Let \(G\) be a smooth connected linear algebraic group over a field \(k\) and let \(X\) be a \(G\)-torsor. Let \(d\) be the index of \(X\) as a \(k\)-variety, i.e., the greatest common divisor of the degrees \([L:k]\), where \(L\) runs through finite field extensions such that \(X\) has an \(L\)-point. A question of B. Totaro [Duke Math. J. 121, No. 3, 425–455 (2004; Zbl 1048.11031)] asks whether there exists a finite separable field extension \(F/k\) of degree \(d\) such that \(X\) has an \(F\)-point.
The paper under review proves that Totaro’s question has an affirmative answer if the field \(k\) has characteristic \(\neq 2\) and if \(G\) is an absolutely simple adjoint group of type \(A_1\) or \(A_{2n}\). In terms of algebras with involution, the main theorem of the paper implies the following: Let \(K/k\) be a quadratic separable extension and let \((A,\,\sigma)\) and \((B,\,\tau)\) be central simple \(K\)-algebras with \(K/k\)-involution. If there are finite field extensions \(L_1/k,\cdots, L_m/k\) with \(\gcd\{[L_i:k]\}=d\) such that \((A,\,\sigma)_{L_i}\cong (B,\,\tau)_{L_i}\) for each \(i=1,\cdots, m\), then there exists a separable field extension \(F/k\) with \([F:k]\,|\,d\) such that \((A,\,\sigma)_F\cong (B,\,\tau)_F\).
Note that Totaro’s question in general has a negative answer, as shown in [R. Gordon-Sarney and V. Suresh, Duke Math. J. 167, No. 2, 385–395 (2018; Zbl 1383.14003)].