具有好约化的四元数斜埃尔米特型的特殊酉群的有限性定理
A finiteness theorem for special unitary groups of quaternionic skew-hermitian forms with good reduction
摘要：
给定一个域$K$配备一组离散赋值$V$，我们开发了一个一般的理论，涉及约化性质的四元数$K$ -代数$Q$上的斜厄米特形式的函数域$K(Q)$上的二次型通过森田等价。利用这一点，我们表明，如果$(K,V)$满足一定的条件，则四元数斜交群的特殊酉群的泛覆盖的$K$ -同构类的个数在$V$中的所有赋值处具有好约化的厄米特形式是有限的，并且由取决于$V$的Picard群的商的大小和伽罗瓦上同调中的剩余映射的核和上核的大小的值限定具有有限系数的$K$。作为一个推论，我们证明了一个猜想Chernousov，Rapinchuk，Rapinchuk为这种类型的群体。
英文摘要：
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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