Automorphisms and holomorphic differentials in characteristic p |
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Authors: | Robert C Valentini Manohar L Madan |
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Institution: | Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 USA |
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Abstract: | Let G be a group of automorphisms of a function field F of genus gF over an algebraically closed field K. The space of holomorphic differentials of F is a gF? dimensional K-space. In response to a query of Hecke, Chevalley and Weil (Abh. Math. Sem. Univ. Hamburg, 10 (1934), 358–361) completely determined the structure of as representation space for G in the classical case. They carried out the proof for the special case in which F is unramified over the fixed field of G. The case of cyclic ramified extensions had been previously considered by Hurwitz (Math. Ann., 41 (1893), 37–45). Weil (Abh. Math. Sem. Univ. Hamburg, 11 (1935), 110–115) gave a proof in the general case. The treatment in the last two papers is analytical. In characteristic p, the problem is open. If G is cyclic and F is unramified over the fixed field E of G, Tamagawa (Proc. Japan Acad., 27 (1951), 548–551) proved that the representation is the direct sum of one identity representation of degree 1 and gE ? 1 regular representations. The principal object of this paper is an extension of Tamagawa's result to arbitrary cyclic extensions of p-power degree. The number of times an indecomposable representation of given degree occurs in the representation of G on is explicitly determined in terms of gE and the Witt vector characterizing the extension . The paper also contains a purely algebraic proof of the result of Chevalley and Weil for arbitrary cyclic extensions of degree relatively prime to p. Using character theory, it can be extended to arbitrary groups of order relatively prime to the characteristic. |
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