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Two refined notions in quadratic form theory

Connor Cassady

Abstract

We use the Witt index to define and study a refined notion of the local-global principle for isotropy of quadratic forms over a field $k$ and to define and study refined versions of the $m$-invariant of $k$. We also explore connections between these refinements.

Two refined notions in quadratic form theory

Abstract

We use the Witt index to define and study a refined notion of the local-global principle for isotropy of quadratic forms over a field and to define and study refined versions of the -invariant of . We also explore connections between these refinements.
Paper Structure (10 sections, 61 theorems, 63 equations)

This paper contains 10 sections, 61 theorems, 63 equations.

Table of Contents

  1. Notation and preliminaries
  2. Refined local-global principle for isotropy
  3. Counterexamples
  4. $I^n$-neighbors
  5. All quadratic forms
  6. Refining the $m$-invariant
  7. Definition and general results
  8. Refined $m$-invariants of complete discretely valued fields
  9. The refined $m$-invariants separate fields
  10. Connecting $m_{i,j}$ and $\text{LGP}(r,s)$

Key Result

Theorem 1

Let $\ell$ be a field of characteristic $\ne 2$. Assume that $\ell \in \mathscr{A}_i(2)$ for some $i \geq 0$ and $u(\ell) = 2^i$. For any integer $r \geq 1$ let $L_r = \ell(x_1, \ldots, x_r)$, and for $r \geq 2$ let $V_r$ be the set of non-trivial discrete valuations on $L_r$ that are trivial on $L_

Theorems & Definitions (134)

  • Theorem : \ref{['ce to lgp']}
  • Proposition : \ref{['I^n neighbors and lgp']}
  • Theorem : \ref{['varying r,s together']}
  • Theorem : \ref{['upper and lower bounds']}
  • Theorem : \ref{['refined m determine u']}
  • Theorem : Propositions \ref{['m_i,1 of cdvf for small i']} and \ref{['m_i,1 of cdvf for large i']}
  • Proposition : Corollary \ref{['m_i,j separates']}
  • Theorem : \ref{['going-down']}
  • Lemma 1.1
  • proof
  • ...and 124 more