On rigid inner forms

TL;DR

This paper provides an explicit construction of global Galois gerbes for defining global rigid inner forms, making the concept more concrete and computationally accessible, which is essential for Arthur's multiplicity formula.

Contribution

It offers a concrete algorithm to compute local rigid inner forms from global data, enhancing the explicitness of global rigid inner forms compared to previous abstract approaches.

Findings

Global rigid inner forms are almost everywhere unramified.

An explicit algorithm for computing local rigid inner forms is provided.

Global rigid inner forms are as explicit as global pure inner forms, up to class field theory computations.

Abstract

We give an explicit construction of global Galois gerbes constructed more abstractly by Kaletha to define global rigid inner forms. This notion is crucial to formulate Arthur's multiplicity formula for inner forms of quasi-split reductive groups. As a corollary, we show that any global rigid inner form is almost everywhere unramified, and we give an algorithm to compute the resulting local rigid inner forms at all places in a given finite set. This makes global rigid inner forms as explicit as global pure inner forms, up to computations in local and global class field theory.