所做的著名的那样)，或者我们根本不能显示任何解！第三种结果的倾向是丢番图方程主题所享受的持久吸引力的核心。到目前为止，提到的方程只涉及三个变量，这些方程是研究最深入的方程。相比之下，四个或更多变量方程的整数可解性仍然是一个未被驯服的前沿，只有零星的结果和猜测在地图上。这个项目的主要成果之一将是许多猜测的路点将成为既定事实。我将使用的工具植根于解析数论，但也将利用代数几何和下降理论的方法。丢番图方程和方程的基本几何之间有一种有用的相互作用。这种关系提供了一个非常有用的额外杠杆来源，通常会揭示出相当美好的关系。我的研究充分利用了这一观点。

Polynomial equations are extremely commonplace in nature, and can be used to describe a myriad of physical and mathematical phenomena. For example, the theorem of Pythagoras states that a right-angled triangle with sides of lengths a<=b<=c has the property that these lengths always satisfy the quadratic equation a^2+b^2=c^2. It is a very natural step to try and determine under what circumstances a given polynomial equation admits integer solutions. For the Pythagorean equation this was answered completely by Diophantus in 250 AD, from whom we have inherited the term 'Diophantine equations'. Diophantus actually managed to write down the general solution in integers to Pythagoras' equation. For a general Diophantine equation, there are 3 basic possibilities: either we can show that there are infinitely many solutions (as above), or we can show that there are only finitely many solutions (as Wiles famously did for Fermat's equation a^k+b^k=c^k, when k>2), or we have trouble showing anything at all! The propensity for the third outcome lies at the heart of the enduring appeal that the subject of Diophantine equations enjoys.The equations mentioned so far have only involved 3 variables, and these are the equations that have been most closely studied. By contrast the solubility in integers of equations in 4 or more variables is still an untamed frontier, with only a scattering of results and conjectures on the map. One of the major outcomes of this project will be that many of the conjectural waypoints become established fact. The tools that I will use are rooted in analytic number theory, but will also take advantage of methods from algebraic geometry and the theory of descent. There is a useful interplay between Diophantine equations and the underlying geometry of the equation. This sort of connection provides a very useful source of extra leverage, and often reveals quite beautiful relations. My research makes essential use of this point of view.