Third Galois cohomology group of function fields of curves over number fields (Q2190846)

Let \(F\) be a field, \(\ell\) a prime different from the characteristic of \(F\) and \(\mu_{\ell}\) the group of \(\ell\)-th roots of unity. For \(n\geq 1\), let \(H^n(F,\mu_{\ell}^{\otimes n})\) be the \(n\)-th Galois cohomology group with coefficients in \(\mu_{\ell}^{\otimes n}\). Let \(a_1,\ldots,a_n\in F^*\). The cup product \((a_1)\cdots (a_n)\in H^n(F,\mu_{\ell}^{\otimes n})\) is called a \textit{symbol}. Consider \(K\) a number field or a \(p\)-adic field and \(F\) the function field of a curve over \(K\). Assume that \(K\) contains a primitive \(\ell\)-th root of unity. If \(\ell=2\) and \(K\) is a number field, we assume that \(K\) is totally imaginary. The main result of the paper is Theorem 7.7 that establishes that if \(\zeta\in H^3(F,\mathbb{Z}/\ell(2))\) satisfies that \(\zeta\otimes_F(F\otimes_K K_v)\) is trivial for all real places \(v\) of \(K\), then there exist \(a,b,f\in F^*\) such that \(\zeta= [a,b)(f)\). As a corollary (Corollary 7.8), it is obtained that if either \(\ell \neq 2\) or \(K\) has no real places, then, for every element \(\zeta\in H^3(F,\mathbb{Z}/\ell(2))\), there exist \(a,b,c\in F^*\) such that \(\zeta =[a,b)(c)\). Hence every element in \(H^3(F,\mu_{\ell}^{\otimes 3})\) is a symbol. It is also obtained (Corollary 8.3) the following. Let \(K\) be a totally imaginary number field and \(F\) the function field of a curve over \(K\). Let \(q\) be a quadratic form over \(F\) and \(\lambda\in F^*\). If the dimension of \(q\) is greater than or equal to \(5\), then \(q\otimes \langle 1,-\lambda\rangle\) is isotropic. One final result (Theorem 8.4) is the following. Let \(K\) be a totally imaginary number field, \(C\) a smooth projective geometrically integral curve over \(K\). Let \(X\longrightarrow C\) be an admissible quadric fibration. If \(\dim(X)\geq 4\), then the kernel \(CH_0(X/C)\) of \(CH_0(X)\longrightarrow CH_0(C)\) of the Chow groups of \(0\)-cycles modulo rational equivalence, is \(0\). In particular \(CH_0(X)\) is finitely generated.