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

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. 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.

On nilpotent automorphism groups of function fields

“…Remark 3.7. The bound 10(g − 1) is attained only by F of genus 2 such that F is of type (5,10) if char(K) = 2 or (2, 10) if char(K) = 5.…”

“…The following example shows that the bound in Theorem 3.4 is attained by function fields of infinitely many genera. We apply a similar approach as in [5]. Note that F/K(x) is a Kummer extension of degree 2, where (x = ∞), (x = 0) and (x = ζ 2k ), k = 0, 1, 2, 3, are the ramified places of K(x), see [13,Proposition 3.7.3].…”

“…the ramification indices 5 and 10, respectively. That is, F is of type (5,10) satisfying [F : K(z)] = 10(g(F ) − 1).…”