For a given family of number fields, the Gauss problem asks the list of those of class number one. The most famous solution (Heegner, Baker, Stark) tells us that there are exactly 9 imaginary quadratic fields of class number fields. The central leaves are the subvarieties in the Siegel modular variety $A_g$ modulo $p$ in which points share the same geometric $p$-divisible groups. In this talk we will discuss the geometric analog: when a central leaf consists of only one points，and explain ingredients of the proof. This is based on the joint work with Tomoyoshi Ibukiyama and Valentijn Karemaker.