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Universality lifting from a general base field

Vitezslav Kala, Daejun Kim, Seok Hyeong Lee

TL;DR

This work develops a general theory of universality lifting for totally positive definite quadratic forms over totally real fields. It defines $F$-representability of elements by ${\mathcal{O}_F}$-lattices and shows that a universal lift to $K$ exists iff all indecomposables in ${\mathcal{O}_K^{+}}$ are $F$-representable, enabling a finite-rank construction via Chan--Oh criterion sets. The main contributions include a general finiteness theorem for fixed base field $F$ and extension degree $d$, plus a detailed d=2 analysis: a computational classification for real quadratic $F$ with class number 1 and small discriminants, and explicit obstruction/results for extensions by square roots, notably ruling out most extensions unless the base discriminant is small. In particular, the paper proves a finite list of real quadratic base fields admitting universal lifts in several regimes and establishes sharp bounds for extensions by $\sqrt{d}$, with a tight result for $\sqrt{5}$ that $D_F\le 4076$; these findings suggest that universality lifting is a rare phenomenon with concrete, checkable criteria.

Abstract

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification results in the case of relative quadratic extensions.

Universality lifting from a general base field

TL;DR

This work develops a general theory of universality lifting for totally positive definite quadratic forms over totally real fields. It defines -representability of elements by -lattices and shows that a universal lift to exists iff all indecomposables in are -representable, enabling a finite-rank construction via Chan--Oh criterion sets. The main contributions include a general finiteness theorem for fixed base field and extension degree , plus a detailed d=2 analysis: a computational classification for real quadratic with class number 1 and small discriminants, and explicit obstruction/results for extensions by square roots, notably ruling out most extensions unless the base discriminant is small. In particular, the paper proves a finite list of real quadratic base fields admitting universal lifts in several regimes and establishes sharp bounds for extensions by , with a tight result for that ; these findings suggest that universality lifting is a rare phenomenon with concrete, checkable criteria.

Abstract

Given a totally real number field , we show that there are only finitely many totally real extensions of of a fixed degree that admit a universal quadratic form defined over . We further obtain several explicit classification results in the case of relative quadratic extensions.
Paper Structure (9 sections, 16 theorems, 52 equations, 1 table)

This paper contains 9 sections, 16 theorems, 52 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and terminologies
  4. Real quadratic number fields
  5. General properties for lifting problem
  6. Lifting problems for quadratic extensions over PIDs
  7. When $F$ is a real quadratic field of class number $1$
  8. Extension by fixed $\sqrt{e}$
  9. Extension by $\sqrt{5}$

Key Result

Theorem 1.1

Let $F$ be a totally real number field and $d\in{\mathbb N}$. There are at most finitely many totally real fields $K\supseteq F$ with $[K:F]=d$ such that there is an ${\mathcal{O}_F^{}}$-lattice $L$ such that $L\otimes{\mathcal{O}_K^{}}$ is universal.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 25 more