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

2018/10/17 by Korchmáros, Gábor, Montanucci, Maria
#14H05 #14H37 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.07506

Abstract

Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field \mathbbK of odd characteristic p≥ 0, and let \rmAut(X) be the group of all automorphisms of X which fix \mathbbK element-wise. For any a subgroup G of \rmAut(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 \mathbbK. 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 \rmAut(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, 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.

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• Large automorphism groups of ordinary curves of even genus in odd\n characteristic
• On algebraic curves with many automorphisms in characteristic p
• Large automorphism groups of ordinary curves in characteristic 2
• Large p-groups of automorphisms of algebraic curves in characteristic p
• Large 3-groups of automorphisms of algebraic curves in characteristic 3