论文标题
明确的统一界面的单数K3表面的brauer组
Explicit uniform bounds for Brauer groups of singular K3 surfaces
论文作者
论文摘要
令$ k $为一个数字字段。我们给出一个明确的束缚，仅取决于$ [k：\ mathbf {q}] $和néron-severi lattice的判别物，基于Brauer组的大小k3表面$ x/k $，该组对kummorphic of Gocenatip cm ellipt ellipt curves of Isegen of Isegen of Isegen corves corves of Gomorphic of Gemorphic commorphic at Gomorphic commorphic in Gemorphic commorphic comegenationallysmorphic。作为一个应用程序，我们表明，为这种品种设置的Brauer-Manin设置是有效的计算。以GRH为条件，我们还可以使显式绑定仅取决于$ [K：\ Mathbf {Q}] $，并删除椭圆曲线是等于的条件。此外，我们仅取决于$ [k：\ mathbf {q}] $的限制，这是根据$ \ mathbf {c} $的数量的数量，在$ k $上定义的奇异k3表面的同构类别类别，因此证明了有效的Shafarevich Injepenture singular K3 curface的有效版本。
Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$ and the discriminant of the Néron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on $[k:\mathbf{Q}]$ and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on $[k:\mathbf{Q}]$, on the number of $\mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.
