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

  • Srimathy Srinivasan

    School of Mathematics, Institute for Advanced Study, Princeton NJ, 08540, USA
A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction cover
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Abstract

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

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Srimathy Srinivasan, A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction. Doc. Math. 25 (2020), pp. 1171–1194

DOI 10.4171/DM/773