Function Fields Of Curves Research Articles

Relative Brauer Groups of Function Fields of Curves of Genus One

Let Kbe an algebraic function field in one variable over a constant field k. In this paper, we investigate the relative Brauer groups Br(K/k) of Kover kin various cases. When kis a global field, we focus on function fields K = k(C) of genus 1 where Cis the curve of the form y 2 = at 4 + bwith a, b ∈ k − {0}, and we describe the Brauer classes in Br(K/k). More precisely, we show that each algebra in Br(K/k) is a quaternion algebra which can be obtained by taking one of a finite number of the x-coordinates of k-rational points on the Jacobian of the curve C. In particular, for the field ℚ of rational numbers, we determine Br(K/ℚ) precisely in numerous cases and give examples.

Algebraic elements in division algebras over function fields of curves

LetD be a division algebra with centerK a function field of a curveC overk;k(C)=K. We study the maximalk-algebraic subfields ofD. In Theorem 3.1 it is shown that ifD is unramified andC is an elliptic curve thenD contains ak-algebraic splitting field. This enables us to give a new class of counter examples to the Hasse principle for division algebras.

Hasse's principle for simple algebras over function fields of curves. I. Algebras of index 2 and 3; curves of genus 0 and 1.

Article Hasse's principle for simple algebras over function fields of curves. I. Algebras of index 2 and 3; curves of genus 0 and 1. was published on January 1, 1978 in the journal Journal für die reine und angewandte Mathematik (volume 1978, issue 299-300).