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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, 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.
For example, the vertex of each indecomposable module in a block is contained (up to conjugacy) in the defect group of the block, and no proper subgroup of the defect group has that property. Brauer's first main theorem states that the number of blocks of a finite group that have a given p -subgroup as defect group is the same as the ...
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 ...
Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SL n on the space of n by n matrices by left multiplication, then the determinant is an ...
The braid group B 3 is the universal central extension of the modular group. Under this covering, the preimage of the modular group PSL(2, Z) is the braid group on 3 generators, B 3, which is the universal central extension of the modular group. These are lattices inside the relevant algebraic groups, and this corresponds algebraically to the ...
That implies that any two rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two variables over the complex numbers. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular curve.
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