In mathematics the modularity theorem ( formerly called the Taniyama – Shimura – Weil conjecture and several related names ) states that elliptic curves over the field of rational numbers are related to modular forms.

Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem, and Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended his techniques to prove the full modularity theorem in 2001.

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.

The modularity theorem implies a closely related analytic statement: to an elliptic curve E over Q we may attach a corresponding L-series.

The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform ( an eigenvector of all Hecke operators ); this is the Hasse – Weil conjecture, which follows from the modularity theorem.

The now fully proved conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem.

# REDIRECT modularity theorem

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 ).

It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem.

# REDIRECT modularity theorem

* In mathematics, the modularity theorem ( formerly the Taniyama – Shimura conjecture ) establishes a connection between elliptic curves and modular forms.

This partial proof of the Sato – Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves.

A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama – Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura.

# REDIRECT modularity theorem

# REDIRECT modularity theorem

# REDIRECT modularity theorem

For elliptic curves over the rational numbers, the Hasse – Weil conjecture follows from the modularity theorem.

* There is a basic form of modularity: files can be compiled separately and linked together, with control over which functions and data objects are visible to other files via and attributes.

" According to Steven Pinker, EP is " not a single theory but a large set of hypotheses " and a term which " has also come to refer to a particular way of applying evolutionary theory to the mind, with an emphasis on adaptation, gene-level selection, and modularity.

Forth is a simple yet extensible language ; its modularity and extensibility permit the writing of high-level programs such as CAD systems.

In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components ( which may reflect individual physical processes of interest, for example ), or for the purpose of obtaining a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of modularity ( i. e., independence or non-interaction ).

Another advantage of modularity is the ability to spend one's money on only a few core components at first and then later add additional components to one's system.

This modularity of the operating system is at the binary ( image ) level and not at the architecture level.

Another approach is to use a design targeting modularity with interfaces to increase the flexibility of the software with respect to the design.

One of these is modularity: as has been long recognized, plants and animal bodies are modular: they are organized into developmentally and anatomically distinct parts.

They thus reject a body of opinion in cognitive science suggesting that there is evidence for the modularity of language.

In computing, aspect-oriented programming ( AOP ) is a programming paradigm which aims to increase modularity by allowing the separation of cross-cutting concerns.

The massive modularity theory thesis posits that there is a huge number of tremendously interlinked but specialized modules running programs called Darwinian algorithms, or DA.

The stated mission of the vehicle is counter-terrorism for the United States Department of Homeland Security, but could be easily adapted for other uses ; medical response, emergency power and lighting, unlimited modularity.

It is this fundamental modularity of the system combined with its step-by-step ( hence the alternative name ) selection process and an almost unlimited potential for expansion that gives the Strowger system its technical advantage: Previous systems had all been designed for a fixed number of subscribers to be switched directly to each other in a mesh arrangement.

Progress towards SoC is traditionally achieved through modularity of programming and encapsulation ( or " transparency " of operation ), with the help of information hiding.

One reason for XScreenSaver's popularity is the modularity of its design, which makes it easy to add new display modes.

Even within evolutionary psychology there is discussion about the degree of modularity, either as a few generalist modules or as many highly specific modules.

Modular Function Deployment uses QFD to establish customer requirements and to identify important design requirements with a special emphasis on modularity.

Instead of postulating ' pure ' modularity, theorists have opted for a weaker version, domain-specificity implemented in functionally specialised neural circuits and computation ( e. g. Jackendoff and Pinker ’ s ( 2005 ) words, we must investigate language “ not as a monolith but as a combination of components, some special to language, others rooted in more general capacities ” ( p. 223 ).

Thanks to its design and modularity Paris-Bercy is a unique place, allowing a range of technical applications for unlimited sound, lighting and special effects.

The modern evidence for this specificity ( sometimes called modularity ) includes direct recording from single neurons, case studies of patients with brain damage or disease, recording of electrical or magnetic activity in the brain through advanced scanning techniques, and even direct manipulation of brain activity.

He also notes that the characteristics are not an all-or-nothing proposition, but rather each of the characteristics may be manifest in some degree, and that modularity itself is also not a dichotomous construct — something may be more or less modular: " One would thus expect — what anyhow seems to be desirable — that the notion of modularity ought to admit of degrees " ( Fodor, 1996 / 1983: 37 ).

Even in cases where no more than one implementation of each policy will ever be used, decomposing a class into policies can aid the design process, by increasing modularity and highlighting exactly where orthogonal design decisions have been made.

Some characteristics that make software more easily reusable are modularity, loose coupling, high cohesion, information hiding and separation of concerns.

To provide even more modularity with reduced cost, memory and I / O buses ( and the required control and power buses ) were sometimes combined into a single unified system bus.

The M4 carbine and M16 are not ideally suited for all missions, so it was proposed that the modularity of the M16 series would allow a user to replace the upper receiver of an existing weapon with one more suitable to the task.

Its modularity, adaptability, and scalability means that it can be applied to other, more complex devices, for more complex uses.

The short development cycles had led programmers to sacrifice modularity and elegance in the scramble to implement more features.

Furthermore, the whole system is more flexible and has a greater modularity than conventional communication systems.

Their study reveals significant differences in modularity, consistent with a view that distributed teams tend to develop more modular products.

Computers use modularity to overcome changing customer demands and to make the manufacturing process more adaptive to change ( see modular programming ).