Let $q=p^h$ be an odd prime power and let $X/\F_{q^2}$ be a maximal curve of genus at least two. We classify the curves for which the full geometric automorphism group is transitive on the set of $\F_{q^2}$-rational points. We prove that, if $h>1$, then $X$ is the Hermitian curve. If $h=1$, the only additional possibility occurs for $q=5$: the unique $\F_{25}$-maximal genus-three curve, namely the maximal $S_4$-model of the Klein quartic. The proof separates tame and wild actions. In the tame case, genus bounds, signatures of quotient maps, specialization to characteristic zero, low-genus automorphism classifications, and a Hasse--Witt computation reduce the problem to the Klein quartic. In the wild case, the rational points are identified with the Sylow $p$-subgroups of the automorphism group. Noncyclic Sylow subgroups are treated using a finite-group classification theorem together with Henn's large-automorphism classification, while cyclic Sylow subgroups are excluded by local ramification and the Riemann--Hurwitz formula. Combined with the known characteristic-two result, this yields the classification for all prime powers.
Let $\cX$ be a curve of genus $g\ge2$ and zero $p$-rank over an algebraically closed field of odd characteristic $p$. We classify the pairs $(\cX,G)$ for which $G\le\Aut(\cX)$ has no common fixed point and $|G|>24g(g-1)$. For $g\ge4$, the curves are explicit cyclic covers of the projective line, the Hermitian curve, or...
Let $q=p^h>7$ be odd, put $m=(q+1)/2$, and suppose that $i=(m-2)/2$ satisfies $\gcd(i,m)=\gcd(i+2,m)=1$. For the $\mathbf{F}_{q^2}$-maximal function field $\mathcal{F}_i=\mathbf{F}_{q^2}(x,y)$ defined by $y^m=x^i(x^2+1)$, Peter Beelen, Maria Montanucci, Jonathan Niemann, and Luciane Quoos showed that the geometric auto...
Let $X$ be a curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\le G_P$ satisfies $|S|>\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, o...
Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\mathrm{Out}(H)$ and isomorphism classes of the associated small group rings with orbits of $\math...
Let $X$ be a nonsingular projective curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchm\'aros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\ge2$, positive $p$-r...
Let $\mathcal{X}$ be a projective, geometrically irreducible, nonsingular algebraic curve of genus $g\ge2$ over an algebraically closed field of characteristic two. We classify cyclic subgroups of $\Aut(\mathcal{X})$ of order $N\ge2g+1$. Besides Kummer extensions of odd degree ramified over three points, precisely two...
Marco Timpanella· 1 citation· ⚡1
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