Tractability of algebraic function fields in one variable of genus one over global fields

Abstract

A field k of characteristic unequal to 2 is called tractable if for every nonzero a i , b i ∈ k , i = 1 , 2 , 3 , whenever the quaternion algebra ( a i , b j / k ) is split for all i ≠ j and ( a 1 , b 1 / k ) ≅ ( a 2 , b 2 / k ) ≅ ( a 3 , b 3 / k ) , then ( a i , b i / k ) is split. In the present paper, we study tractability of algebraic function fields in one variable over global fields and give specific examples of tractable function fields and intractable function fields of genus one over Q , the rationals.