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Register a new userThe congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
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Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=mathbf{G}(k_v). Let Gamma be an arithmetic lattice in G and let C=C(Gamma) 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 hat{F}_{omega}, 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 Gamma=SL_2(mathcal{O}(S)), where mathcal{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 Gamma on the Bruhat-Tits tree associated with G.
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