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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: ShŇćichi 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 $\operatorname {Aut} (X)\;(\S 3)$ and that an inequality $|\operatorname {Aut} (X)| \leqslant 84(g - 1)g$ holds for an ordinary (i.e. $\gamma = g$) curve $X (\S 4)$.


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Article copyright: © Copyright 1987 American Mathematical Society