On the reduced unramified Witt group of the product of two conics

We investigate the reduced unramified Witt group of the product of two smooth projective conics \(X_1\) $X_1$ , \(X_2\) $X_2$ over a field. In some particular cases, this group denoted as \(W(X_1,X_2)\) $W(X_1,X_2)$ turns out to be very small (zero or \({\mathbb {Z}}/2{\mathbb {Z}}\) $\mathbb{Z}/2\mathbb{Z}$ ). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of \(W(X_1,X_2)\) $W(X_1,X_2)$ in terms of 2-fold Pfister forms \(\pi _1\) ${\pi }_1$ , \(\pi _2\) ${\pi }_2$ associated with the conics. These conditions and constructions of the corresponding nonzero elements in \(W(X_1,X_2)\) $W(X_1,X_2)$ depend on \({\text {ind}}(\pi _1+\pi _2)\) $\text{ind}({\pi }_1+{\pi }_2)$ . Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.