Tame division algebras of prime period over function fields of p-adic curves

Let F be a field finitely generated and of transcendence degree one over a p-adic field, and let ℓ ≠ p be a prime. Results of Merkurjev and Saltman show that H2(F, µℓ) is generated by ℤ/ℓ-cyclic classes. We prove the “ℤ/ℓ-length” in H2(F, µℓ) is less than the ℓ-Brauer dimension, which Salt-man showed to be three. It follows that all F-division algebras of period ℓ are crossed products, either cyclic (by Saltman’s cyclicity result) or tensor products of two cyclic F-division algebras. Our result was originally proved by Suresh when F contains the ℓ-th roots of unity µℓ. © 2014, Hebrew University of Jerusalem.