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

TL;DR

This work introduces and develops the notion of $n$-ADC lattices, connecting representation of lattices by lattices to the classical concepts of $n$-universality and $n$-regularity. It provides a comprehensive local classification over non-archimedean fields for $n\ge 2$, including explicit descriptions via BONGs and maximal lattices, and establishes a global local–global framework: a lattice is globally $n$-ADC iff it is locally $n$-ADC and $n$-regular. For rank $n+1$ lattices over algebraic number fields, $n$-ADC is equivalent to being $\,\mathcal{O}_{F}$-maximal of class number one, leading to finite classifications in the totally real and quaternary cases; the paper also delivers a precise counting formula $B(m,n)$ for the number of $n$-ADC lattices of a given rank over local fields and enumerates explicit families in the quaternary $\\mathbb{Z}$-lattice setting. Overall, the results connect depth of local structure with global arithmetic, enabling finiteness and concrete classifications in a broad setting of integral quadratic lattices.

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.

Theorems & Definitions (130)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10