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- Congruence_subgroup abstract "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 2x2 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 principal congruence subgroup, Γ(N). Formally a congruence subgroup is one that contains Γ(N) for some N, 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 {I,-I} 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 {e}) 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 2x2 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_subgroup wikiPageExternalLink item?id=PMIHES_1967__33__59_0.
- Congruence_subgroup wikiPageExternalLink item?id=PMIHES_1974__44__241_0.
- Congruence_subgroup wikiPageID "382694".
- Congruence_subgroup wikiPageRevisionID "562482683".
- Congruence_subgroup hasPhotoCollection Congruence_subgroup.
- Congruence_subgroup subject Category:Group_theory.
- Congruence_subgroup subject Category:Modular_forms.
- Congruence_subgroup type Abstraction100002137.
- Congruence_subgroup type Form106290637.
- Congruence_subgroup type LanguageUnit106284225.
- Congruence_subgroup type ModularForms.
- Congruence_subgroup type Part113809207.
- Congruence_subgroup type Relation100031921.
- Congruence_subgroup type Word106286395.
- Congruence_subgroup comment "In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.".
- Congruence_subgroup label "Congruence subgroup".
- Congruence_subgroup label "Kongruenzuntergruppe".
- Congruence_subgroup sameAs Kongruenzuntergruppe.
- Congruence_subgroup sameAs m.021td6.
- Congruence_subgroup sameAs Q5161000.
- Congruence_subgroup sameAs Q5161000.
- Congruence_subgroup sameAs Congruence_subgroup.
- Congruence_subgroup wasDerivedFrom Congruence_subgroup?oldid=562482683.
- Congruence_subgroup isPrimaryTopicOf Congruence_subgroup.