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The braid group B 3 is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group SL 2 (R) → PSL 2 (R). Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of B 3 modulo its center ; equivalently, to the ...
Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group () ().
In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G (Ballester-Bolinches, Esteban-Romero & Asaad 2010, pp. 24–25). Kenkichi Iwasawa proved that a p-group G is an Iwasawa group if and ...
Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups ...
The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N for some positive integer N, which is defined to be
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m -fold product of upper half-planes H {\displaystyle {\mathcal {H}}} satisfying a certain kind of functional equation .
The modular function is a continuous group homomorphism from G to the multiplicative group of positive real numbers. A group is called unimodular if the modular function is identically 1 {\displaystyle 1} , or, equivalently, if the Haar measure is both left and right invariant.
In mathematics, a Picard modular group, studied by Picard , is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and J is a hermitian form on L of signature (2, 1). Picard modular groups act on the unit sphere in C 2 and the quotient is called a Picard modular surface.