Fix a prime number \(p\) and let \(k\) be a field of characteristic different from \(p\). Let \(X\) be a regular curve over \(k\) with function field \(k(X)\). For a Brauer class \(\alpha\) in the \(p\)-primary torsion part of the Brauer group \(\mathrm{Br}(k(X))\), the Faddeev index \(F(\alpha)\) is defined as the maximum of the indices of Brauer classes \(\beta\in\mathrm{Br}(k(X))\{p\}\) which have the same residues as \(\alpha\) at all points of \(X\) B. È. Kunyavskiĭ et al. [Trans. Am. Math. Soc. 358, No. 6, 2579–2610 (2006; Zbl 1101.16013)].
In the paper under review, the author first studies the Faddeev index over the rational function field \(k(t)\) and shows that if the ramification locus of \(\alpha\) on \(\mathbb{A}^1\) is composed of \(n\) rational points, then \(F(\alpha)\leq p^{[\frac{n+1}{2}]}\). This upper bound is shown to be optimal under some assumptions on the Brauer group \(\mathrm{Br}(k)\). In the special case where \(n=3\) and \(\alpha\) is of exponent \(p=2\), it is known that both of the two values \(2,\,4\) are possible for \(F(\alpha)\). The author gives here a criterion for \(F(\alpha)=2\) using the language of quadratic forms. He then applies this criterion to study the existence of rational points on the intersection of two three-dimensional quadrics in certain cases.
When \(X\) is a complete smooth geometrically irreducible curve, it is proved that the Faddeev index can take any prescribed value, provided that the Brauer classes of exponent \(p\) over \(k\) can have arbitrarily large index.
In the last section, the author introduces the notion of Faddeev cyclic length and computes it for some Brauer classes when the field \(k\) contain all \(p\)-primary roots of unity.