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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ p$-ranks and automorphism groups of algebraic curves


Author: Shōichi Nakajima
Journal: Trans. Amer. Math. Soc. 303 (1987), 595-607
MSC: Primary 14H30
DOI: https://doi.org/10.1090/S0002-9947-1987-0902787-6
MathSciNet review: 902787
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Abstract: Let $ X$ be an irreducible complete nonsingular curve of genus $ g$ over an algebraically closed field $ k$ of positive characteristic $ p$. If $ g \geqslant 2$, the automorphism group $ \operatorname{Aut} (X)$ of $ X$ is known to be a finite group, and moreover its order is bounded from above by a polynomial in $ g$ of degree four (Stichtenoth). In this paper we consider the $ p$-rank $ \gamma $ of $ X$ and investigate relations between $ \gamma $ and $ \operatorname{Aut} (X)$. We show that $ \gamma $ affects the order of a Sylow $ p$-subgroup of $ \operatorname{Aut} (X)\;(\S3)$ and that an inequality $ \vert\operatorname{Aut} (X)\vert \leqslant 84(g - 1)g$ holds for an ordinary (i.e. $ \gamma = g$) curve $ X\,(\S4)$.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0902787-6
Article copyright: © Copyright 1987 American Mathematical Society