Transcendental Brauer groups of products of CM elliptic curves
Transcendental Brauer groups of products of CM elliptic curves
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CM 椭圆曲线乘积的超越布劳尔群
DOI：
10.1112/jlms/jdv058
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Rachel Newton
中科院分区：
文献类型：
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作者：
Rachel Newton
设L是数域，E/L是椭圆曲线，其复数乘为虚二次域K的整数环OK.利用类域理论和Skorobogatov和Zarhin的结果计算了交换曲面E×E的Brauer群的超越部分.奇数阶挠率的结果也适用于K3曲面Kum（E×E）的Brauer群。我们明确地描述了椭圆曲线E/Q与OK的复数乘法，使得E×E的Brauer群包含一个奇数阶超越元.我们证明了这样一个元对Kum（E×E）上的弱逼近产生Brauer-Manin阻碍，而Brauer群的代数部分不产生阻碍.
Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers OK of an imaginary quadratic field K . We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface E×E . The results for the odd‐order torsion also apply to the Brauer group of the K3 surface Kum(E×E) . We describe explicitly the elliptic curves E/Q with complex multiplication by OK such that the Brauer group of E×E contains a transcendental element of odd order. We show that such an element gives rise to a Brauer–Manin obstruction to weak approximation on Kum(E×E) , while there is no obstruction coming from the algebraic part of the Brauer group.