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The \(u\)-invariant of the function fields of \(p\)-adic curves. (English) Zbl 1208.11053
The \(u\)-invariant \(u(F)\) of a field \(F\) is defined to be the maximum of dimensions of anisotropic quadratic forms over \(F\), it is defined to be \(\infty\) if the maximum does not exist. It is still open in general whether the finiteness of \(u(F)\) implies that for the rational function field \(F(t)\).
Denote by \(K\) the function field of a \(p\)-adic curve with \(p \neq 2\). In a previous paper [Publ. Math., Inst. Hautes Étud. Sci. 88, 129–150 (1998; Zbl 0972.11020)], the same authors proved that \(u(K) \leq 10\).
In the paper under review they showed that \(u(K) \leq 8\). They have given certain sufficient conditions for \(F\) to have \(u(F) \leq 8\) (Proposition 4.4), used a local-global theorem for \(H^3(K, \mathbb Z/2\mathbb Z)\) (Theorem 3.4) to verify one of the conditions for \(K\), and reached the result.
On the other hand, they do not show us the existence of anisotropic quadratic forms of rank \(8\). If a \(p\)-adic curve \(C\) has a \(\mathbb Q_p\)-rational point, anisotropic forms over \(\mathbb Q_p\) remind anisotropic over the function field \(K\) of \(C\), and one sees \(u(K) = 8\). In general, it does not seem to be clear the existence of anisotropic quadratic forms of rank \(8\).

MSC:
11E08 Quadratic forms over local rings and fields
11E04 Quadratic forms over general fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
Citations:
Zbl 0972.11020
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References:
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