2007/10/22 by A. W. Mason, Mason, A. W., Alexander Premet +5
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0710.4009
openalex publication_date 2007/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a global field and let kv be the completion of k with respect to v, a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of kv-rank 1. Let G=G(kv). Let Γbe an arithmetic lattice in G and let C=C(Γ) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is Fω, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Γ=SL2(O(S)), where O(S) is the ring of S-integers in k, with S=\v\, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Γon the Bruhat-Tits tree associated with G.