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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).

##### MSC:
 14H30 Coverings of curves, fundamental group 14H37 Automorphisms of curves 14H45 Special algebraic curves and curves of low genus
##### Keywords:
orbifold curves; automorphisms; modular curves; Fermat curves
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##### References:
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