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The Chevalley-Weil formula for orbifold curves. (English) Zbl 1400.14079
Let \(f:X \rightarrow Y\) be a Galois cover of compact Riemann surfaces of genus \(g_X\) and \(g_Y\) respectively. Let \(G=\mathrm{Aut}_X(Y)\) be the corresponding Galois group. The group \(G\) acts by pull-back on the space of holomorphic differential 1-forms of \(X\) giving a \(g_X\)-dimensional complex representation \(\rho_f: G^{\mathrm{op}} \rightarrow\mathrm{GL}(H^0(X,\Omega_X))\), called “canonical representation” of \(G\).
Let \(\rho\) be an irreducible representation of \(G\). C. Chevalley and A. Weil gave in [Abh. Math. Semin. Univ. Hamb. 10, 358–361 (1934; JFM 60.0098.01)] a formula for the multiplicity of \(\rho\) inside \(\rho_f\) in terms of \(g_Y\) and the ramification structure of \(f\). This formula is usually called “Chevalley-Weil formula”.
This result was generalized in the more general setting of algebraic curves over an algebraically closed field of characteristic \(p \geq 0\) provided that \(p \nmid |G|\), and then also by S. Nakajima [Invent. Math. 75, 1–8 (1984; Zbl 0612.14017)] to any coherent sheaf and any ramified cover of algebraic varieties over an any algebraically closed field.
In this paper, the Chevalley-Weil formula is generalized to ramified Galois covers of “orbifold curves” (also known as “Deligne-Mumford curves” or “stacky curves”) under the assumption that the ramification locus is disjoint from the locus of orbifold points in \(Y\). The author analyzes the problem over the complex field \(\mathbb{C}\), but every argument applies over arbitrary algebraically closed field whose characteristic does not divide \(|G|\) (see Section 3).
Applications are also provided in the study of Fermat curves \(F_N\) (See Section 6).

14H30 Coverings of curves, fundamental group
14H37 Automorphisms of curves
14H45 Special algebraic curves and curves of low genus
Full Text: DOI arXiv
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