arXiv subjects

Yuri Zarhin

Publications and source records attributed to Yuri Zarhin.

2 recordsLinked to original sources

arXiv2016

Anabelian geometry and descent obstructions on moduli spaces

We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we prove some partial results that indicate that the finite descent obstruction suffices. We also show how this sufficiency implies the same for all hyperbolic curves.

math.NT↗

arXiv2007

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological Brauer-Grothendieck group of $X$ and $Br_0(X)$ the image of $Br(k)$ in $Br(X)$. We write $Br_1(X)$ for the subgroup of elements in $Br(X)$ that become trivial after replacing $k$ by its algebraic closure. We prove that $Br(X)/Br_0(X)$ is finite if $X$ is a $K3$ surface. When $X$ is (a principal homogeneous space of) an abelian variety over $k$ then we prove that $Br(X)/Br_1(X)$ is finite.

math.NT↗