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arXiv · 1906.01414

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

Srimathy Srinivasan
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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.

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Srimathy Srinivasan. 2019-06-04. A finiteness theorem for special unitary groups of quaternionic skew-hermitian forms with good reduction. https://arxiv.org/abs/1906.01414

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