Abstract
Let be a reductive algebraic group defined over a number field , and let be a finite set of valuations of containing all archimedean ones. Let , and let be an -arithmetic subgroup of . Let and , where each is a torus of of maximal -rank. We prove that if admits a closed -orbit, then either or is a singleton, and we describe the closed -orbits in both cases. We apply this result to prove that if a collection of decomposable homogeneous forms takes discrete values at , where is the ring of -integers of , then there exists a homogeneous form such that , , for all . Our result is also new in the simplest case of one real homogeneous form when and
Citation
George Tomanov. "Values of decomposable forms at -integral points and orbits of tori on homogeneous spaces." Duke Math. J. 138 (3) 533 - 562, 15 June 2007. https://doi.org/10.1215/S0012-7094-07-13836-5
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