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In that sense a modular equation becomes the equation of a modular curve. Such equations first arose in the theory of multiplication of elliptic functions (geometrically, the n 2 -fold covering map from a 2- torus to itself given by the mapping x → n · x on the underlying group) expressed in terms of complex analysis .
The first step is relatively slow but only needs to be done once. Modular multiplicative inverses are used to obtain a solution of a system of linear congruences that is guaranteed by the Chinese Remainder Theorem. For example, the system X ≡ 4 (mod 5) X ≡ 4 (mod 7) X ≡ 6 (mod 11) has common solutions since 5,7 and 11 are pairwise coprime ...
To find from a given value =, it takes the following steps: Find the modular square root ().This step is quite easy when is a prime, irrespective of how large is.; Solve a quadratic equation associated with the modular square root of = + +.
The first step sets m to 12 ⋅ 47 mod 100 = 64. The second step sets t to (12 + 64 ⋅ 17) / 100. Notice that 12 + 64 ⋅ 17 is 1100, a multiple of 100 as expected. t is set to 11, which is less than 17, so the final result is 11, which agrees with the computation of the previous section.
As modular forms also satisfy a certain kind of functional equation with respect to the group action of the modular group, this Fourier series may be expressed in terms of =. So if f {\displaystyle f} is a modular form, then there are coefficients c ( n ) {\displaystyle c(n)} such that f ( z ) = ∑ n ∈ N c ( n ) q n {\displaystyle f(z)=\sum ...
A modular form for G of weight k is a function on H satisfying the above functional equation for all matrices in G, that is holomorphic on H and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G. The C-vector spaces of modular and cusp forms of weight k are denoted M k (G) and S k (G), respectively.
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Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).