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In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2×2 integer matrices of determinant 1, such that the off-diagonal entries are ''even''. An important class of congruence subgroups is given by reduction of the ring of entries: in general given a group such as the special linear group SL(n, Z) we can reduce the entries to modular arithmetic in Z/NZ for any N >1, which gives a homomorphism :''SL''(''n'', Z) → ''SL''(''n'', Z/''N''·Z) of groups. The kernel of this reduction map is an example of a congruence subgroup – the condition is that the diagonal entries are congruent to 1 mod ''N,'' and the off-diagonal entries be congruent to 0 mod ''N'' (divisible by ''N''), and is known as a , Γ(''N''). Formally a congruence subgroup is one that contains Γ(''N'') for some ''N'',〔Lang (1976) p.26〕 and the least such ''N'' is the ''level'' or ''Stufe'' of the subgroup. In the case ''n=2'' we are talking then about a subgroup of the modular group (up to the quotient by taking us to the corresponding projective group): the kernel of reduction is called Γ(N) and plays a big role in the theory of modular forms. Further, we may take the inverse image of any subgroup (not just ) and get a congruence subgroup: the subgroups Γ0(N) important in modular form theory are defined in this way, from the subgroup of mod ''N'' 2×2 matrices with 1 on the diagonal and 0 below it. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' respected by the subgroup, and so some general idea of what 'congruence' means. ==Congruence subgroups and topological groups== Are all subgroups of finite index actually congruence subgroups? This is not in general true, and ''non-congruence subgroups'' exist. It is however an interesting question to understand when these examples are possible. This problem about the classical groups was resolved by . It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base of neighbourhoods of is the set of subgroups of finite index; and there is another topology defined in the same way using only congruence subgroups. We can ask whether those are the same topologies; equivalently, if they give rise to the same completions. The subgroups of finite index give rise to the completion of Γ as a profinite group. If there are essentially fewer congruence subgroups, the corresponding completion of Γ can be bigger (intuitively, there are fewer conditions for a Cauchy sequence to comply with). Therefore the problem can be posed as a relationship of two compact topological groups, with the question reduced to calculation of a possible kernel. The solution by Hyman Bass, Jean-Pierre Serre and John Milnor involved an aspect of algebraic number theory linked to K-theory. The use of adele methods for automorphic representations (for example in the Langlands program) implicitly uses that kind of completion with respect to a congruence subgroup topology - for the reason that then all congruence subgroups can then be treated within a single group representation. This approach - using a group G(A) and its single quotient G(A)/G(Q) rather than looking at many G/Γ as a whole system - is now normal in abstract treatments. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Congruence subgroup」の詳細全文を読む スポンサード リンク
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