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Sergey Rybakov

Publications and source records attributed to Sergey Rybakov.

12 recordsLinked to original sources
arXiv2026

Twisted Deligne modules and abelian varieties over finite fields

Isogeny classes of abelian varieties over finite fields were described by Tate and Honda. Deligne proved that the category of ordinary abelian varieties over a finite field is equivalent to the category of ordinary Deligne modules. Centeleghe and Stix extended Deligne's results to the whole category of abelian varieties, but, if the base finite field is not prime, then the target category is not related to Deligne modules. In this paper we introduce generalized and twisted Deligne modules and give a more direct generalization of the Deligne theorem.

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arXiv2024

Generalized Kummer surfaces over finite fields

In this paper, we prove a refinement of the Katsura theorem on finite group actions on abelian surfaces such that the quotient is birational to a $K3$ surface. As an application, we compute traces of Frobenius on the Neron--Severi groups of supersingular generalized Kummer surfaces over finite fields.

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arXiv2024

Principal polarizations on products of abelian varieties over finite fields

We refine and generalize the results of K. E. Lauter and E. W. Howe on principal polarizations on products of abelian varieties over finite fields. Firstly, we study the reasons for the absence of an irreducible principal polarization in the isogeny class of the product of an ordinary and a supersingular abelian variety. Secondly, we provide a necessary condition for the existence of a principal polarization on an abelian variety in the isogeny class of the product of a geometrically simple abelian surface and an elliptic curve. As an application, we prove that this abelian threefold or its quadratic twist is a Jacobian.

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arXiv2021

Lattices in Tate modules

Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A \in \operatorname{M}_{2g}(\mathbb{Z})$ such that each Tate module $T_\ell X$ has a $\mathbb{Z}_\ell$-basis on which the action of $u$ is given by $A$, and similarly for the covariant Dieudonn\'e module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$.

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arXiv2017

Minimal cubic surfaces over finite fields

Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $Γ$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and $5$ of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.

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arXiv2013

Finite group subschemes of abelian varieties over finite fields

Let $A$ be an abelian variety over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by the Weil polynomial $f_A$. We assume that $f_A$ is separable. For a given prime number $\ell\neq\mathrm{char}\, k$ we give a classification of group schemes $B[\ell]$, where $B$ runs through the isogeny class, in terms of certain Newton polygons associated to $f_A$. As an application we classify zeta functions of Kummer surfaces over $k$.

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arXiv2013

DG-modules over de Rham DG-algebra

For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over de Rham DG-algebras. Positselski proved that for a smooth algebraic variety $X$ over a field $k$ of characteristic zero the coderived category of DG-modules over $Ω^\bullet_{X/k}$ is equivalent to the unbounded derived category of quasi-coherent right ${\mathscr D}_X$-modules. We prove that our functors correspond to the functors of the same name for ${\mathscr D}_X$-modules under Positselski equivalence.

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arXiv2012

The groups of points on abelian surfaces over finite fields

Let $A$ be an abelian surface over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ of degree 4. We give a classification of the groups of $k$-rational points on varieties from this class in terms of $f_A$.

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arXiv2010

The groups of points on abelian varieties over finite fields

Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of $k$-rational points on varieties from this class in terms of Newton polygons of $f_A(1-t)$.

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