The Mass of Shifted Lattices and Class Numbers of Inhomogeneous Quadratic Polynomials

In this paper, we investigate class numbers of shifted quadratic lattices L+u/c with u∈ L and odd conductor c∈ℕ. For a lattice L whose genus only contains one class, we determine a lower bound for the number of classes in the genus of L+u/c depending on c. As a result, we obtain an explicit bound c_0 such that any such shifted lattice with one class in its genus must have conductor smaller than c_0, restricting the possible choices of such L+u/c to a finite set.