The mass of shifted lattices and class numbers of inhomogeneous quadratic polynomials

2023/05/29 by Bringmann, Kathrin, He, Zilong, Kane, Ben
#11E08 #11E12 #11E41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2305.17909

Abstract

In this paper, we investigate class numbers of shifted quadratic lattices L+\frac\boldsymboluc with \boldsymbolu∈ 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+\frac\boldsymboluc depending on c. As a result, we obtain an explicit bound c₀ such that any such shifted lattice with one class in its genus must have conductor smaller than c₀, restricting the possible choices of such L+\frac\boldsymboluc to a finite set.

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