Some Classical Problems in Number Theory via the Theory of K3 Surfaces

Abstract

The theory of the elliptic modular function plays an important role in many situations in number theory. The elliptic modular function is obtained as a one-to-one correspondence between the parameter space of the family of elliptic curves (given by the Weierstrass normal form) and its period domain (i.e., the complex upper half plane). The K3 surface is considered to be a two-dimensional counterpart of the elliptic curve. So, if we consider a family of algebraic K3 surfaces with some normal form, we can obtain its modular function. We call it a K3 modular function (see [18, 19], some mathematical physicists call it a mirror map for K3 surfaces).