A METHOD OF COMPUTING THE CONSTANT FIELD OBSTRUCTION TO THE HASSE PRINCIPLE FOR THE BRAUER GROUPS OF GENUS ONE CURVES

作者：Han Ilseop^*

来源：Journal of the Korean Mathematical Society, 2016, 53(6): 1431-1443.

DOI：10.4134/JKMS.j150572

• Brauer groups
• Hasse principle
• function fields of genus 1

Let k be a global field of characteristic unequal to two. Let C: y(2) - f(x) be a nonsingular projective curve over k, where f(x) is a quartic polynomial over k with nonzero discriminant, and K = k(C) be the function field of C. For each prime spot p on k, let (k) over cap (p) denote the corresponding completion of k and (k) over cap (p) (C) the function field of C x (k) (k) over cap (p) .
h : Br(K) -> Pi Br-p((k) over cap (p)(C))
where p ranges over all the prime spots of k. In this paper, we explicitly describe all the constant classes (coming from Br(k)) lying in the kernel of the map h, which is an obstruction to the Hasse principle for the Brauer groups of the curve. The kernel of h can be expressed in terms of quaternion algebras with their prime spots. We also provide specific examples over Q, the rationals, for this kernel.

• 出版日期2016-11