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Concept
Congruence subgroup
Résumé
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, in which the off-diagonal entries are even. 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' 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 are essentially congruence subgroups.
Congruence subgroups of 2×2 matrices are fundamental objects in the classical theory of modular forms; the modern theory of automorphic forms makes a similar use of congruence subgroups in more general arithmetic groups.
The simplest interesting setting in which congruence subgroups can be studied is that of the modular group .
If is an integer there is a homomorphism induced by the reduction modulo morphism . The principal congruence subgroup of level in is the kernel of , and it is usually denoted . Explicitly it is described as follows:
This definition immediately implies that is a normal subgroup of finite index in . The strong approximation theorem (in this case an easy consequence of the Chinese remainder theorem) implies that is surjective, so that the quotient is isomorphic to Computing the order of this finite group yields the following formula for the index:
where the product is taken over all prime numbers dividing .
If then the restriction of to any finite subgroup of is injective. This implies the following result:
If then the principal congruence subgroups are torsion-free.
The group contains and is not torsion-free.
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Forme automorphe
droite|vignette|500x500px|La fonction êta de Dedekind est une forme automorphe dans le plan complexe. Une forme automorphique, en analyse harmonique et théorie des nombres, est une fonction d'un groupe topologique G à valeurs dans le corps des nombres complexes (ou un espace vectoriel complexe) qui est invariante sous l'action d'un sous-groupe discret du groupe topologique et qui vérifie certaines conditions de dérivabilité et de croissance à l'infini.
Arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory.
Courbe modulaire
En théorie des nombres et en géométrie algébrique une courbe modulaire désigne la surface de Riemann, ou la courbe algébrique correspondante, construite comme quotient du demi-plan de Poincaré H sous l'action de certains sous-groupes Γ d'indice fini dans le groupe modulaire. La courbe obtenue est généralement notée Y(Γ). On appelle Γ le niveau de la courbe Y(Γ). Depuis Gorō Shimura, on sait que ces courbes admettent des équations à coefficients dans un corps cyclotomique, qui dépend du niveau Γ.