2019/06/04 by Srinivasan, Srimathy
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1906.01414
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.