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Local information of ADC quadratic lattices over algebraic number fields

Zilong He

TL;DR

This work advances the arithmetic of ADC quadratic lattices by delivering a comprehensive local classification for unary, binary, and ternary lattices over arbitrary non-archimedean fields, and extends to algebraic number fields with global implications. It leverages BONG invariants to derive compact, invariant-driven formulas for local densities and masses, showing that these invariants completely determine local densities in broad settings. For binary lattices, the authors prove finiteness of primitive ADC lattices and construct infinite families of non-primitive ADC lattices, while for ternary lattices they analyze codeterminant spinor-exception sets and spinor-norm groups to establish sufficient conditions for indefinite lattices to be globally ADC. The reformulated density/mass framework unifies dyadic and non-dyadic cases and provides explicit computational tools, enriching the landscape of universal/regular ADC theory with practical density and class-number insights.

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.

Local information of ADC quadratic lattices over algebraic number fields

TL;DR

This work advances the arithmetic of ADC quadratic lattices by delivering a comprehensive local classification for unary, binary, and ternary lattices over arbitrary non-archimedean fields, and extends to algebraic number fields with global implications. It leverages BONG invariants to derive compact, invariant-driven formulas for local densities and masses, showing that these invariants completely determine local densities in broad settings. For binary lattices, the authors prove finiteness of primitive ADC lattices and construct infinite families of non-primitive ADC lattices, while for ternary lattices they analyze codeterminant spinor-exception sets and spinor-norm groups to establish sufficient conditions for indefinite lattices to be globally ADC. The reformulated density/mass framework unifies dyadic and non-dyadic cases and provides explicit computational tools, enriching the landscape of universal/regular ADC theory with practical density and class-number insights.

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.
Paper Structure (7 sections, 70 theorems, 98 equations, 5 tables)

This paper contains 7 sections, 70 theorems, 98 equations, 5 tables.

Table of Contents

  1. Introduction
  2. A brief overview of BONGs theory
  3. Representations of quadratic lattices over non-archimedean local fields
  4. ADC lattices over non-dyadic local fields
  5. ADC lattices over dyadic local fields
  6. Proof of Main Results
  7. Reformulation of the formulas for local densities and masses

Key Result

Theorem 1.2

Suppose $\hbox{rank}\, M \in \{1,2\}$. Let $\mathfrak{p}\in \Omega_{F}\backslash \infty_{F}$.

Theorems & Definitions (144)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 134 more