Large automorphism groups of curves of zero $p$-rank in odd characteristic

Saeed Tafazolian

Home/Archive/arXiv

arXiv · 2609.33185

Large automorphism groups of curves of zero $p$-rank in odd characteristic

Saeed Tafazolian

View original source ↗Read source document ↗Export metadata ↓

Abstract

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 the Ree curve. We determine their full automorphism groups and the possible subgroups $G$, including the central extensions. The exceptional $A_7$ action in characteristic $5$ determines the Hermitian curve of degree $6$. The cases of genus $2$ and $3$ are treated separately. These results give an odd-characteristic counterpart to the classification of large automorphism groups of zero $2$-rank curves. As an application, we obtain a new characterization of the generalized Suzuki curve in which the fixed-point hypothesis is no longer required.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Saeed Tafazolian. 2026-09-27. Large automorphism groups of curves of zero $p$-rank in odd characteristic. https://arxiv.org/abs/2609.33185

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rationality problem for norm one tori for small non-Galois extensions: Symmetric, alternating and linear group cases including stably rational ${\rm PSL}_2(\mathbb{F}_{2^d})$ cases with $d=1,2,3$

Let $k$ be a field, $K/k$ be a separable field extension of degree $n$ and $L/k$ be the Galois closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$ with $[G:H]=n$. Then $H$ is core-free in $G$, i.e. $\bigcap_{σ\in G} σ^{-1}Hσ=\{1\}$. Let $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ be the norm one torus of $K/k$ with ${\rm dim}\, T=n-1$. We fix $G$ as an abstract group and let $H$ range over all core-free subgroups of $G$ up to conjugacy. For each of the following $15$ groups $G\simeq S_3$, $S_4$, $S_5$, $S_6$, $A_4$, $A_5$, $A_6$, ${\rm GL}_2(\mathbb{F}_q)$ $(q=3,4,5)$, ${\rm SL}_2(\mathbb{F}_q)$ $(q=3,5,7)$, ${\rm PSL}_2(\mathbb{F}_q)$ $(q=7,8)$, by using GAP computations with the aid of PARI/GP, we determine, for any core-free subgroup $H\leq G$ up to conjugacy, whether $T$ is stably $k$-rational, not stably but retract $k$-rational, or not retract $k$-rational. The resulting $175$ cases ($15$ are Galois, $160$ are non-Galois) split into $7$, $7$, $161$ cases respectively. We also give alternative proofs for the known cases (i) $H=\{1\}$, i.e. $K/k$ is Galois, and (ii) $[K:k]=[G:H]=n\leq 15$ with $G\not\simeq 9T27\simeq {\rm PSL}_2(\mathbb{F}_8)$. The $6$ stably $k$-rational cases occur for $G\simeq S_3\simeq {\rm PSL}_2(\mathbb{F}_{2})$, $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$, ${\rm PSL}_2(\mathbb{F}_8)$. Except for the stably $k$-rational case $d=1$ and $H=\{1\}$, for $G\simeq {\rm PSL}_2(\mathbb{F}_{2^d})$ with $d=1,2,3$, we prove that $T$ is stably $k$-rational if and only if $T$ is retract $k$-rational if and only if $(C_2)^d\leq H\leq (C_2)^d\rtimes C_{2^d-1}$, and conjecture that this holds for any $d\geq 1$. When $k$ is a global field and $T$ is retract $k$-rational, e.g. $(G,H)\simeq ({\rm PSL}_2(\mathbb{F}_7),D_4)$, $({\rm PSL}_2(\mathbb{F}_8),(C_2)^3)$, $({\rm PSL}_2(\mathbb{F}_8),(C_2)^3\rtimes C_7)$, the Hasse norm principle holds for $K/k$.

math.AG↗

arXiv2026

The Gersten conjecture for $p$-adic étale Tate twists and the $p$-adic cycle class map

We prove the Gersten conjecture for $p$-adic étale Tate twists for a smooth scheme $X$ in mixed characteristic in the Nisnevich topology. Our main observation is that, while $p$-adic étale Tate twists are not $\mathbb A^1$-invariant, for the proof of the Gersten conjecture it suffices that they satisfy the $\mathbb P^1$-bundle formula. This fits nicely with the emphasis on the projective bundle formula in non $\mathbb A^1$-invariant motivic cohomology recently developed by Elmanto-Morrow and Annala-Hoyois-Iwasa. Furthermore, identifying $p$-adic étale Tate twists with the syntomic cohomology defined by Bhatt-Morrow-Scholze, the result generalises the Gersten conjecture for logarithmic deRham-Witt sheaves due to Gros-Suwa to arbitrary characteristic. We also extend work of Schmidt-Strunk on the Gersten conjecture for presheaves of spectra to the non-$\mathbb A^1$-invariant situation. In the second part of the article, we revisit the cycle class map from thickened zero-cycles on the special fiber of $X$ to étale cohomology with coefficients in $p$-adic étale Tate twists previously studied in [26]. This cycle class map is important in the study of zero-cycles on smooth projective varieties over local fields and the approach to the cycle class map which we use in this article is more conceptual and, in contrast to the approach in loc. cit., works for arbitrary finite residue fields.

math.AG↗

arXiv2026

The multiplicity of a Mori Dream Space: applications to polar duality and the classification of toric $\mathbb{Q}$-Fano varieties

We extend the multiplicity of fake weighted projective spaces, as introduced by Averkov, Kasprzyk, Lehmann and Nill in 2021, to complete toric varieties and Mori Dream Spaces, defining it as the order of the torsion subgroup of the divisor class group. It equals the degree of the canonical $1$-covering, a finite surjective morphism étale in codimension one. We classify toric $\Q$-Fano varieties with prescribed weight matrix and Gorenstein index by admissible subgroups of a finite abelian group, realized as the deck transformation groups of their universal $1$-coverings. The classification allows arbitrary class-group rank without assuming $\Q$-factoriality, and yields an intrinsic intermediate $1$-covering, the \emph{unitary} one, controlling the Gorenstein index. We outline a computational classification strategy based on weight matrices and finite subgroup enumeration. In the Gorenstein Fano case, a perfect pairing of weight groups expresses polar duality as an inclusion-reversing subgroup correspondence. For iso-weighted polar partners over $\C$, the mirror correspondence admits an internal description within the anticanonical family of their common universal $1$-covering, through invariant subfamilies and their quotient actions. Finally, we derive divisibility relations and multiplicity bounds, also applicable to Mori Dream Spaces whose canonical toric ambient varieties admit suitable $\Q$-Fano completions.

math.AG↗