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