Diophantine geometry via analytic number theory

多项式方程在自然界中非常普遍，可以用来描述无数的物理和数学现象。例如，毕达哥拉斯定理指出，边长为a<=b<=c的直角三角形具有这样的性质，即这些边长总是满足二次方程a^2+b^2=c^2。这是一个很自然的步骤，试图确定在什么情况下，一个给定的多项式方程允许整数解。对于毕达哥拉斯方程，丢番图在公元250年完全回答了这个问题，我们从他那里继承了“丢番图方程”这个术语。丢番图用整数形式写出了毕达哥拉斯方程的通解。对于一般的丢芬图方程，有三种基本的可能性：或者我们可以证明有无限多个解（如上所述），或者我们可以证明只有有限多个解（正如怀尔斯对费马方程a^k+b^k=c^k所做的著名的那样，当k>2时），或者我们很难证明任何东西！第三种结果的倾向，是丢番图方程这门学科经久不衰的魅力所在。到目前为止提到的方程只涉及3个变量，这些方程是最仔细研究的。相比之下，4个或更多变量的整数方程的溶解度仍然是一个未驯服的边界，在地图上只有散射的结果和猜想。这个项目的主要成果之一将是许多推测的路径点成为既定的事实。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。这种联系提供了一个非常有用的额外杠杆来源，并经常揭示相当美丽的关系。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.