Toward discretization of Nevanlinna theory

内万林纳理论是建立在经典微积分基础上的数学理论。然而，这个理论有那些数学的方面，如统计力学或算术几何，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，只有在一个人成功地通过充分利用其统计或算术性质，把我们的问题变成一个好的形式之后，才能应用微分几何。为什么内瓦布林纳理论有这样的性质？这个问题激发了我的研究计划。我的目的是构建背景几何来解释奈万林纳理论的这种性质的起源。我的研究的指导原则来自统计力学和算术几何（Arakelov几何）。Vojta提出了所谓的Vojta字典，介于Nevanlinna理论和丢番图近似之间。定义在数域上的射影变的有理点集合，被认为是超越同态曲线变成由其复点组成的复变的丢芬图类比。在我的项目中，我问了一个问题：“全纯曲线的微分的丢番图类比是什么？”对于这个问题，我的回答是将奈凡林纳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。在丢番图设置中，相应的表述成为有理点导数的定义方程。这样定义的丢番图式的“分化”并不是一个绝对的概念。只有给出了近似目标，这个定义才有意义。在奈万林纳理论中，绝对微分服从相对定律，这是对数导数引理的结果。在丢番图的背景下，我们应该考虑到有限的地方。通过扩展数字的Minkovski/ bombierri - valer几何，提出了丢芬图近似中分支计数函数的定义。然后提出了具有截断计数函数的Schmidt子空间定理。这个版本的海表温度使我们能够建立一些类似于“abc猜想”的猜想，这似乎与原来的猜想有很大的不同。导数的定义包含了方程求根计数的规则。在超越全纯曲线或有理点集合的情况下，计数规则应该建立在一些非平凡统计量的基础上。事实上，无论目标的维数如何，全纯曲线的截断计数函数都应该是1级。如果我们以与奈万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。我们在项目结束时开始这个方向。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