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Strongly branched coverings of closed Riemann surfaces


Author: Robert D. M. Accola
Journal: Proc. Amer. Math. Soc. 26 (1970), 315-322
MSC: Primary 30.45
DOI: https://doi.org/10.1090/S0002-9939-1970-0262485-4
MathSciNet review: 0262485
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Abstract: Let $ b:{W_1} \to {W_2}$ be a $ B$-sheeted covering of closed Riemann surfaces of genera $ {p_1}$ and $ {p_2}$ respectively. $ b$ is said to be strongly branched if $ {p_1} > {B^2}{p_2} + {(B - 1)^2}$. If $ {M_2}$ is the function field on $ {W_1}$ obtained by lifting the field from $ {W_2}$ to $ {W_1}$, then $ {M_2}$ is said to be a emphstrongly branched subfield if the same condition holds.

If $ {M_1}$ admits a strongly branched subfield, then there is a unique maximal one. If $ {M_2}$ is this unique one and $ f$ is a function in $ {M_1}$ so that

$\displaystyle (B - 1)o(f) < ({p_1} - B{p_2}) + (B - 1)$

then $ f \in {M_2}$, where $ o(f)$ is the order of $ f$. (This is a generalization of the hyperelliptic situation.) These results are applied to groups of automorphisms of $ {W_1}$ to obtain another generalization of the hyperelliptic case.

References [Enhancements On Off] (What's this?)

  • [1] Walter L. Baily, Jr., On the automorphism group of a generic curve of genus $ > 2$, J. Math. Kyoto Univ. 1 (1961/2), 101-108; correction, p. 325. MR 26 #121. MR 0142552 (26:121)
  • [2] A. V. Dorodnov, On the structure of fields of algebraic functions, Kazan State Univ. Sci Survey Conf., 1962, Izdat. Kazan Univ., Kazan, 1963, pp. 21-22. (Russian) MR 32 #5649. MR 0188210 (32:5649)
  • [3] K. Hensel and G. Landsberg, Theorie der algebraischen Funktionen einer Variablen und ihrer Anwendung auf algebraische Kurven und Abelsche Integrale, Chelsea, New York, 1965. MR 33 #272. MR 0192045 (33:272)
  • [4] Helmut Röhrl, Unbounded covering of Riemann surfaces and extensions of rings of meromorphic functions, Trans. Amer. Math. Soc. 107 (1963), 320-346. MR 26 #6397. MR 0148900 (26:6397)
  • [5] Robert Walker, Algebraic curves, Dover, New York, 1962. MR 26 #2438. MR 0144897 (26:2438)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0262485-4
Keywords: Riemann surface, coverings of closed Riemann surfaces, function fields, linear series, automorphism, strongly branched coverings, strongly branched subfields
Article copyright: © Copyright 1970 American Mathematical Society

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