## 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:

 1.1e+09 Quadratic forms over local rings and fields 110000 Quadratic forms over general fields 1.1e+82 Algebraic theory of quadratic forms; Witt groups and rings

Zbl 0972.11020
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### References:

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