Conjugacy classes of series in positive characteristic and Witt vectors
Let \(k\) be an algebraic closure of the finite field \(\mathbb{F}_p\), and \(K=k((t))\) the field of formal Laurent series over \(k\). In this context, Oort's conjecture states that every cyclic field extension \(L|K\) of finite degree should lift smoothly to mixed characteristic. Note that a choice of uniformizer \(s\in L\) yields the isomorphism \(L\cong k((s))\), and then a finite-order continuous \(k\)-automorphism of \(L\) corresponds to a power series \(\sigma(s)\in ks\) having finite order for the composition law. Hence, Oort's conjecture can be reformulated as the statement that every invertible power series of order \(n\) for the composition law in \(ks\) can be lifted to a series of order \(n\) for the composition law in \(RS\), where \(R\) is a mixed characteristic discrete valuation ring with residue field \(k\). Oort's conjecture is known to hold for all tamely-ramified extensions, so the essential case is that of totally ramified cyclic extensions of degree \(p^m\). The case \(m=1\) was established using global methods by \textit{T. Sekiguchi, F. Oort}, and \textit{N. Suwa} in [``On the deformation of Artin-Schreier to Kummer'', Ann. Sci. Éc. Norm. Supér. (4) 22, No. 3, 345--375 (1989; Zbl 0714.14024)], and the \(m=1,2\) cases were obtained via local methods in [\textit{B. Green} and \textit{M. Matignon}, ``Lifting Galois covers of smooth curves'', Compos. Math. 113, 237--272 (1998; Zbl 0923.14006)]. Recently, A. Obus and S. Wewers have announced a proof of the \(m=3\) case: [{A. Obus}, ``The (local) lifting problem for curves'', \texttt{arXiv:1105.1530}]. In order to establish the conjecture in general, it suffices to lift a single representative from each conjugacy class of series of order \(p^m\) in \(ks\). The present paper contributes to this area by providing (Theorem 5.6) an explicit bijection between the set \(\mathcal{Y}_m\) of conjugacy classes of series of order \(p^m\) and the orbits of a certain group action on a particular set of Witt vectors \(\mathcal{A}_m\subset W_m(K)\). Using Artin-Schreier-Witt theory and work of Kanesaka-Sekiguchi, the author shows that the subset \(\mathcal{A}_m\) is in bijection with the set of \(p^m\)-cyclic totally ramified extensions of \(K\) (with a chosen generator of the Galois group). Moreover, these bijections preserve the ramification data, which can be read off from the Witt vectors in \(\mathcal{A}_m\). Hence, the main result provides an explicit description in terms of Witt vectors (including the ramification breaks) of the series of order \(p^m\) that must be lifted in order to establish Oort's conjecture.
scientific article; zbMATH DE number 5620650
scientific article