Geodesic
An approximation principle for congruence subgroups
Journal of the European Mathematical Society, Tome 20 (2018) no. 5, pp. 1075-1138

Voir la notice de l'article provenant de la source EMS Press

The motivating question of this paper is roughly the following: given a flat group scheme G over Zp​, p prime, with semisimple generic fiber GQp​​, how far are open subgroups of G(Zp​) from subgroups of the form X(Zp​)Kp​(pn), where X is a subgroup scheme of G and Kp​(pn) is the principal congruence subgroup Ker(G(Zp​)→G(Z/pnZ))? More precisely, we will show that for GQp​​ simply connected there exist constants J≥1 and ε>0, depending only on G, such that any open subgroup of G(Zp​) of level pn admits an open subgroup of index ≤J which is contained in X(Zp​)Kp​(p⌈εn⌉) for some proper, connected algebraic subgroup X of G defined over Qp​. Moreover, if G is defined over Z, then ε and J can be taken independently of p.
DOI : 10.4171/jems/783
Classification : 20-XX, 22-XX
Keywords: Lattices in Lie groups, uniform pro-p groups, Lie algebras 
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     author = {Tobias Finis and Erez Lapid},
     title = {An approximation principle for congruence subgroups},
     journal = {Journal of the European Mathematical Society},
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     year = {2018},
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Tobias Finis; Erez Lapid. An approximation principle for congruence subgroups. Journal of the European Mathematical Society, Tome 20 (2018) no. 5, pp. 1075-1138. doi: 10.4171/jems/783

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