# A finiteness theorem for special unitary groups of quaternionic skew-hermitian forms with good reduction

## TL;DR

This paper establishes a finiteness theorem for certain special unitary groups over quaternionic skew-hermitian forms with good reduction, linking algebraic properties to geometric and cohomological invariants.

## Contribution

It develops a general theory connecting reduction properties of skew-hermitian forms over quaternion algebras to quadratic forms, proving a conjecture for these groups.

## Key findings

- Finiteness of isomorphism classes of universal coverings with good reduction
- Bound depends on Picard group quotient and Galois cohomology kernels
- Proves a conjecture of Chernousov, Rapinchuk, Rapinchuk

## Abstract

Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion $K$-algebra $Q$ to quadratic forms over the function field $K(Q)$ obtained via Morita equivalence. Using this we show that if $(K,V)$ satisfies certain conditions, then the number of $K$-isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in $V$ is finite and bounded by a value that depends on size of a quotient of the Picard group of $V$ and the size of the kernel and cokernel of residue maps in Galois cohomology of $K$ with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.