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Compactifications of S-arithmetic Quotients for the Projective General Linear Group

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 188)

Abstract

Let F be a global field, let S be a nonempty finite set of places of F which contains the archimedean places of F, let \(d\geqslant 1\), and let \(X =\prod _{v\in S} X_v\) where \(X_v\) is the symmetric space (resp., Bruhat-Tits building) associated to \({{\mathrm{PGL}}}_d(F_v)\) if v is archimedean (resp., non-archimedean). In this paper, we construct compactifications \(\Gamma \backslash \bar{X}\) of the quotient spaces \(\Gamma \backslash X\) for S-arithmetic subgroups \(\Gamma \) of \({{\mathrm{PGL}}}_d(F)\). The constructions make delicate use of the maximal Satake compactification of \(X_v\) (resp., the polyhedral compactification of \(X_v\) of Gérardin and Landvogt) for v archimedean (resp., non-archimedean). We also consider a variant of \(\bar{X}\) in which we use the standard Satake compactification of \(X_v\) (resp., the compactification of \(X_v\) due to Werner).

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Dedicated to Professor John Coates on the occasion of his 70th birthday.

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Acknowledgements

The work of the first two authors was supported in part by the National Science Foundation under Grant No. 1001729. The work of the third author was partially supported by the National Science Foundation under Grant Nos. 1401122/1661568 and 1360583, and by a grant from the Simons Foundation (304824 to R.S.).

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Correspondence to Romyar Sharifi .

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Fukaya, T., Kato, K., Sharifi, R. (2016). Compactifications of S-arithmetic Quotients for the Projective General Linear Group. In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_5

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