Indecomposability and cyclicity in the p-primary part of the Brauer group of a p-adic curve

Abstract

We obtain two results about the p-primary part of the Brauer group of a p-adic curve X. First, assuming enough roots of unity and that X has good reduction, we construct indecomposable K(X)-division algebras of period p² and index p³. Second, for an elliptic curve X with split multiplicative reduction, we show that all order p elements of $\text{Br}(X)$ are $\mathbb{Z}/p$-cyclic, and that if moreover X is defined over ${\mathbb{Q}}_p,$ that all order p^r elements of $\text{Br}(X)$ are $\mathbb{Z}/p^r$-cyclic for $r\ge 1.$

Keywords:

• Brauer group
• cyclic algebras
• indecomposable division algebras
• p-adic curves

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16K (primary)
• 14G (secondary)