The original form of the theorem, contained in a third-century AD book The Mathematical Classic of Sun Zi ( 孫子算經 ) by Chinese mathematician Sun Tzu and later generalized with a complete solution called Da yan shu ( 大衍术 ) in a 1247 book by Qin Jiushao, the Shushu Jiuzhang ( 數書九章 Mathematical Treatise in Nine Sections ) is a statement about simultaneous congruences ( see modular arithmetic ).

* Hilbert modular form

A holomorphic function in the upper half plane which is invariant under linear fractional transformations with integer coefficients and determinant 1 is called a modular form.

That is, a holomorphic function is a modular form if

The cusp form idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as ' discrete spectrum ', contrasted with the ' continuous spectrum ' from Eisenstein series.

The Langlands program seeks to attach an automorphic form or automorphic representation ( a suitable generalization of a modular form ) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field.

This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found ( which by the modularity theorem itself is now known to be a number called the conductor ), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer q-expansion, followed if need be by an isogeny.

Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form.

While potentially lacking the physical presence of desirable analog sound generation, real voltage manipulation, knobs, sliders, cables, and LEDs, software modular synthesizers offer the infinite variations and visual patching at a more affordable price and in a compact form factor.

A modular synthesizer has a case or frame into which arbitrary modules can be plugged ; modules are usually connected together using patch cords and a system may include modules from different sources, as long as it fits the form factors of the case and uses the same electrical specifications.

Although this method requires about p modular multiplications, rendering it impractical, theorems about primes and modular residues form the basis of many more practical methods.

Jini ( pronounced like genie i. e. ), also called Apache River, is a network architecture for the construction of distributed systems in the form of modular co-operating services.

A well-known example is the Taniyama – Shimura conjecture, now the modularity theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form ( in such a way as to preserve the associated L-function ).

Amongst his most original contributions were: his " Conjecture II " ( still open ) on Galois cohomology ; his use of group actions on Trees ( with H. Bass ); the Borel-Serre compactification ; results on the number of points of curves over finite fields ; Galois representations in ℓ-adic cohomology and the proof that these representations have often a " large " image ; the concept of p-adic modular form ; and the Serre conjecture ( now a theorem ) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Macro-modular origami is a form of modular origami in which finished assemblies are themselves used as the building blocks to create larger integrated structures.

LXI-compliant instruments offer the size and integration advantages of modular instruments without the cost and form factor constraints of card-cage architectures.

In mathematics, a modular form is a ( complex ) analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition.

A modular function is a modular form, without the condition that f ( z ) be holomorphic at infinity.

The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form

Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n / 2.

Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve.

The Jacobian of the modular curve can ( up to isogeny ) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2.

Mk 41 VLS adopts a modular design concept, which result in different versions that vary in size and weight.

As a corollary he proved the celebrated Ramanujan-Petersson conjecture for modular forms of weight greater than one ; weight one was proved in his work with Serre.

The Clinchfield layout measured and was noteworthy for a number of reasons: 1 ) it was highly portable because of its modular construction, 2 ) showed the potential of high scenery to track ratios possible in 1: 160, 3 ) used a unique aluminum frame and ( at the time revolutionary ) styrofoam construction to cut down on weight and 4 ) was highly prototypical for the era.

Because of these functional equations the eta function is a modular form of weight 1 / 2 and level 1 for a certain character of order 24 of the metaplectic double cover of the modular group, and can be used to define other modular forms.

and is a modular form of weight 12.

The discriminant is a modular form of weight 12.

This theta function is a modular form of weight n / 2 ( on an appropriately defined subgroup ) of the modular group.

Δ is a modular form of weight twelve by the above, and one of weight four, so that its third power is also of weight twelve.

The quotient is therefore a modular function of weight zero ; this means j has the absolutely invariant property that

Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to homotheties, as well as moderate growth at infinity ; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight.

Given a modular form f ( z ) of weight k, the mth Hecke operator acts by the formula

In the classical elliptic modular form theory, the Hecke operators T < sub > n </ sub > with n coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product.

In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional.

The modular inverse k < sup >− 1 </ sup > mod q is the second most expensive part, and it may also be computed before the message hash is known.

In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n. That is, for every integer a coprime to n, there is an integer k such that g < sup > k </ sup > ≡ a ( mod n ).

Using modular exponentiation by repeated squaring, the running time of this algorithm is O ( k log < sup > 3 </ sup > n ), where k is the number of different values of a we test ; thus this is an efficient, polynomial-time algorithm.

# Use the extended Euclidean algorithm to compute k < sup >− 1 </ sup >, the modular multiplicative inverse of k mod 2 < sup > w </ sup >, where w is the number of bits in a word.

where N < sub > 1 </ sub >< sup >− 1 </ sup > denotes the modular multiplicative inverse of N < sub > 1 </ sub > modulo N < sub > 2 </ sub > and vice-versa for N < sub > 2 </ sub >< sup >− 1 </ sup >; the indices k < sub > a </ sub > and n < sub > a </ sub > run from 0 ,..., N < sub > a </ sub >− 1 ( for a

: where a, b, c, d are integers satisfying ad − bc = 1, i. e., defines f to be a modular form of order k.

The second variable might be given in terms of the parameter m, or as the elliptic modulus k, where k < sup > 2 </ sup > = m, or in terms of the modular angle α, where m = sin < sup > 2 </ sup > α.

The more general Ramanujan – Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent ( k − 1 )/ 2 where k is the weight of the form.

Using fast algorithms for modular exponentiation, the running time of this algorithm is O ( k · log < sup > 3 </ sup > n ), where k is the number of different values of a we test.

In ordinary representation theory, the number of simple modules k ( G ) is equal to the number of conjugacy classes of G. In the modular case, the number l ( G ) of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime p, the so-called p-regular classes.

Then given integers a, b, c, d with ad − bc = 1 ( thus belonging to the modular group ), with c chosen so that c = kq for some integer k > 0, define

This ring is not commutative unless k equals the field ( see modular arithmetic ).