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The purpose of this article is to show that the Castelnuovo theory for abelian varieties, developed by G. Pareschi and M. Popa, can be infinitesimalized. More precisely, we prove that an irreducible principally polarized abelian variety has a finite scheme in extremal position, in the sense of Castelnuovo theory for abelian varieties, if, and only if, it is a Jacobian and the scheme is contained in a unique Abel-Jacobi curve.
We prove that the torsion points of an abelian variety are equidistributed over the corresponding berkovich space with respect to the canonical measure.
Let $X$ be a polarized abelian variety over a field $K$. Let $O$ be a ring with an involution that acts on $X$ and this action is compatible with the polarization. We prove that the natural action of $O$ on $(X times X^t)^4$ is compatible with a certain principal polarization.
These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ c
We study endomorphisms of abelian varieties and their action on the l-adic Tate modules. We prove that for every endomorphism one may choose a basis of each Tate module such that the corresponding matrix has rational entries and does not depend on l.
A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $mathcal{B}$ of $G$ contains a commutative subgroup $mathcal{A}subset mathcal{B}$ such that $mathcal{A}$ is normal in $mathcal{B}$ and the index $[mat