Congruence subgroup

In mathematics , congruence subsets are a class of arithmetically defined discrete subsets of the general linear group .

In the theory of module forms , congruence subgroups to the module group are often considered. ${\displaystyle \mathrm {SL} (2,\mathbb {Z} )}$

definition

Be

${\displaystyle G\subset \mathrm {GL} (n,\mathbb {C} )}$

an over- defined algebraic group and a natural number. Then ${\displaystyle \mathbb {Q} }$ ${\displaystyle N}$

${\displaystyle \Gamma :=\ker(p_{N}\colon G(\mathbb {Z} )\rightarrow \mathrm {GL} (n,\mathbb {Z} /N\mathbb {Z} ))}$

a congruence subgroup . (Here the restriction of the "reduction modulo N" denotes .) ${\displaystyle p_{N}}$ ${\displaystyle G(\mathbb {Z} )}$

Arithmetic groups

Congruence subgroups are (by construction) arithmetic groups . For each arithmetic group contains a congruence subgroup. ${\displaystyle n\geq 3}$ ${\displaystyle \Gamma \subset \mathrm {SL} (n,\mathbb {C} )}$

General rings

Be a commutative ring . A congruence subgroup is the core of homomorphism ${\displaystyle R}$

${\displaystyle SL(R)\to SL(R/I)}$

for an ideal . ${\displaystyle I\subset R}$

Congruence subgroup problem

The congruence subgroup problem asks whether every normal subgroup for a commutative ring is in a congruence subgroup. ${\displaystyle R}$ ${\displaystyle SL(R)}$

literature

1. ^ Madabusi S. Raghunathan : The congruence subgroup problem. In: Sundararaman Ramanan (Ed.): Proceedings of the Hyderabad Conference on Algebraic Groups (December 1989). Manoj Prakashan, Madras 1991, ISBN 81-231-0090-6 , pp. 465-494.
2. ^ Madabusi S. Raghunathan: The congruence subgroup problem. In: Proceedings of the Indian Academy of Sciences. Mathematical Sciences. Vol. 114, No. 4, 2004, pp. 299-308, doi : 10.1007 / BF02829437 .