15 June 2007 Values of decomposable forms at S-integral points and orbits of tori on homogeneous spaces
George Tomanov
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Duke Math. J. 138(3): 533-562 (15 June 2007). DOI: 10.1215/S0012-7094-07-13836-5


Let G be a reductive algebraic group defined over a number field K, and let S be a finite set of valuations of K containing all archimedean ones. Let G=vSG(Kv), and let Γ be an S-arithmetic subgroup of G. Let RS and TR=vRTv, where each Tv is a torus of G(Kv) of maximal Kv-rank. We prove that if G/Γ admits a closed TRπ(g)-orbit, then either R=S or R is a singleton, and we describe the closed TR-orbits in both cases. We apply this result to prove that if a collection of decomposable homogeneous forms fvKv[x1,,xn],vS, takes discrete values at On, where O is the ring of S-integers of K, then there exists a homogeneous form gO[x1,,xn] such that fv=αvg, αvKv*, for all vS. Our result is also new in the simplest case of one real homogeneous form when K=Q and O=Z


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George Tomanov. "Values of decomposable forms at S-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


Published: 15 June 2007
First available in Project Euclid: 18 June 2007

zbMATH: 1121.22004
MathSciNet: MR2322686
Digital Object Identifier: 10.1215/S0012-7094-07-13836-5

Primary: 22E40
Secondary: 11K60 , 37D99

Rights: Copyright © 2007 Duke University Press


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Vol.138 • No. 3 • 15 June 2007
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