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Failures of integral Springer's Theorem

Nicolas Daans, Vítězslav Kala, Jakub Krásenský, Pavlo Yatsyna

TL;DR

The paper investigates integral representations under Springer’s paradigm (ISTR) for positive definite quadratic $\mathcal{O}_K$-lattices over totally real $K$ and odd-degree totally real extensions $L/K$. It combines local integral results, global considerations, and finiteness arguments to show that while ISTR can fail in this setting, the obstructions are finitely many when the base form is fixed or the extension degree is restricted, and it provides explicit infinite families of failures (notably over the cubic field $K_{49}$ with $Q=\langle 1,1,1,8k+5\rangle$). The results include precise finiteness theorems for rank $n\ge 5$ and analogous (though subtler) statements for ranks $3$–$4$, highlighting a clear boundary between indefinite and positive definite cases. The work also supplies concrete counterexamples across $\mathbb{Z}$ and various totally real fields, illustrating that ISTR fails frequently in the positive definite setting and informing the lifting problem for universal quadratic lattices.

Abstract

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.

Failures of integral Springer's Theorem

TL;DR

The paper investigates integral representations under Springer’s paradigm (ISTR) for positive definite quadratic -lattices over totally real and odd-degree totally real extensions . It combines local integral results, global considerations, and finiteness arguments to show that while ISTR can fail in this setting, the obstructions are finitely many when the base form is fixed or the extension degree is restricted, and it provides explicit infinite families of failures (notably over the cubic field with ). The results include precise finiteness theorems for rank and analogous (though subtler) statements for ranks –, highlighting a clear boundary between indefinite and positive definite cases. The work also supplies concrete counterexamples across and various totally real fields, illustrating that ISTR fails frequently in the positive definite setting and informing the lifting problem for universal quadratic lattices.

Abstract

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.
Paper Structure (4 sections, 13 theorems, 6 equations)

This paper contains 4 sections, 13 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Finiteness results
  4. Failures of ISTR

Key Result

Theorem 1

Let $K$ be a field, $Q \in K[X_1, \ldots, X_n]$ a quadratic form over $K$, $L/K$ a field extension of odd degree, $a \in K$. If $a\in Q(L)$, then $a\in Q(K)$.

Theorems & Definitions (28)

  • Theorem : Springer, 1952 Springer
  • Theorem : see \ref{['C:finitely-many-degree-d-failures']} and \ref{['P:finiteness-HKK']}
  • Theorem : see \ref{['C:infinite-family-Springer-failure']}
  • Theorem 2.1: Local Integral Springer's Theorem for representations, XuSpringerHeSpringer
  • Theorem 2.2: Integral Springer's Theorem for representations by indefinite forms, XuSpringerHeSpringer
  • Theorem 2.3: Local-global principle for integral representation by indefinite forms
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 18 more