Counting rational points in arithmetic varieties by the determinant

By the slope method in Arakelov geometry, we can construct a
family of hypersurfaces which cover the rational points of bounded height on
an arithmetic variety, but do not contain the generic point of this variety.
By estimating some invariants of Arakelov geometry, we can control the
number and the maximal degree of this family of auxiliary hypersurfaces
explicitly. In this talk, I will explain the method of studying the problem
of counting rational points by using methods from Arakelov geometry.