A. W. Mason

Congruence hulls in SLn

Abstract Let D be a Dedekind ring. For a large class of subgroups S of SLn(D) (which includes, for example, all subgroups of finite index and all non-central, normal subgroups) it is proved that there is a (unique) smallest congruence subgroup of SLn(D) containing S. This is called the congruence hull of S and is denoted by C(S). Let A be a Dedekind ring of arithmetic type and let A∗ denote its units. From well-known results of Bass, Liehl, Milnor, Serre and Vaserstein it follows that, when A∗ is infinite or n≥3, every subgroup S of SLn(A) containing a non-central, normal subgroup has a congruence hull. In addition the index |C(S):S| divides m, the number of roots of unity in A. When n = 2 and A∗ is finite the situation is completely different. Using previous results of the author it is proved that, for any such A and any finitely generated group G, there exists a subgroup S of SL2(A), normal in its congruence hull, such that G is isomorphic to C(S), modulo S.

THE MINIMUM INDEX OF A NON-CONGRUENCE SUBGROUP OF SL2 OVER AN ARITHMETIC DOMAIN

AbstractLetA be an arithmetic Dedekind ring with only finitely many units. It is known that (i)A = ℤ, the ring of rational integers, (ii)A =Od, the ring of integers of the imaginary quadratic field

Serre’s generalization of Nagao’s theorem: An elementary approach

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The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

, whered is a square-free positive integer, or (iii)A=C=C(C,P,k), the coordinate ring of the affine curve obtained by removing a closed pointP from a smooth projective curveC over afinite fieldk. serre has shown that, in comparison with other low rank arithmetic groups, the groups SL2(A) have “many” non-congruence subgroups.Let ncs (A) denote the smallest index of a non-congruence subgroup of SL2(A). It is well-known that ncs(ℤ) = 7. Grunewald and Schwermer have proved that, with 4 exceptions, ncs(Od) = 2. In this paper we prove that ncs(C)=2, for “most”, but not all,C.

THE STABILIZERS IN A DRINFELD MODULAR GROUP OF THE VERTICES OF ITS BRUHAT–TITS TREE: AN ELEMENTARY APPROACH

Abstract Let ν be a (discrete) valuation of the rational function field k(t), where k is a field, and let 𝒞ν be the intersection of the valuation rings of all the valuations of k(t), other than that of ν. It is well-known that either ν = νπ, the valuation determined by some irreducible polynomial π in k[t], or ν = ν∞, the “valuation at infinity”. In this paper we prove that GL 2(𝒞ν), where ν = νπ, is the fundamental group of a certain tree of groups. The tree has finitely many vertices and its terminal vertices correspond with the elements of the ideal class group of 𝒞ν. This extends a previous result of Nagao for the special case ν = ν∞. In this case 𝒞ν = k[t] and Nagao proves that GL 2(k[t]) is an amalgamated product of a pair of groups. As a consequence we show that, when the degree of π is at least 4, GL 2(𝒞ν) has a free, non-cyclic quotient whose kernel contains (for example) all the unipotent matrices. This represents a two-dimensional anomaly.

Nonrational Genus Zero Function Fields and the Bruhat–Tits Tree

Let C be a smooth projective curve over a field k. For each closed point Q of C let C = C(C,Q, k) be the coordinate ring of the affine curve obtained by removing Q from C. Serre has proved that GL2(C) is isomorphic to the fundamental group, π1(G,T ), of a graph of groups (G, T ), where T is a tree with at most one non-terminal vertex. Moreover the subgroups of GL2(C) attached to the terminal vertices of T are in one-one correspondence with the elements of Cl(C), the ideal class group of C. This extends an earlier result of Nagao for the simplest case C = k[t]. Serre’s proof is based on applying the theory of groups acting on trees to the quotient graph X = GL2(C)\X, where X is the associated Bruhat-Tits building. To determine X he makes extensive use of the theory of vector bundles (of rank 2) over C. In this paper we determine X using a more elementary approach which involves substantially less algebraic geometry. The subgroups attached to the edges of T are determined (in part) by a set of positive integers S, say. In this paper we prove that S is bounded, even when Cl(C) is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of GL2(C), involving unipotent and elementary matrices.

On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

Let be a function field of genus with a finite constant field . Choose a place of of degree and let be the arithmetic Dedekind domain consisting of all elements of that are integral outside . An explicit formula is given (in terms of , and ) for the minimum index of a non-congruence subgroup in SL. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL. The minimum index of a normal non-congruence subgroup is also determined.

Elliptic points of the Drinfeld modular groups

Abstract Let k be a global field and let kυ be the completion of k with respect to υ a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of kυ -rank 1. Let G = G(kυ ). Let Г be an arithmetic lattice in G and let C = C(Г) 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 Г by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is , 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 , where is the ring of S-integers in k, with S = {υ}, 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 Г on the Bruhat-Tits tree associated with G.