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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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.

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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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