Editing Congruence subgroup (section)

{{Short description|Matrix group}}
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 is the subgroup of [[invertible matrix|invertible]] {{nowrap|2 × 2}} integer matrices of [[determinant]] 1 in which the off-diagonal entries are <em>even</em>. More generally, the notion of '''congruence subgroup''' can be defined for [[arithmetic subgroup]]s of [[algebraic group]]s; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is [[residually finite]]. An important question regarding the algebraic structure of arithmetic groups is the '''congruence subgroup problem''', which asks whether all subgroups of finite [[Index of a subgroup|index]] are essentially congruence subgroups.

Congruence subgroups of {{nowrap|2 × 2}} matrices are fundamental objects in the classical theory of [[modular form]]s; the modern theory of [[automorphic form]]s makes a similar use of congruence subgroups in more general arithmetic groups.