Large odd prime power order automorphism groups of algebraic curves in any characteristic (Q2285233)

Let \(\mathcal{X}\) be a (projective, geometrically irreducible, non-singular) algebraic curve of genus \(g \geq 2\) defined over an algebraically closed field \(K\) of characteristic \(p \geq 0\), and \(d \neq p\) and odd prime. If \(\mathcal{X}\) is a Riemann surface, \textit{R. Zomorrodian} [Trans. Am. Math. Soc. 288, 241--255 (1985; Zbl 0565.20029); Glasg. Math. J. 29, 237--244 (1987; Zbl 0626.20035)] proved that the size of any \(d\)-subgroup of \(\mathrm{Aut}(\mathcal{X})\) is upper-bounded by \(9(g-1)\), and such a bound can be attained only if \(d =3\) and \(g-1\) a power of \(3\) bigger than or equal to \(9\). The first result of this paper is to extend the Zomorrodian bound to any characteristic (Theorems 3.1 and 3.2). This paper combines deep results from group theory with classical tools from algebraic geometry and Galois Theory, chiefly the Riemann-Hurwitz formula and the Riemann-Roch Theorem. The paper is also self contained, and the necessary background is provided in Section 2. The rest of the paper is devoted to a deep study of the so-called extremal \(3\)-Zomorrodian curves, that is to say, curves defined over a field \(K\) of characteristic \(p \neq 3\) such that \(\mathrm{Aut}(\mathcal{X})\) contains a \(3\)-Sylow \(3\)-subgroup \(G\) of size equal to \(9(g-1)\) with \(g = 3^h+1\). Let \(Z \leq Z(G)\) be a subgroup of order \(3\). The key observation here is that the quotient curve \(\bar{\mathcal{X}} = \mathcal{X}/Z\) is either an extremal \(3\)-Zomorrodian curve of genus \(\bar{g} = 3^{h-1}+1\) or an elliptic curve \(\mathcal{E}\) with \(j\)-invariant equal to \(0\). If the latter condition holds, \(\mathcal{X}\) is said to be of \textit{elliptic type}. The rest of the paper is devoted to the study of extremal \(3\)-Zomorrodian curves, starting from the ones of elliptic type (Sections 5-6-7) , with many deep results regarding the structure of \(G\). More in detail, \(Z(G)\) is either of order \(3\) or elementary abelian of order \(9\), and in both cases, \(G\) is completely determined in terms of generators and their relations for \(g \geq 82\) (Theorems 5.5 and 5.6). An infinite family of \(3\)-extremal Zomorrodian curves of elliptic type is provided in Section \(6\), while in Section 7 the case of low genus extremal curves (namely, \(g =10,28\)) is dealt with. Finally, Section \(8\) provides an infinite family of non-elliptic type extremal 3-Zomorrodian curves. The function field \(L\) of such a curve is obtained as the unique maximal unramified abelian \(3\)-extension of the function field \(K(x,y)\) with \( x^9+y^6+y^3 =0\), which is the function field of na elliptic type \(3\)-extremal curve of genus \(10\).