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...
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Ramanujan's congruences, congruences for the partition function, p(n), first discovered by Ramanujan in 1919 Congruence subgroup, a subgroup defined by...
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In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector...
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Modular group (section Congruence subgroups)
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...
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Arithmetic group (redirect from Arithmetic subgroup)
integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture...
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In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...
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Maass wave form (section Congruence subgroups)
( N ) {\displaystyle \Gamma (N)} principal congruence subgroup of level N {\displaystyle N} . A subgroup Γ ⊆ S L 2 ( Z ) {\displaystyle \Gamma \subseteq...
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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...
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In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let G {\displaystyle...
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Discrete group (redirect from Discrete subgroup)
group is a discrete subgroup such that the Haar measure of the quotient space is finite. crystallographic point group congruence subgroup arithmetic group...
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)} ; they belong to a more general class of finite-index subgroups, congruence subgroups. Any order in a quaternion algebra over Q {\displaystyle \mathbb...
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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...
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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 ...
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Sylow theorems (redirect from Sylow subgroup)
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...
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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...
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7) triangle group, after quotienting by the center. The principal congruence subgroup defined by an ideal I ⊂ Z [ η ] {\displaystyle I\subset \mathbb {Z}...
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arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties...
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action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal I = ⟨ η − 2 ⟩ {\displaystyle I=\langle...
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Cooper found a general approach that used the underlying modular congruence subgroup Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} , while G. Almkvist has...
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example), the lattice of congruences is modular (Kearnes & Kiss 2013). Lattice-theoretic information about the lattice of subgroups can sometimes be used...
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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...
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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...
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group Complete group Complex reflection group Congruence subgroup Continuous symmetry Frattini subgroup Growth rate Heisenberg group, discrete Heisenberg...
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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...
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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...
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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...
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Semigroup (redirect from Semigroup congruence)
semigroup congruence ~ induces congruence classes [a]~ = {x ∈ S | x ~ a} and the semigroup operation induces a binary operation ∘ on the congruence classes:...
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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 +...
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{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...
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Local zeta function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modular curve Ramanujan–Petersson...
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