On the Hasse principle for Witt groups
Let \(k\) be a global field of characteristic different from 2. For a field \(F\), let \(W(F)\) denote the Witt group of quadratic forms over \(F\). The set of places of \(k\) is denoted by \(P(k)\). The classical Hasse principle for the Witt group states that the natural map \(W(k)\to \prod_{\nu\in P(k)} W(k_ \nu)\) is injective, where \(k_ \nu\) denotes the completion of \(k\) at \(\nu\). This can be used to prove that the map \(W(K)\to \prod_{\nu\in P(k)} W(K_ \nu)\) is injective for the rational function field \(K= k(T)\) with \(K_ \nu= k_ \nu (T)\). In this paper, we prove that if \(K\) is the function field of a smooth projective curve \(X/k\) such that \(X(k)\neq \emptyset\), and \(K_ \nu= k_ \nu (X)\), then the kernel of the natural map \(W(K)\to \prod_{\nu\in P(k)} W(K_ \nu)\) is finite. In fact we show that this obstruction to the Hasse principle is a subgroup of the 2-torsion group of the Tate- Shafarevich group of \(J\) where \(J\) is the Jacobian of \(\overline {X}= X\times_ k \overline{k}\). We also give examples to show that this obstruction is non-trivial.
scientific article; zbMATH DE number 778942
scientific article