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On classic $n$-universal quadratic forms over dyadic local fields

Zilong He

TL;DR

This work characterizes classic $n$-universal lattices over dyadic local fields by developing a dyadic universality criterion based on Bases Of Norm Generators (BONGs) and invariants, and by identifying a minimal testing set that certifies $n$-universality for arbitrary $n$. It provides parity-aware conditions for even and odd $n$, with explicit tester lattices and equivalences to a general representation framework, highlighting where maximal-lattice strategies fail in the classic integral setting. The results yield concrete local tests and have global implications via local–global representations, including constraints on discriminants and implications for sums of squares on number fields. Overall, the paper advances the local theory of universal lattices in the dyadic setting and strengthens the bridge to global representation questions.

Abstract

Let $ n $ be an integer and $ n\ge 2 $. A classic integral quadratic form over local fields is called classic $ n $-universal if it represents all $n$-ary classic integral quadratic forms. We determine the equivalent conditions and minimal testing sets for classic $ n $-universal quadratic forms over dyadic local fields.

On classic $n$-universal quadratic forms over dyadic local fields

TL;DR

This work characterizes classic -universal lattices over dyadic local fields by developing a dyadic universality criterion based on Bases Of Norm Generators (BONGs) and invariants, and by identifying a minimal testing set that certifies -universality for arbitrary . It provides parity-aware conditions for even and odd , with explicit tester lattices and equivalences to a general representation framework, highlighting where maximal-lattice strategies fail in the classic integral setting. The results yield concrete local tests and have global implications via local–global representations, including constraints on discriminants and implications for sums of squares on number fields. Overall, the paper advances the local theory of universal lattices in the dyadic setting and strengthens the bridge to global representation questions.

Abstract

Let be an integer and . A classic integral quadratic form over local fields is called classic -universal if it represents all -ary classic integral quadratic forms. We determine the equivalent conditions and minimal testing sets for classic -universal quadratic forms over dyadic local fields.
Paper Structure (8 sections, 62 theorems, 95 equations)

This paper contains 8 sections, 62 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sufficient conditions for $N{\to\!\!\! -} M$
  4. Classic $n$-universality conditions for even $n$
  5. Classic $n$-universality conditions for odd $n$
  6. Proof of Theorems \ref{['thm:classicalnuniversaldyadic']} and \ref{['thm:necessarycon']}
  7. Proof of Theorem \ref{['thm:classicalnuniversaldyadic15theorem']}
  8. Applications to global representations

Key Result

Theorem 1.1

Let $n$ be an integer and $n\ge 2$. Let $M\cong \prec a_{1},\ldots,a_{m}\succ$ be a classic integral $\mathcal{O}_{F}$-lattice relative to some good BONG and $R_{i}=\mathrm{ord}\, (a_{i})$ for $1\le i\le m$. Then $M$ is classic $n$-universal if and only if $m\ge n+3$ and the following conditions hol

Theorems & Definitions (126)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 116 more