A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction

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.