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Register a new userOn matrices of endomorphisms of abelian varieties
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Authors
Yuri G. Zarhin
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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.
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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.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
We show that the infinitesimal deformations of the Brill--Noether locus $W_d$ attached to a smooth non-hyperelliptic curve $C$ are in one-to-one correspondence with the deformations of $C$. As an application, we prove that if a Jacobian $J$ deforms together with a minimal cohomology class out the Jacobian locus, then $J$ is hyperelliptic. In particular, this provides an evidence to a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class. Finally, we also study simultaneous deformations of Fano surfaces of lines and intermediate Jacobians.
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 $ell=p$ case) that is based on a quaternion trick. In fact, a a slightly stronger version of those theorems with finite coefficients is proven.
We prove that the torsion points of an abelian variety are equidistributed over the corresponding berkovich space with respect to the canonical measure.
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