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Relates rational elliptic curves to modular forms
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way
Modularity_theorem
17th-century conjecture proved by Andrew Wiles in 1994
known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques
Fermat's_Last_Theorem
1995 publication in mathematics
Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
British mathematician who proved Fermat's Last Theorem
Hilbert modular forms. In 1986, upon reading Ken Ribet's seminal work on Fermat's Last Theorem, Wiles set out to prove the modularity theorem for semistable
Andrew_Wiles
Mathematical concept
elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts
Modular_elliptic_curve
Analytic function on the upper half-plane with a certain behavior under the modular group
of number theory and the formulation of the modularity theorem in particular made it clear that modular forms are deeply implicated. Yutaka Taniyama
Modular_form
Conjecture in number theory
them Fermat's Last Theorem and the now-proven Taniyama–Weil (or Taniyama–Shimura) conjecture, now known as the modularity theorem (although this implies
Serre's_modularity_conjecture
36 mathematical problems stated in 1955
conjecture (now known as the modularity theorem), which states that every elliptic curve over the rational numbers is modular. This conjecture played a major
Taniyama's_problems
Result concerning properties of Galois representations associated with modular forms
that the Modularity theorem implied FLT. The origin of the name is from the ε part of "Taniyama-Shimura conjecture + ε ⇒ Fermat's last theorem". Suppose
Ribet's_theorem
modularity theorem is a theorem about modular tensor categories. It asserts that two different formulations of the modularity condition of a modular tensor
Bruguières_modularity_theorem
A prime p divides a^p–a for any integer a
little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic
Fermat's_little_theorem
Topics referred to by the same term
module or modular in Wiktionary, the free dictionary. Module, modular and modularity may refer to the concept of modularity. They may also refer to: Modular design
Module
Branch of number theory
and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning
Algebraic_number_theory
theory) Modularity theorem (number theory) Mordell–Weil theorem (number theory) Multiplicity-one theorem (group representations) Nagell–Lutz theorem (elliptic
List_of_theorems
residual modularity was one of the starting inputs in Wiles's proof of the modularity of semistable elliptic curves, and hence of Fermat's Last Theorem. In
Langlands–Tunnell_theorem
Japanese mathematician (1930–2019)
conjecture (later known as the modularity theorem) in the 1950s played a key role in the proof of Fermat's Last Theorem by Andrew Wiles in 1995. In 1990
Goro_Shimura
Computation modulo a fixed integer
important theorems relating to modular arithmetic: Carmichael's theorem Chinese remainder theorem Euler's theorem Fermat's little theorem (a special
Modular_arithmetic
Proposition in mathematics that is unproven
19th century, and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics, and prior
Conjecture
Unproved conjecture in mathematics
curves over Q {\displaystyle \mathbb {Q} } , as a consequence of the modularity theorem in 2001.[citation needed] Finding rational points on a general elliptic
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Branch of algebraic geometry
the modularity theorem) relating elliptic curves to modular forms. This connection would ultimately lead to the first proof of Fermat's Last Theorem in
Arithmetic_geometry
Algebraic variety
century. Manin–Drinfeld theorem Moduli stack of elliptic curves Modularity theorem Shimura variety, a generalization of modular curves to higher dimensions
Modular_curve
American mathematician (1964-)
mathematician, known for his role in proving the modularity theorem for elliptic curves. His research interest is in modular forms and Galois representations. Diamond
Fred_Diamond
Class of mathematical functions
as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem. The addition
Weierstrass_elliptic_function
Type of monoidal category
the modular group representation, the Bruguières modularity theorem, the Verlinde formula, the rank-finiteness theorem, the Schauenburg-Ng theorem, and
Modular_tensor_category
Theorem on modular exponentiation
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers
Euler's_theorem
Type of lattice in mathematical order theory
universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction
Modular_lattice
the universal deformation space. A key step in Wiles's proof of the modularity theorem was to study the relation between universal deformation rings and
Deformation_ring
Algebraic curve in mathematics
geometry) Modularity theorem Moduli stack of elliptic curves Nagell–Lutz theorem Riemann–Hurwitz formula Wiles's proof of Fermat's Last Theorem Sarli, J
Elliptic_curve
program modularity theorem Pythagorean triple Pell's equation Elliptic curve Nagell–Lutz theorem Mordell–Weil theorem Mazur's torsion theorem Congruent
List_of_number_theory_topics
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
Japanese mathematician
curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro
Yutaka_Taniyama
Carlos Vinuesa, 2010) Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008) Green–Tao theorem (Ben J. Green and Terence
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
