ABELIAN -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\,Y/\text{Br}_{1}\,Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd torsion of $\text{Br}Y/\text{Br}_{1}Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $\ell$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Néron–Severi lattice, $\#(\text{Br}Y/\text{Br}_{1}Y)[\ell ^{\infty }]$ is bounded by a constant that depends only on $\ell$, $r$, and the discriminant.

Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

Letkbe a number field andTak-torus. Consider a family of torsors underT, i.e. a morphismf:X→ ℙ1kfrom a projective, smoothk-varietyXto ℙ1k, the generic fibreXη→ η of which is a smooth compactification of a principal homogeneous space underT⊗kη. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation forX, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional ifk=Qand all non-split fibres offare defined overQ. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

Global Duality, Signature Calculus and the Discrete Logarithm Problem

We develop a formalism for studying the discrete logarithm problem for the multiplicative group and for elliptic curves over finite fields by lifting the respective group to an algebraic number field and using global duality. One of our main tools is the signature of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of its ramification at certain places. We then develop signature calculus, which generalizes and refines the index calculus method. Finally, using some heuristics, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem. This relates the discrete logarithm problem to some very well-known problems in algebraic number theory and arithmetic geometry.