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Arithmetic Springer theorem and $n$-universality under field extensions

Zilong He

TL;DR

The paper advances the theory of quadratic forms by establishing norm principles for integral and relative integral spinor norms over dyadic local fields via BONGs, enabling a unified treatment of representations, isometries, and their behavior under field extensions. Leveraging these local results, it proves an Arithmetic Springer theorem for indefinite lattices over number fields (valid for odd extensions and rank at least $3$) and resolves lifting problems for $n$-universal lattices, with explicit criteria in both non-dyadic and dyadic settings. The approach combines local BONG-based invariant analysis with global adelic arguments to connect genus and spinor-genus properties across extensions, and it clarifies when $n$-universality persists under inflation and descent. These results provide a comprehensive framework for understanding representations and isometries of lattices under field extensions and offer precise lifting conditions for universality in both local and global contexts.

Abstract

Based on BONGs theory, we prove the norm principle for integral and relative integral spinor norms of quadratic forms over general dyadic local fields, respectively. By virtue of these results, we further establish the arithmetic version of Springer's theorem for indefinite quadratic forms. Moreover, we solve the lifting problems on $n$-universality over arbitrary local fields.

Arithmetic Springer theorem and $n$-universality under field extensions

TL;DR

The paper advances the theory of quadratic forms by establishing norm principles for integral and relative integral spinor norms over dyadic local fields via BONGs, enabling a unified treatment of representations, isometries, and their behavior under field extensions. Leveraging these local results, it proves an Arithmetic Springer theorem for indefinite lattices over number fields (valid for odd extensions and rank at least ) and resolves lifting problems for -universal lattices, with explicit criteria in both non-dyadic and dyadic settings. The approach combines local BONG-based invariant analysis with global adelic arguments to connect genus and spinor-genus properties across extensions, and it clarifies when -universality persists under inflation and descent. These results provide a comprehensive framework for understanding representations and isometries of lattices under field extensions and offer precise lifting conditions for universality in both local and global contexts.

Abstract

Based on BONGs theory, we prove the norm principle for integral and relative integral spinor norms of quadratic forms over general dyadic local fields, respectively. By virtue of these results, we further establish the arithmetic version of Springer's theorem for indefinite quadratic forms. Moreover, we solve the lifting problems on -universality over arbitrary local fields.
Paper Structure (6 sections, 80 theorems, 98 equations)

This paper contains 6 sections, 80 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Spinor norms and invariants in terms of BONGs
  3. Liftings over local fields
  4. Norm principles for spinor norms
  5. Arithmetic Springer theorem
  6. Liftings of $n$-universality

Key Result

Theorem 1

Let $V$ and $U$ be quadratic spaces over $F$. Suppose that $[E:F]$ is odd. If $V\otimes_{F} E$ represents $U\otimes_{F} E$, then $V$ represents $U$.

Theorems & Definitions (153)

  • Theorem : Springer Theorem
  • Theorem 1.1: Arithmetic Springer Theorem
  • Remark 1.2
  • Definition 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • ...and 143 more