Cyclicity and indecomposability in the Brauer group of a $p$-adic curve

TL;DR

This paper investigates the cyclicity and indecomposability of classes in the Brauer group of p-adic curves, revealing conditions for cyclicity and constructing indecomposable algebras under specific circumstances.

Contribution

It identifies when all Brauer classes are cyclic and constructs indecomposable division algebras over function fields of p-adic curves.

Findings

Not all classes in the Brauer group are cyclic in general.

All order n elements are cyclic if the curve has good reduction and n is prime to p.

Existence of indecomposable division algebras with specific period and index over certain function fields.

Abstract

For a $p$-adic curve $X$, we study conditions under which all classes in the $n$-torsion of $Br(X)$ are $\mathbb{Z}/n$-cyclic. We show that in general not all classes are $\mathbb{Z}/n$-cyclic classes. On the other hand, if $X$ has good reduction and $n$ is prime to $p$, of if $X$ is an elliptic curve over $\mathbb{Q}_p$ with split multiplicative reduction and $n$ is a power of $p$, then we prove that all order $n$ elements of $Br(X)$ are $\mathbb{Z}/n$-cyclic. Finally, if $X$ has good reduction and its function field $K(X)$ contains all $p^2$-th roots of $1$, we show the existence of indecomposable division algebras over $K(X)$ with period $p^2$ and index $p^3$.