Linkage property of Witt ring and torsion quadratic forms with trivial Witt invariant
\textit{R. Elman} and \textit{T. Y. Lam} [Invent. Math. 21, 125-137 (1973; Zbl 0267.10029)] established some bounds on the \(u\)-invariant of a field \(F\) depending on whether the powers of the fundamental ideal \(IF\) of the Witt ring \(WF\) are linked or their intersections with the torsion ideal \(W_tF\) of the Witt ring are linked. Thus, if \(I^2F\) is linked, then the \(u\)-invariant assumes only the values 0, 1, 2, 4 or 8, and if \(I^3F\) is torsion free, then \(u\leq 4\). Also, for a formally real field with \(I^nF\cap W_tF\) linked for all \(n\), the values of the \(u\)-invariant are 0, 2, 4, 8, 16 or 18. The present author obtains some further results in this direction. First, instead of the \(u\)-invariant he considers the \(cu\)-invariant, which is defined to be the maximal dimension of anisotropic torsion forms \(\varphi\) with trivial Witt invariant \(c(\varphi)=1\). He proves that, when \(I^3F\) is linked, then \(cu(F)\) assumes only the values 0, 1, 2, 8 or 16, and if additionally \(I^4F\) is torsion free, then \(cu(F)=0,1,2\) or 8. The second result assumes that for \(n\geq 3\) any torsion form in \(I^nF\) is congruent to an \(n\)-fold Pfister form \(\text{mod } I^{n+1}F\). Then the \(cu\)-invariant of a formally real field \(F\) assumes one of the values 0, 2, 8, 16 or 18, and if additionally \(I^4F\) is torsion free, then only 0, 2 or 8 are possible.
maximal dimension of anisotropic torsion forms with trivial Witt invariant
scientific article; zbMATH DE number 927037
scientific article