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 ![]() ![]() ![]() ![]() |
[2] | J.-P. Serre, "Le problème des groupes de congruence pour ![]() |
[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.
Congruence subgroup problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_subgroup_problem&oldid=21407