Toward discretization of Nevanlinna theory

内瓦林纳理论是一门建立在古典微积分基础上的数学。然而，这一理论具有统计力学或算术几何等数学方面的特点，这使得经典微分几何的应用成为一个困难的问题。更准确地说，只有当一个人通过最大限度地利用它的统计或算术性质，成功地将我们的问题放在一个好的形式之后，才能应用微分几何。为什么纳瓦布林纳理论具有这样的性质？这个问题激励了我的研究项目。我的目的是构建背景几何学，解释内瓦林纳理论的这种性质的起源。我研究的指导原则来自统计力学和算术几何(阿拉克洛夫几何)。伏伊达在纳瓦林纳理论和丢番图近似之间提出了所谓的伏伊塔词典。定义在数域上的射影簇的有理点集被认为是超越h-…的丢番图模拟更多的全纯曲线转化为由其复数点组成的复数簇。在我的项目中，我问了这样一个问题：“全纯曲线的微分的丢番图类比是什么？”我对这个问题的回答是，将Nevanlinna理论中关于对数导数的引理解释为全纯曲线导数的定义方程。丢番图集上的相应语句成为有理点的导数的定义方程。这样定义的“区分”的丢番图类比并不是一个绝对的概念。这一定义只有在给定近似目标之后才有意义。在Nevanlinna理论中，绝对微分得到相对定律，这是对数导数引理的结果。在丢番图设置中，我们应该考虑有限的位置。通过推广数的Minkovski/Bombierri-Vaaler几何，我提出了丢番图近似下的分支计数函数的定义。然后，我提出了一个带截断计数函数的Schmidt子空间定理。这一版本的SST使我们能够建立一些与“ABC猜想”等价的猜想，而ABC猜想似乎与原始猜想有很大的不同。导数的定义包含了计算方程根数的规则。在超越全纯曲线或有理点集的情况下，计数规则应基于一些非平凡的统计量。实际上，阿贝尔变种的全纯曲线的截断计数函数应该是1级的，而与目标的维度无关。如果我们以与纳瓦林纳理论相反的方式引入丢番图对导数的定义，就会出现这样的问题。我们在这个项目结束时就开始了这个方向。较少

Nevanlinna theory is a mathematics which is based on classicical calculus. However, this theory has aspects of those Mathematics such as statistical mechanics or arithmetic geometry and this makes the application of classical Differential geometry a difficult issue. More precisely, one can apply differential geometry only after one is successful in putting our problem in a good form by making best use of its statistical or arithmetic nature. Why does Nevablinna theory have such a nature? This question motivated my research project. I aimed at constructing background geometry explaining the origin of such nature of Nevanlinna theory. The guiding principle of my study came from statistical mechanics and arithmetic geometry (Arakelov geometry). Vojta proposed the so called Vojta's dictionary between Nevanlinna theory and Diophantine approximation. The set of rational points of projective varieties defined over a number field is considered to be the Diophantine analogue of transcendental h … More olomorphic curves into the complex variety consisting of its complex points. In my project I asked the question "What is the Diophantine analogue of differentiation of holomorphic curves?" My answer to this queestio is to interpret the lemma on logarithmic derivative in Nevanlinna theory as a defining equation of the derivative of a holomorphic curve. The corresponding statement in Diophantine setting becomes the defining equation of derivatives of rational points. The Diophantine Analogue of "differentiation" thus defined is not an absolute concept. This is defined becomes meaningful only after the target of approximation is given. In Nevanlinna theory, the absolute differentiation obays the relative law, which is the consequence of Lemma on logarithmic derivative. In Diophantine setting we should take finite places into account. I proposed a definition of ramification counting function in Diophantine approximation by extending the Minkovski/Bombierri-Vaaler geometry of numbers. I then proposed a Schmidt Subspace Theorem with truncated counting function. This version of SST enables us to establish some conjectures which is equivalent to the "abc conjecture" which seems to be quite different from the original conjecture. The definition of derivatives contain the rule of counting roots of equations. In the case of transcendental holomorphic curve or the set of rational points, the rule of counting should be based on some non-trivial statistics. In fact the truncated counting function for holomorphic curves in Abelian varieties should be of level 1 regardless of the target's dimension. Such kinds of question arises if we import the Diophantine definition of derivatives in the opposite way to Nevanlinna theory. We started this direction at the end of this project. Less