Large odd prime power order automorphism groups of algebraic curves in any characteristic

Gábor Korchmáros, Maria Montanucci

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p ≥ 0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G| ≤ 9(g − 1) where the extremal case can only be obtained for d = 3 and g ≥ 10. We prove Zomorrodian’s result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups.
Original languageEnglish
JournalJournal of Algebra
Volume547
Pages (from-to)312-344
ISSN0021-8693
DOIs
Publication statusPublished - 2020

Keywords

  • Algebraic curves
  • Algebraic function fields
  • Positive characteristic
  • Automorphism groups

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