prime number one less than a power of 2. modular form Modular form modularity theorem The modularity theorem (which used to be called the Taniyama–Shimura
Glossary_of_number_theory
Mathematical method in functional analysis
the modular operator and J the modular conjugation. In Takesaki (2003, pp. 5–17), there is a self-contained proof of the main commutation theorem of Tomita-Takesaki:
Tomita–Takesaki_theory
Plane algebraic curve
with integer coefficients. The famous modularity theorem tells us that all elliptic curves over Q are modular. Mappings also arise in connection with
Classical_modular_curve
Mordell–Weil theorem Oka–Weil theorem Siegel–Weil formula Shafarevich–Weil theorem Taniyama–Shimura–Weil conjecture, now proved as the modularity theorem Weil
List of things named after André Weil
List_of_things_named_after_André_Weil
Concept in mathematics
Theorem 6.4. Farb & Margalit 2012, Theorem 6.15 and Theorem 6.12. Farb & Margalit 2012, Theorem 6.11. Ivanov 1992, Theorem 4. Ivanov 1992, Theorem 1
Mapping class group of a surface
Mapping_class_group_of_a_surface
View of mathematicians to consolidate two or more theories into a more generalized one
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
Unifying theories in mathematics
Unifying_theories_in_mathematics
American mathematician
Michigan and at Columbia University. Conrad and others proved the modularity theorem, also known as the Taniyama-Shimura Conjecture. He proved this in
Brian_Conrad
inequality on height Serre group Serre's modularity conjecture Serre's multiplicity conjectures Serre's open image theorem Serre's property FA Serre relations
List of things named after Jean-Pierre Serre
List_of_things_named_after_Jean-Pierre_Serre
Type of mathematical function
{\frac {r}{k}}\right).} Generalized Riemann hypothesis L-function Modularity theorem Artin conjecture Special values of L-functions Dirichlet, Peter Gustav
Dirichlet_L-function
Family of hypersurfaces in algebraic geometry
to have relationships with mirror symmetry and extensions of the modularity theorem. The Dwork family is given by the equations x 1 n + x 2 n + ⋯ + x
Dwork_family
About simultaneous modular congruences
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then
Chinese_remainder_theorem
include the Green–Tao theorem (2004), existence of bounded gaps between arbitrarily large primes (2013), and the modularity theorem (2001). The AKS primality
History_of_mathematics
Theorem about the range of an analytic function
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after
Picard_theorem
Group in arithmetic geometry
extended this to modular elliptic curves over the rationals of analytic rank at most 1. (The modularity theorem later showed that the modularity assumption
Tate–Shafarevich_group
Theorem on the orders of subgroups
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
little theorem, which states that a p ≡ a ( mod p ) {\displaystyle a^{p}\equiv a{\pmod {p}}} for every prime number p and every integer a (see modular arithmetic)
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Concept in modular arithmetic
extended Euclidean algorithm, Euler's theorem may be used to compute modular inverses. According to Euler's theorem, if a is coprime to m, that is, gcd(a
Modular multiplicative inverse
Modular_multiplicative_inverse
Quadratic imaginary number fields with unique factorisation
In number theory, the Heegner theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers
Stark–Heegner_theorem
Theorem on prime numbers
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers
Wilson's_theorem
Connection on a vector bundle
been intensively studied in recent years, in connection with the modularity theorem and its extensions). Thus, the base space of the bundle is taken to
Gauss–Manin_connection
Studies linear representations of finite groups over fields of positive characteristic
|G|, then modular representations are completely reducible, as with ordinary (characteristic 0) representations, by virtue of Maschke's theorem. In the
Modular_representation_theory
Topics referred to by the same term
theorem, formerly known as Serre's conjecture Serre's conjecture II, concerning the Galois cohomology of linear algebraic groups Serre's modularity conjecture
Serre's_conjecture
Meromorphic function on the complex plane
Generalized Riemann hypothesis Dirichlet L-function Automorphic L-function Modularity theorem Artin conjecture Special values of L-functions Explicit formulae for
L-function
Connects non-singular algebraic curves with compact Riemann surfaces
In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents
Belyi's_theorem
Mathematical function associated to algebraic varieties
Hasse–Weil conjecture follows from the modularity theorem: each elliptic curve E over Q {\displaystyle \mathbb {Q} } is modular. The Birch and Swinnerton-Dyer
Hasse–Weil_zeta_function
Theorem in transcendental number theory
Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: Lindemann–Weierstrass theorem—if α1
Lindemann–Weierstrass_theorem
Theorems that help decompose a finite group based on prime factors of its order
specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow
Sylow_theorems
Curves of genus > 1 over the rationals have only finitely many rational points
Faltings' theorem is a result in arithmetic geometry, according to which a non-singular algebraic curve of genus greater than 1 over the field Q {\displaystyle
Faltings'_theorem
Number divisible only by 1 and itself
but not a field. Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if a ≢ 0 {\displaystyle
Prime_number
