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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 group ...
The matrices [e 1, ..., e n] are divisible by all non-zero linear forms in the variables X i with coefficients in the finite field F q. In particular the Moore determinant [0, 1, ..., n − 1] is a product of such linear forms, taken over 1 + q + q 2 + ... + q n – 1 representatives of ( n – 1)-dimensional projective space over the field.
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.
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 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z). [2]In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms M(Γ) is generated in weight at most 3 when is the congruence subgroup () of prime level N in SL(2, Z) using the theory of toric modular forms. [3]
A modular function is a function that is invariant with respect to the modular group, but without the condition that it be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic : they are holomorphic on the complement of a set of isolated points, which are poles of the function.
It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL ± (2, R), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group, however.
When the field F has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra F[G] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when F is sufficiently large: each block is a full matrix algebra over F, the ...