论文标题
奇数扭转brauer元素和对角四分之一表面的算术数字字段
Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields
论文作者
论文摘要
我们在对Brauer元素的局部评估中使用最新进展来研究Brauer组的{\ IT奇数}扭转元素在对角线四分之一表面上{\ it nutyary}数字字段上所起的作用。我们表明，在局部字段上，如果Brauer元素的顺序是奇怪的，并且与残基特征相关，则其在局部点上诱导的评估图是恒定的。在数字字段上，我们对方程系数的足够条件，该方程式温和且易于检查，因此奇数扭转不会阻碍弱近似。我们还注意到了一种系统的方法，可以在$ \ q_2 $上产生$ k3 $的表面，并以良好的减少和Brauer Group的非平凡的2个扭力元素，而Swan Coventor Zero则是零。
We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also note a systematic way to produce $K3$ surfaces over $\Q_2$ with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.
