Arithmetic Differential Equations

在一系列的论文中，PI发展了一种常微分方程（ode）的算术模拟。在该理论中，实数自变量用素数代替，函数用整数代替，函数的导数算子用费马商算子代替。这一理论的一部分被扩展为二维的算术偏微分方程理论。该理论的应用是由PI在NSF资助的先前工作中发现的。PI建议沿着以下思路继续探索这一理论的应用：a)算术偏微分方程在有限Diophantine结果中的应用，用于Shimura变量和Abelian变量之间的对应；b)二维算术偏微分方程在局部类场论中的应用，以及经典模形式的傅里叶（或Serre-Tate）展开系数之间的同余；c)在高属曲线上的算术偏微分方程的构造。函数与数的类比在现代数论的发展中起着关键作用。函数理论的基本工具之一是微分方程理论。我们有理由希望，这一理论的一个算术类比将对数论问题产生有益的影响。PI在以前的工作中发展了微分方程的算术模拟。他建议在丢番图几何、类场论和模形式中寻找这一理论的新应用。

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.