Arithmetic Differential Equations

在一系列论文中，PI发展了一种常微分方程式的算术模拟。在这个理论中，用素数代替独立的实变量，用整数代替函数，用费马商运算符代替函数上的导算子。这一理论的一部分被扩展到二维算术偏微分方程组(PDE)理论。这一理论的应用是由PI在之前由NSF资助的工作中发现的。PI建议沿着以下路线继续探索这一理论的应用：a)算术常数在Shimura簇和Abel簇之间对应的有限丢番图结果中的应用；b)二维算术偏微分方程组在局部类场理论中的应用；另一方面，在经典模形式的傅立叶(或Serre-Tate)展开的系数之间的同余方面的应用；c)在更高亏格曲线上构造算术偏微分方程组。函数和数字之间的类比在现代数论的发展中起着关键的作用。函数论的基本工具之一是微分方程式理论。我们有理由希望这一理论的算术类比将对数论问题产生有用的影响。在以前的工作中，PI发展了一种微分方程式的算术模拟。他建议寻找这一理论在丢番图几何、类域理论和模形式中的新应用。

In a series of papers the PI developed an arithmetic analogue of ordinary differential equations (ODEs). In this theory the independent real variable is replaced by a prime number, functions are replaced by integer numbers, and the derivative operator on functions is replaced by a Fermat quotient operator. Part of this theory was extended to a theory of arithmetic partial differential equations (PDEs) in two dimensions. Applications of this theory were found by the PI in previous work funded by the NSF. The PI proposes to continue exploring the applications of this theory along the following lines: a) applications of arithmetic ODEs to finiteness Diophantine results for correspondences between Shimura varieties and Abelian varieties,b) applications of two dimensional arithmetic PDEs to local class field theory on the one hand and, on the other, to congruences between coefficients of Fourier (or Serre-Tate) expansions of classical modular forms, and c) construction of arithmetic PDEs on higher genus curves.The analogy between functions and numbers plays a key role in the development of modern number theory. One of the basic tools in function theory is the theory of differential equations. It is reasonable to hope that an arithmetic analogue of this theory will have a useful impact on number theoretical questions. The PI has developed, in previous work, an arithmetic analogue of differential equations. He proposes to find new applications of this theory to Diophantine geometry, class field theory, and modular forms.