Let \(K/k\) be a finite extension of number fields, \(D\) a quaternion algebra over \(k\), \(\mathfrak D\) a maximal order of \(D\), and \(\mathfrak D'\) a maximal order of \(D_K = D \otimes_kK\). For a skew-hermitian form \(h\) over \(\mathfrak D\), denote by \(h'\) the lifting of \(h\) to \(\mathfrak D'\) via scalar extension.
Let \(h_1,h_2\) be skew-hermitian forms over \(\mathfrak D\) which are in the same genus but different spinor genera. In this paper, the author considers the question of whether it follows, under suitable conditions on the forms and/or the extension \(K/k\), that \(h_1'\) and \(h_2'\) are in different spinor genera over \(\mathfrak D'\).
It is shown that this question has an affirmative answer when the forms under consideration are unimodular and \([K:k]\) is odd, and when the forms have discriminant relatively prime to 2 and \(K/k\) is a Galois extension of odd degree. An example is given which shows that the answer is in general negative for arbitrary forms and odd-degree extensions. The author speculates that it is likely to be affirmative for arbitrary forms and odd-degree Galois extensions. The obstruction to proving this more general result lies in the intractability to date of the computation of the image of the local spinor norm mapping without imposing some technical restrictions. The author also determines conditions under which the number of spinor genera in a given genus is non-decreasing under liftings from \(\mathfrak D\) to \(\mathfrak D'\). Results for the corresponding questions for integral quadratic forms were obtained by the reviewer and J. S. Hsia [Bull. Am. Math. Soc. 81, 942–943 (1975; Zbl 0314.10017); Am. J. Math. 100, 523–538 (1978; Zbl 0393.10021)], and by D. R. Estes and J. S. Hsia [Jap. J. Math., New Ser. 16, No. 2, 341–350 (1990; Zbl 0725.11019)].