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...

Ramanujan's congruences, congruences for the partition function, p(n), first discovered by Ramanujan in 1919 Congruence subgroup, a subgroup defined by...

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector...

0, or 1, so these subgroups are torsion-free groups. (There are other torsion-free subgroups.) The principal congruence subgroup of level 2, Γ(2), is...

integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture...

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...

( N ) {\displaystyle \Gamma (N)} principal congruence subgroup of level N {\displaystyle N} . A subgroup Γ ⊆ S L 2 ( Z ) {\displaystyle \Gamma \subseteq...

a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term...

In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let G {\displaystyle...

group is a discrete subgroup such that the Haar measure of the quotient space is finite. crystallographic point group congruence subgroup arithmetic group...

)}  ; they belong to a more general class of finite-index subgroups, congruence subgroups. Any order in a quaternion algebra over Q {\displaystyle \mathbb...

the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z). In 2003, Lev Borisov and Paul Gunnells showed that the...

groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup. Example-1 Groups of order...

Conjecturally, arithmetic lattices in higher-rank groups have the congruence subgroup property but there are many lattices in S O ( n , 1 ) , S U ( n ...

6 and the relations are generated in weight at most 12 when the congruence subgroup has nonzero odd weight modular forms, and the corresponding bounds...

7) triangle group, after quotienting by the center. The principal congruence subgroup defined by an ideal I ⊂ Z [ η ] {\displaystyle I\subset \mathbb {Z}...

Cooper found a general approach that used the underlying modular congruence subgroup Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} , while G. Almkvist has...

arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties...

action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal I = ⟨ η − 2 ⟩ {\displaystyle I=\langle...

example), the lattice of congruences is modular (Kearnes & Kiss 2013). Lattice-theoretic information about the lattice of subgroups can sometimes be used...

SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic...

Study in Princeton a year long program on "Pro-finite groups and the congruence subgroup problem". In 2006, he got an honorary degree from the University...

principal congruence subgroup of the Hilbert modular group of the field Q(√5). The quotient of the Hilbert modular group by its level 2 congruence subgroup is...

theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a...

generalise this to universal algebra, normal subgroups need to be replaced by congruence relations. A congruence on an algebra A {\displaystyle A} is an equivalence...

semigroup congruence ~ induces congruence classes [a]~ = {x ∈ S | x ~ a} and the semigroup operation induces a binary operation ∘ on the congruence classes:...

group Complete group Complex reflection group Congruence subgroup Continuous symmetry Frattini subgroup Growth rate Heisenberg group, discrete Heisenberg...

i\tau })} is invariant under Γ ( 5 ) {\displaystyle \Gamma (5)} , a congruence subgroup of the modular group. Also for positive real numbers a , b ∈ R +...

{N}{d}}r_{d}\equiv 0{\pmod {24}},} then ηg is a weight k modular form for the congruence subgroup Γ0(N) (up to holomorphicity) where k = 1 2 ∑ 0 < d ∣ N r d . {\displaystyle...

Local zeta function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modular curve Ramanujan–Petersson...