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In mathematics, the modular group is the projective special linear group (,) of matrices with integer coefficients and determinant, such that the matrices and are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations .
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.
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.
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 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 ...
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 .
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]
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