Raynaud-Tamagawa theta divisors and new-ordinariness of ramified coverings of curves
Let \((X,D_X)\) be a smooth pointed stable curve of genus \(g_X\) over an algebraic closed field \(k\) of characteristic \(p>0\). Define the Hasse-Witt invariant as \(\sigma_X:=\dim_{\mathbb{F}_p}(H_{et}^1(X,\mathbb{F}_p))\). We call the curve \(X\) \emph{ordinary} if \(g_X=\sigma_X\). Consider a multi-admissible cover \((Z,D_Z)\to (X,D_X)\) over \(k\) as in [\textit{Y. Yang}, Publ. Res. Inst. Math. Sci. 54, No. 3, 649--678 (2018; Zbl 1439.14102)]. The cover \((Z,D_Z)\) is called \emph{new-ordinary} if \(\sigma_Z-\sigma_X=g_Z-g_X\). In [\textit{S. Nakajima}, Adv. Stud. Pure Math. 2, 69--88 (1983; Zbl 0529.14016)], Nakajima consider the case when \(D_X=\emptyset\) and \(X\) is generic. He found that if the cover \(Z\) corresponds to a prime-to-\(p\) cyclic quotient of the tame fundamental group \(\Pi_X\) of \(X\) then \(Z\) is ordinary. The main result of the paper under review gives necessary and sufficient conditions for a cover \((Z,D_Z)\to (X,D_X)\) to be new-ordinary when \((X,D_X)\) is generic and the cover corresponds to a prime-to-\(p\) cyclic quotient of the tame fundamental group \(\Pi_{(X,D_X)}\). For the proof, the author relate new-ordinariness to the existence of Raynaud-Tamagawa theta divisors, see [\textit{M. Raynaud}, Bull. Soc. Math. Fr. 110, 103---125 (1982; Zbl 0505.14011)] and [\textit{A. Tamagawa}, Math. Sci. Res. Inst. Publ. 41, 47--105 (2003; Zbl 1073.14035)].
scientific article; zbMATH DE number 7396018
scientific article