Specializations of Brauer classes over algebraic function fields

Abstract

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