On $n$-ADC integral quadratic lattices over algebraic number fields

Abstract

In the paper, we extend the ADC property to the representation of quadratic lattices by quadratic lattices, which we define as $n$-ADC-ness. We explore the relationship between $n$-ADC-ness, $n$-regularity, and $n$-universality for integral quadratic lattices. Also, for $n\ge 2$, we give necessary and sufficient conditions for an integral quadratic lattice over arbitrary non-archimedean local fields to be $n$-ADC. Moreover, we show that over any algebraic number field $F$, an integral ${\mathcal{O}}_F$-lattice with rank $n+1$ is $n$-ADC if and only if it is ${\mathcal{O}}_F$-maximal of class number one.