Specializations of Brauer classes over algebraic function fields

Abstract:

Let $F$ be either a number field or a field finitely generated of transcendence degree $\ge 1$ over a Hilbertian field of characteristic 0, let $F(t)$ be the rational function field in one variable over $F$, and let $\alpha \in \operatorname {Br} (F(t))$. It is known that there exist infinitely many $a\in F$ such that the specialization $t\to a$ induces a specialization $\alpha \to \overline {\alpha }\in \operatorname {Br} (F)$, where $\overline {\alpha }$ has exponent equal to that of $\alpha$. Now let $K$ be a finite extension of $F(t)$ and let $\beta =\operatorname {res} _{K/F(t)}(\alpha )$. We give sufficient conditions on $\alpha$ and $K$ for there to exist infinitely many $a\in F$ such that the specialization $t\to a$ has an extension to $K$ inducing a specialization $\beta \to \overline {\beta }\in \operatorname {Br} (\overline {K})$, $\overline {K}$ the residue field of $K$, where $\overline {\beta }$ has exponent equal to that of $\beta$. We also give examples to show that, in general, such $a\in F$ need not exist.