A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction
Srimathy SrinivasanSchool of Mathematics, Institute for Advanced Study, Princeton NJ, 08540, USA
Given a field equipped with a set of discrete valuations , we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion -algebra to quadratic forms over the function field obtained via Morita equivalence. Using this we show that if satisfies certain conditions, then the number of -isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in is finite and bounded by a value that depends on size of a quotient of the Picard group of and the size of the kernel and cokernel of residue maps in Galois cohomology of with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.
Cite this article
Srimathy Srinivasan, A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction. Doc. Math. 25 (2020), pp. 1171–1194DOI 10.4171/DM/773