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.
