Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)$.


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

  • [1] H. Hasse, Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper, J. Reine Angew. Math. 172 (1934), 37-54 = Math. Abh. Band 2, 133-150.
  • [2] H. Hasse and E. Witt, Zyklische unverzweigte Erweiterungskörper von Primzahlgrad $ p$ über einem algebraischen Funktionenkörper der Charakteristik $ p$, Monatsh. Math. Phys. 43 (1936), 477-492=Math. Abh. Band 2, 202-217. MR 1550551
  • [3] H.-W. Henn, Funktionenkörper mit grosser Automorphismengruppe, J. Reine Angew. Math. 302 (1978), 96-115. MR 511696 (80a:14012)
  • [4] M. L. Madan, On a theorem of M. Deuring and I. R. Šafarevič, Manuscripta Math. 23 (1977), 91-102. MR 0460335 (57:329)
  • [5] D. J. Madden and R. C. Valentini, The group of automorphisms of algebraic function fields, J. Reine Angew. Math. 343 (1983), 162-168. MR 705883 (85i:11096)
  • [6] S. Nakajima, Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants, Math. Z. 190 (1985), 559-566. MR 808922 (87g:14024)
  • [7] P. Roquette, Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik, Math. Z. 117 (1970), 157-163. MR 0279100 (43:4826)
  • [8] I. R. Šafarevič, On $ p$-extensions, Amer. Math. Soc. Transl. (2) 4 (1954), 59-72.
  • [9] J-P. Serre, Corps locaux, Hermann, Paris, 1968. MR 0354618 (50:7096)
  • [10] B. Singh, On the group of automorphisms of a function field of genus at least two, J. Pure Appl. Algebra 4 (1974), 205-229. MR 0360600 (50:13047)
  • [11] H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik, I, II, Arch. Math. 24 (1973), 527-544, 615-631.
  • [12] D. Subrao, The $ p$-rank of Artin-Schreier curves, Manuscripta Math. 16 (1975), 169-193. MR 0376693 (51:12868)
  • [13] F. J. Sullivan, $ p$-torsion in the class group of curves with too many automorphisms, Arch. Math. 26 (1975), 253-261. MR 0393035 (52:13846)
  • [14] A. Wiman, Über die hyperelliptischen Curven und diejenigen von Geschlechte $ P = 3$ welche eindeutigen Transformationen in sich zulassen, Bihang Till. Kongl. Svenska Vetenskaps-Akademiens Hadlingar 21 (1895-96), 1-23.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14H30

Retrieve articles in all journals with MSC: 14H30


Additional Information

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

American Mathematical Society