Congruence subgroup problem

Is every subgroup of finite index in , where is the ring of integers in an algebraic number field (cf. Algebraic number theory) and is a connected linear algebraic group defined over , a congruence subgroup? This is the classical statement of the congruence problem. A contemporary version of it is based on the concept of the congruence kernel, which is a measure of the deviation from a positive solution. Let and be the completions of the groups of -points of in the topologies defined by all subgroups of finite index and all congruence subgroups of , respectively. Then there is a surjective continuous homomorphism . The kernel of is called the congruence kernel and is denoted by . The positive solution of the classical congruence problem is equivalent to proving . In its modern form, the congruence problem is that of computing the congruence kernel .

If , where is the ring of integers, it was known already in the 19th century that the congruence problem has a negative solution for . For , it was proved in 1965 that every subgroup of finite index in is a congruence subgroup (see [1]). After this, the congruence problem was solved [1] for , , and , , where denotes the symplectic group. The results are as follows for these groups; only for totally imaginary fields , in which case the congruence kernel is isomorphic to the (cyclic) group of roots of unity in . It turned out that the same result holds for simply-connected Chevalley groups other than (see [3]). The condition of being simply connected is essential, because it follows from the strong approximation theorem (cf. Linear algebraic group) that the congruence kernel of a non-simply-connected semi-simple group is infinite. For every non-semi-simple group , , where is a maximal semi-simple subgroup of ; in particular, for a solvable group .

A more general form of the congruence problem is obtained by replacing by the ring

where is any finite set of inequivalent norms of the field containing all Archimedean norms. In this situation, the congruence kernel, denoted by , depends in an essential way on (see [4], [5]).

References

[1]	H. Bass, J. Milnor, J.-P. Serre, "Solution of the congruence subgroup problem for () and ()" Publ. Math. IHES , 33 (1967) pp. 421–499
[2]	J.-P. Serre, "Le problème des groupes de congruence pour " Ann. of Math. , 92 (1970) pp. 489–527
[3]	H. Matsumoto, "Sur les sous-groupes arithmétiques des groupes semi-simples dépolyés" Ann. Sci. Ecole Norm. Sup. (4) , 2 (1969) pp. 1–62
[4]	V.P. Platonov, "Algebraic groups" J. Soviet Math. , 4 : 5 (1975) pp. 463–482 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 11 (1974) pp. 5–37
[5]	M. Raghunathan, "On the congruence subgroup problem" Publ. Math. IHES , 46 (1946) pp. 107–161

Comments

The congruence problem is usually called the congruence subgroup problem.