arXiv subjects

Jordan Rizov

Publications and source records attributed to Jordan Rizov.

5 recordsLinked to original sources

arXiv2006

Moduli Stacks of Polarized K3 Surfaces in Mixed Characteristic

In this note we define moduli functors of (primitively) polarized K3 spaces. We show that they are representable by Deligne-Mumford stacks over Spec(Z). Further, we look at K3 spaces with a level structure. Our main result is that the moduli functors of K3 spaces with a primitive polarization of degree 2d and a level structure are representable by smooth algebraic spaces over open parts of Spec(Z). To do this we use ideas of Grothendieck, Deligne, Mumford, Artin and others. These results are the starting point for the theory of complex multiplication for K3 surfaces and the definition of Kuga-Satake abelian varieties in positive characteristic given in our Ph.D. thesis.

math.AG↗

arXiv2006

Kuga-Satake Abelian Varieties in Positive Characteristic

Kuga and Satake associate with every polarized complex K3 surface (X,L) a complex abelian variety called the Kuga-Satake abelian variety of (X,L). We use this construction to define morphisms between moduli spaces of polarized K3 surface with certain level structures and moduli spaces of polarized abelian varieties with level structure over C. In this note we study these morphisms. We prove first that they are defined over finite extensions of Q. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

math.AG↗

arXiv2005

Complex Multiplication for K3 Surfaces

In this note we prove analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. This is done by studying the field of definition of the period morphism for complex K3 surfaces. More precisely we relate the moduli spaces of primitively polarized K3 surfaces with level structures over $\Q$, constructed using algebraic stacks, to the canonical model of the Shimura variety associated to $\SO(2,19)$.

math.AG↗

arXiv2005

Non-Emptiness of the Height Strata of the Moduli Stack of Polarized K3 Surfaces

In this paper we consider the following problem: For a given natural number $d$ and a prime $p$ determine all Newton polygons of polarized K3 surfaces of degree 2d over fields of characteristic $p$. This is an analogue of the Manin problem for Newton polygons of abelian varieties. This question is equivalent to determining the non-empty height strata of the moduli stack $\M_{2d}\otimes \F_p$ of K3 surfaces with a polarization of degree 2d over $\F_p$. We prove here that if $d$ is large enough and prime to $p$, then the height strata of $\M_{2d}\otimes \F_p$ are non-empty.

math.AG↗

arXiv2005

Fields of Definition of Rational Points on Varieties

Let $X$ be a scheme over a field $K$ and let $M_X$ be the intersection of all subfields $L$ of $\bar K$ such that $X$ has a $L$-valued point. In this note we prove that for a variety $X$ over a field $K$ finitely generated over its prime field one has that $M_X = K$

math.NT↗