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Isotropy of quadratic forms over function fields of \(p\)-adic curves. (English) Zbl 0972.11020

The \(u\)-invariant of a field \(k\) of characteristic \(\neq 2\) is defined to be the supremum of the dimensions of anisotropic quadratic forms over \(k\). It is still an open question whether in general the finiteness of \(u(k)\) implies the finiteness of the \(u\)-invariant of the rational function field \(k(t)\). This wasn’t even known in the case \(k={\mathbb Q}_p\) until Merkurjev and independently J. Van Geel and the reviewer [J. Ramanujan Math. Soc. 13, 85-110 (1998; Zbl 0922.11032)] established the finiteness in the case of a function field of transcendence degree one over a local non-dyadic field \(k\). The bound obtained in the latter paper for such a field \(K\) was \(u(K)\leq 22\), and the proof was based on a result by D. J. Saltman [J. Ramanujan Math. Soc. 12, 25-47 (1997; Zbl 0902.16021)]; Correction to that paper in ibid. 13, 125-130 (1998; Zbl 0920.16008)], who proved that any finite set of elements in \(H^2K\) split over a common biquadratic ground field extension (here, \(H^nK= H^n(\text{Gal}(K_s/K), {\mathbb Z}/2{\mathbb Z})\) denotes the \(n\)-th Galois cohomology group with \(\operatorname {mod} 2\) coefficients, where \(K_s\) denotes a separable closure of \(K\)).
In the present paper, the bound is lowered to \(u(K)\leq 10\) (it can be readily shown that \(u(K)\geq 8\)). This is done by showing that any finite set of elements in \(H^3K\) split over a common quadratic ground field extension, where the main step is to kill the ramification of any element in \(H^3K\) on a regular proper model \({\mathcal X}\) of a quadratic extension \(L\) of \(K\) and by invoking a result of K. Kato [J. Reine Angew. Math. 366, 142-183 (1986; Zbl 0576.12012)] stating that the unramified cohomology group \(H_{nr}^3(L/{\mathcal X}, {\mathbb Z}/2{\mathbb Z})\) is zero. A refinement of the arguments yields that in fact \(H^3K\) consists of symbols.
Two applications are given. First, let \(\pi : Q\to C\) be an admissible quadric fibration where \(C\) is a smooth, projective, geometrically integral curve over \(k\), and let \(CH_0(Q/C)\) be the kernel of the induced homomorphism \(\pi_* : CH_0(Q)\to CH_0(C)\). Colliot-Thélène and Skorobogatov asked whether \(\dim (Q)\geq 4\) implies that \(CH_0(Q/C)=0\). The results in the present paper plus an argument given in the article by Van Geel and the reviewer provide an affirmative answer. As a second application, following ideas of Serre and using a result by Kato, a description of Cayley algebras over \(k(X)\) is obtained, where \(k\) is a non-dyadic \(p\)-adic field and \(X\) a smooth irreducible curve over \(k\).

MSC:

11E04 Quadratic forms over general fields
12G05 Galois cohomology
11E08 Quadratic forms over local rings and fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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References:

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