Branches in the Bruhat-Tits tree for local fields of even characteristic

TL;DR

This paper extends the analysis of branches in the Bruhat-Tits tree to local fields of even characteristic, aiding in understanding maximal orders and solving the selectivity problem using new concepts related to Artin-Schreier extensions.

Contribution

It introduces an analogous concept to quadratic defect for Artin-Schreier extensions and generalizes the analysis of branches beyond pure quaternion orders in even characteristic fields.

Findings

Extended computations for branches in even characteristic fields.

Characterized an analog of quadratic defect for Artin-Schreier extensions.

Provided a generalized Hilbert symbol for arbitrary generators.

Abstract

We extend our previous computations for the relative positions of branches of quaternions to the case of local fields of even characteristic. This is a key step to understand the set of maximal orders containing a given suborder, which is useful, for instance, to compute relative spinor images, thus solving the selectivity problem. In our previous work, the results where given in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin-Schreier extensions. It is no longer useful to restrict our attention to orders generated by pure quaternions, as a separable quadratic extension contains no non-trivial element of null trace. In this work we state our result for an arbitrary pair of generators, for which we discuss a more general version of the Hilbert symbol in this context.