Exercise concerning quaternion algebras

Here’s a fun problem that came up in a talk by Jacob Tsimerman on Monday concerning some joint work with Andrew Snowden:

Problem: Let D/\mathbf{Q}(t) be a quaternion algebra such that the specialization D_t splits for almost all t. Then show that D itself is split.

As a comparison, if you replace \mathbf{Q} by \overline{\mathbf{Q}}, then although the condition that D_t splits becomes empty, the conclusion is still true, by Tsen’s theorem.

This definitely *feels* like the type of question which should have a slick solution; can you find one?

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2 Responses to Exercise concerning quaternion algebras

  1. Perhaps? I confess that it is not obvious to me that it does.

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