Local information of ADC quadratic lattices over algebraic number fields

TL;DR

This paper investigates the structure, counting formulas, and density of representations for ADC quadratic lattices over local fields and algebraic number fields, providing explicit formulas and characterizations.

Contribution

It offers new explicit formulas for local densities, classifies primitive and non-primitive ADC lattices, and characterizes spinor exceptions over local and algebraic fields.

Findings

Finite primitive positive definite ADC lattices under certain conditions

Explicit formulas for local densities and masses

Characterization of spinor exceptions and norm groups

Abstract

In the paper, we mainly determine the structures, counting formulas, and density sets of representations for binary and ternary ADC quadratic lattices over arbitrary non-archimedean local fields. In the binary case, we show that under certain conditions, there are finitely many primitive positive definite ADC lattices and infinitely many non-primitive ones. We also provide concise formulas for local densities and masses using invariants from BONGs theory, and show that these invariants completely determine the local densities over arbitrary non-archimedean local fields. Moreover, we compute the corresponding local quantities for ADC lattices over algebraic number fields. In the ternary case, we characterize the codeterminant set of spinor exceptions and integral spinor norm groups for ADC lattices over arbitrary non-archimedean local fields. Based on these results, we further establish some sufficient conditions on indefinite ADC lattices over algebraic number fields.