where Δ is the modular discriminant.

The real part of the modular discriminant as a function of q.

In particular the modular discriminant of Weierstrass can be defined as

The modular discriminant Δ is defined as

The discriminant is a modular form of weight 12.

The modular discriminant is defined as

and the modular discriminant.

Explicitly it is the modular discriminant

Suppose that E is a modular elliptic curve, then we can perform a level descent modulo primes ℓ dividing one of the exponents δ < sub > p </ sub > of a prime dividing the discriminant.

where with and is the Dedekind eta function and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form.

where is the modular discriminant.

Another example, is the L-function of the modular discriminant Δ

where Δ is the modular function on G. With this involution, it is a *- algebra.

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.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

Assuming normalization Δ ( 1 )= 0, the odd Laplacian is just the Δ operator, and the modular vector field vanishes.

* The Dedekind eta function η ( τ ), a modular form

Using this approach, for fixed z the Weierstrass functions become modular functions of τ.

The summations after the coefficients 60 and 140 are the first two Eisenstein series, which are modular forms when considered as functions G < sub > 4 </ sub >( τ ) and G < sub > 6 </ sub >( τ ), respectively, of τ = ω < sub > 2 </ sub >/ ω < sub > 1 </ sub > with Im ( τ ) > 0.

Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ + 1 and τ ↦-1 / τ.

In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.

The two transformations τ → τ + 1 and τ → τ < sup >− 1 </ sup > together generate a group called the modular group, which we may identify with the projective special linear group.

The roots of the cubics can be exactly given by quotients of the Dedekind eta function η ( τ ), a modular function involving a 24th root, and which explains the 24 in the approximation.

The corresponding modular invariants j ( τ ) are the singular moduli, coming from an older terminology in which " singular " referred to the property of having non-trivial endomorphisms rather than referring to a singular curve.

The modular function j ( τ ) is algebraic on imaginary quadratic numbers τ: these are the only algebraic numbers in the upper half-plane for which j is algebraic.

For example, the famous Ramanujan function τ ( n ) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a < sub > 1 </ sub > = 1.

Up to normalization, there is a unique modular form of weight 4: the Eisenstein series G < sub > 4 </ sub >( τ ).

The idea behind our design is modular units, or panelization.

Still another approach to the changeable letter type of sign is a modular unit introduced by Merritt Products, Azusa, Calif..

However, his construction of the Gothic alphabet is based upon an entirely different modular system.

Most games use a standardized and unchanging board ( chess, Go, and backgammon each have such a board ), but many games use a modular board whose component tiles or cards can assume varying layouts from one session to another, or even while the game is played.

The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.

This addition-based modular sum is used in SAE J1708.

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 ).

He then had little more to publish on the subject ; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

Another important feature of the Doom engine is its modular data files, which allow most of the game's content to be replaced by loading custom WAD files.

Because the Eurocard system provided for so many modular card sizes and because connector manufacturers have continued to create new connectors which are compatible with this system, it is a popular mechanical standard which is also used for innumerable " one-off " applications.

The Eurocard standard is also the basis of the " Eurorack " format for modular electronic music synthesizers, popularized by Doepfer and other manufacturers.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the " parameter " ( as defined above ), while the backslash indicates that it is the modular angle.

Dualis is a strictly modular partial low-floor car, with all doors in the low-floor sections.

Flying cars fall into one of two styles ; integrated ( all the pieces can be carried in the vehicle ), or modular ( the aeronautical sections are left at the airport when the vehicle is driven ).

* The Aerocar 2000 is a modular design currently in development by Ed Sweeney, owner of one of Moulton Taylor's Aerocars.

This is a modular design, in development.

Freenet is modular and features an API called Freenet Client Protocol ( FCP ) for other programs to use to implement services such as message boards, file sharing, or online chat.

Article 16 of Gauss ' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory ; this special case has very different properties.

Windows Server 2008 builds on the technological and security advances first introduced with Windows Vista, and is significantly more modular than its predecessor, Windows Server 2003.

* The ring of p-adic integers is the inverse limit of the rings Z / p < sup > n </ sup > Z ( see modular arithmetic ) with the index set being the natural numbers with the usual order, and the morphisms being " take remainder ".

* The ( 3-dimensional ) metaplectic group is a double cover of SL < sub > 2 </ sub >( R ) playing an important role in the theory of modular forms.

This approach is simple and modular, but has the disadvantage that the model itself can be expensive to store, and also that it forces a single model to be used for all data being compressed, and so performs poorly on files containing heterogeneous data.