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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$p$-ranks and automorphism groups of algebraic curves
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by ShŇćichi Nakajima PDF
Trans. Amer. Math. Soc. 303 (1987), 595-607 Request permission


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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 303 (1987), 595-607
  • MSC: Primary 14H30
  • DOI:
  • MathSciNet review: 902787