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Relations between the genera and between the Hasse-Witt invariants of Galois coverings of curves. (English) Zbl 0557.14017
Let C be a (smooth, projective) curve defined over a field K, and let \(G\subset Aut(C)\) be a finite group of automorphisms of C. For any subgroups \(H\leq G\), let \(g_ H\) denote the genus of the quotient curve C/H, and let \(\epsilon_ H=(1/| H|)\sum_{h\in H}h\quad \in {\mathbb{Q}}[G]\) denote the norm idempotent associated to H. In this paper we prove theorem \(1: \sum_{H\leq G}r_ H\epsilon_ H=0\quad (r_ H\in {\mathbb{Q}})\quad \Rightarrow \quad \sum r_ Hg_ H=0\) and show that this includes two theorems of R. D. M. Accola [Proc. Am. Math. Soc. 25, 598-602 (1970; Zbl 0212.425)] as special cases. - If \(char(K)=p\neq 0\), then a similar result holds for the Hasse-Witt invariants \(\sigma_ H\) of the quotient curves C/H: Theorem 2. \(\sum r_ H\epsilon_ H=0\quad \Rightarrow \quad \sum r_ H\sigma_ H=0.\) In particular, the analogues of Accola’s theorems are valid for Hasse-Witt invariants.

MSC:
14H30 Coverings of curves, fundamental group
14L30 Group actions on varieties or schemes (quotients)
14H25 Arithmetic ground fields for curves
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