Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

We describe the Galois action on the middle $\ell$-adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface $A$ with Mukai vector $v$. We show this action is determined by the action on $H^2_{\'et}(A_{\bar{k}},\mathbb{Q}_\ell(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat{A})[3]$, which depends on $v$. This generalizes the analysis carried out by Hassett and Tschinkel [HT13] over $\mathbb{C}$. As a consequence, we prove that over number fields, $K_2(A)$ and $K_2(\hat{A})$ are not in general derived equivalent. The points of $G_A(v)$ correspond to involutions of $K_A(v)$. Over $\mathbb{C}$, they are known to be symplectic and contained in the kernel of the map $\mathrm{Aut}(K_A(v))\to \mathrm{O}(H^2(K_A(v),\mathbb{Z}))$. We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$. When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed loci in fourfolds $K_A(0,l,s)$ over $\mathbb{C}$ where $l$ is a $(1,3)$-polarization, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$.Comment: 44 pages. v2: added results in positive characteristic and removed a hypothesis from the main result

Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type

We study some combinatorial aspects of the fixed loci of symplectic involutions acting on hyperk\"ahler varieties of Kummer type. Given an abelian surface $A$ with a $(1,d)$-polarization $L$, there is an isomorphism $K_{d-1}A\cong K_{\hat{A}}(0,\hat{l},-1)$ between a hyperk\"ahler of Kummer type that parametrizes length $d$ points on $A$ and one that parametrizes degree $d-1$ line bundles supported on curves in $|\hat{L}|$, where $\hat{L}$ is a $(1,d)$-polarization on $\hat{A}$. We examine the bijection this isomorphism gives between isolated points in the fixed loci of $[-1_A]$ when $d$ is odd, which has a combinatorics related to theta characteristics. Along the way, we give numerical values for a formula of \cite{KMO} counting the number of components of a symplectic involution acting on a Kummer-type variety.Comment: 22 pages, 2 figure

On abelian varieties whose torsion is not self-dual

We construct infinitely many abelian surfaces $A$ defined over the rational numbers such that, for $\ell\leqslant 7$ prime, the $\ell$-torsion subgroup of $A$ is not isomorphic as a Galois module to the $\ell$-torsion subgroup of the dual abelian surface. We do this by analyzing the action of the Galois group on the $\ell$-adic Tate module and its reduction modulo $\ell$.Comment: 22 page

A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold

In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi also showed that over $\mathbb{C}$ each of these Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau threefold $X$. We show that these derived equivalences may also be constructed over $\mathbb{Q}$ and give sufficient conditions for $X$ to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi-Yau variety over $\mathbb{C}$