Growth of class numbers of positive definite ternary unimodular Hermitian lattices over imaginary number fields (Q514375)

Let \(E=\mathbb Q(\sqrt{-m})\), with \(m\) a squarefree positive integer, be an imaginary quadratic field with ring of integers \(\mathcal O\) and discriminant \(d_E\). An exact formula for the class numbers of positive definite binary and ternary unimodular Hermitian \(\mathcal O\)-lattices has been given by \textit{K.-i. Hashimoto} and \textit{H. Koseki} [Tohoku Math. J. (2) 41, No. 2, 171--216 (1989; Zbl 0668.10030)]. Their formulas involve \(d_E\), the number of prime divisors of \(d_E\), the class number of \(E\), Dirichlet characters and Bernoulli numbers. In the paper under review, the authors give an elementary argument to produce a lower bound for the class number \(h(L)\) of positive definite ternary unimodular Hermitian \(\mathcal O\)-lattices which is expressed in terms of \(d_E\) only. The main ingredients in the argument are the observation that such a lattice must be locally \(2\)-universal and a construction of certain positive definite binary Hermitian \(\mathcal O\)-lattices. As a consequence, one obtains that \(\text{lim}\,\text{inf}_{m\rightarrow\infty} \frac{h(L)}{\sqrt{|d_E|}} >0\).