Searcharxiv⌕ Search

arXiv subjects

Toshiyuki Katsura

Publications and source records attributed to Toshiyuki Katsura.

At least 19 recordsLinked to original sources
arXiv2026

Mordell--Weil groups of quasi-elliptic surfaces in characteristic 3

In characteristic $p =3$, we establish a concrete relationship between the torsion rank of the elementary abelian group of global sections of quasi-elliptic surfaces over the projective line ${\mathbf P}^1$ and the $p$-rank of certain hyperelliptic curves, thereby explaining a mysterious relationship between these two invariants previously observed by I. Dolgachev. As an application, we determine the torsion ranks of the Mordell--Weil groups of a class of quasi-elliptic surfaces.

math.AG↗
arXiv2026

Isogeny graphs associated to Moret-Bailly families of supersingular abelian surfaces

For an odd prime $p$ and any prime $\ell\neq p$, we study finite directed graphs arising from the set of all equivalence classes of Moret-Bailly families of abelian surfaces in characteristic $p$ together with relative $(\ell,\ell)$-isogenies. We relate these graphs to the space of algebraic modular forms on an inner form of $GSp_4/\mathbb{Q}$ that is compact modulo its center and, using the Jacquet-Langlands correspondence, estimate the eigenvalues of their adjacency matrices. We further investigate a Sarnak-Xue type theorem in this setting, providing a first step toward the study of cut-off phenomena for these graphs.

math.NT↗
arXiv2025

Kummer surfaces and quadratic line complexes in characteristic two

In this paper, we study the classical theory of quadratic line complexes and Kummer surfaces. A quadratic line complex is the intersection of the Grassmannian $G(2,4)$ and a quadric hypersurface in ${\bf P}^5$, and a Kummer surface is the quotient of the Jacobian of a curve of genus 2 by the inversion. F. Klein discovered a relationship between a quadratic line complex and a curve of genus 2, its Jacobian and the associated Kummer surface. This theory holds in any characteristic not equal to two. However the situation in characteristic two is entirely different. The purpose of this paper is to give an analogue in characteristic 2 of this classical theory.

math.AG↗
arXiv2024

The 2-divisibility of divisors on K3 surfaces in characteristic 2

We show that K3 surfaces in characteristic 2 can admit sets of $n$ disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each $n=8,12,16,20$. More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only $n=8$ is possible.

math.AG↗
arXiv2024

Normal forms for quasi-elliptic Enriques surfaces and applications

We work out normal forms for quasi-elliptic Enriques surfaces and give several applications. These include torsors and numerically trivial automorphisms, but our main application is the completion of the classification of Enriques surfaces with finite automorphism groups started by Kondo, Nikulin, Martin and Katsura-Kondo-Martin.

math.AG↗
arXiv2023

Decomposed Richelot isogenies of Jacobian varieties of hyperelliptic curves and generalized Howe curves

We advance previous studies on decomposed Richelot isogenies (Katsura--Takashima (ANTS 2020) and Katsura (ArXiv 2021)) which are useful for analysing superspecial Richelot isogeny graphs in cryptography. We first give a characterization of decomposed Richelot isogenies between Jacobian varieties of hyperelliptic curves of any genus. We then define generalized Howe curves, and present two theorems on their relationships with decomposed Richelot isogenies. We also give new examples including a non-hyperelliptic (resp.\,hyperelliptic) generalized Howe curve of genus 5 (resp.\,of genus 4).

math.AG↗
arXiv2021

Coble surfaces in characteristic two

We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such surfaces.

math.AG↗
arXiv2021

Decomposed Richelot isogenies of Jacobian varieties of curves of genus 3

For a nonsingular projective curve $C$ of genus 3 defined over an algebraically closed field of characteristic $p > 2$, we give a necessary and sufficient condition that the Jacobian variety $J(C)$ has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.

math.AG↗
arXiv2020

Counting Richelot isogenies between superspecial abelian surfaces

Castryck, Decru, and Smith used superspecial genus-2 curves and their Richelot isogeny graph for basing genus-2 isogeny cryptography, and recently, Costello and Smith devised an improved isogeny path-finding algorithm in the genus-2 setting. In order to establish a firm ground for the cryptographic construction and analysis, we give a new characterization of {\em decomposed Richelot isogenies} in terms of {\em involutive reduced automorphisms} of genus-2 curves over a finite field, and explicitly count such decomposed (and non-decomposed) Richelot isogenies between {\em superspecial} principally polarized abelian surfaces. As a corollary, we give another algebraic geometric proof of Theorem 2 in the paper of Castryck et al.

math.AG↗
arXiv2020

On the multicanonical systems of quasi-elliptic surfaces

We consider the multicanonical systems $\vert mK_{S}\vert$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 2. We show that for any $m \geq 6$ $\vert mK_{S}\vert$ gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces.

math.AG↗
arXiv2019

Classification of Enriques surfaces with finite automorphism group in characteristic 2

We classify supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 8 types according to their dual graphs of all $(-2)$-curves (nonsigular rational curves). We give examples of these Enriques surfaces together with their canonical coverings. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished.

math.AG↗
arXiv2019

K3 surfaces with 9 cusps in characteristic p

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such surface admits a triple covering by an abelian surface. Conversely, we determine which abelian surfaces with order three automorphisms give rise to K3 surfaces. We also investigate how K3 surfaces with 9 cusps hit the supersingular locus.

math.AG↗
arXiv2017

Zariski K3 surfaces

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine different approaches such as quotients by the group scheme $α_p$, Kummer surfaces, and automorphisms of elliptic and hyperelliptic curves.

math.AG↗
arXiv2015

On Enriques surfaces in characteristic 2 with a finite group of automorphisms

Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a one dimensional family of classical and supersingular Enriques surfaces with the automorphism group isomorphic to the symmetric group of degree five.

math.AG↗
arXiv2015

Lefschetz pencils on a certain hypersurface in positive characteristic

We show that on a certain hypersurface in P3 there is a (q3 + q2 + q + 1)q+1-symmetric configuration (resp. a ((q3 + 1)(q2 + 1)q+1, (q3 + 1)(q + 1)q2+1)) -configuration) made up of the rational points over Fq (resp. over Fq2) and the lines over Fq (resp. the lines over Fq2). We also determine the structure of a Lefschetz pencil made by using a line from these lines.

math.AG↗
arXiv2015

The Chern class map on abelian surfaces

We examine the Chern class map ${c}_{1}: {\rm NS}(S)/p{\rm NS}(S) \rightarrow {\rm H}^{1}(S, Ω^{1}_{S})$ for an abelian surface $S$ in characteristic $p \geq 3$, and give a basis of the kernel $c_{1}$ for the superspecial abelian surface.

math.AG↗