Gives conditions for the solvability of quadratic equations modulo prime numbers
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations
Quadratic_reciprocity
American mathematician
Topological Modular Forms (PDF) Ando, Matthew; Hopkins, Michael J.; Strickland, Neil P. (2001), "Elliptic spectra, the Witten genus and the theorem of the
Michael_J._Hopkins
over the rationals has bad reduction somewhere. 2001 Proof of the modularity theorem for elliptic curves is completed. PDF Miscellaneous Diophantine Equations
Timeline_of_abelian_varieties
French mathematician
question of how the modularity theorem for elliptic curves over the rational numbers (which led to the proof of Fermat's Last Theorem) can be extended to
Vincent_Pilloni
In mathematics, the Schauenbug–Ng theorem is a theorem about the modular group representations of modular tensor categories proved by Siu-Hung Ng and
Schauenburg–Ng_theorem
On areas of rational right triangles
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a
Tunnell's_theorem
Algorithm for fast modular multiplication
is congruent to ab can be expressed by applying the Euclidean division theorem: a b = q N + r , {\displaystyle ab=qN+r,} where q is the quotient ⌊ a b
Montgomery modular multiplication
Montgomery_modular_multiplication
Theorem classifying finite simple groups
classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is
Classification of finite simple groups
Classification_of_finite_simple_groups
Two theorems about families of holomorphic functions
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after
Montel's_theorem
diagonals Euclid–Euler theorem, characterizing even perfect numbers Euler's theorem, on modular exponentiation Euler's partition theorem relating the product
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Sub-field of mathematics
linear factors. Sanderson's theorem Dickson, Leonard Eugene (1911), "A Fundamental System of Invariants of the General Modular Linear Group with a Solution
Modular_invariant_theory
Orientation-preserving mapping class group of the torus
In mathematics, the modular group is the projective special linear group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2
Modular_group
In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Topics referred to by the same term
Reciprocity theorem may refer to: Quadratic reciprocity, a theorem about modular arithmetic Cubic reciprocity Quartic reciprocity Artin reciprocity Weil
Reciprocity_theorem
mathematics, Müger's theorem asserts that the Drinfeld center of every spherical fusion category is a modular tensor category. Müger's theorem was introduced
Müger's_theorem
Type of theorem in automorphic forms
converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states
Converse_theorem
Indian mathematician (born 1968)
thesis was published in the Duke Mathematical Journal. He proved Serre's modularity conjecture with Jean-Pierre Wintenberger, published in Inventiones Mathematicae
Chandrashekhar_Khare
Mapping function
(A)+\mu (B)} The above property is called modularity and the argument below proves that additivity implies modularity. Given A {\displaystyle A} and B , {\displaystyle
Sigma-additive_set_function
Australian-American mathematician
Mathematicians, where he gave a lecture entitled "30 years of modularity since Fermat's Last Theorem." Calegari was awarded the 2026 Cole Prize in Number Theory
Frank_Calegari
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to
Minkowski's_theorem
Topics referred to by the same term
Thue's theorem may refer to the following mathematical theorems named after Axel Thue: Thue equation has finitely many solutions in integers. Thue's lemma
Thue's_theorem
Mathematical group based upon a finite number of elements
started with Camille Jordan's theorem that the projective special linear group PSL(2, q) is simple for q ≠ 2, 3. This theorem generalizes to projective groups
Finite_group
Topics referred to by the same term
several theorems of Helmut Hasse that are sometimes called Hasse's theorem: Hasse norm theorem Hasse's theorem on elliptic curves Hasse–Arf theorem Hasse–Minkowski
Hasse's_theorem
Commutative group (mathematics)
structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees
Abelian_group
Belgian mathematician
main theorem. In 1968, he also worked with Jean-Pierre Serre; their work led to important results on the l-adic representations attached to modular forms
Pierre_Deligne
Graded lattice with modular maximal chain
lattice that has a maximal chain of elements, each of which obeys a certain modularity relationship. The definition encapsulates many of the nice properties
Supersolvable_lattice
Division with remainder of integers
ambiguity, Euclidean division. The theorem is frequently referred to as the division algorithm (although it is a theorem and not an algorithm), because its
Euclidean_division
Type of vector space
the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators
Hecke_algebra
Collection of loosely coupled services used to build computer applications
teams to develop, deploy, and scale services independently, improving modularity, scalability, and adaptability. However, it introduces additional complexity
Microservices
Algorithm for public-key cryptography
λ(pq)). This is part of the Chinese remainder theorem, although it is not the significant part of that theorem. Although the original paper of Rivest, Shamir
RSA_cryptosystem
Formula concerning prime numbers
prime field for more details. Because the modulus is prime, Lagrange's theorem applies: a polynomial of degree k can only have at most k roots. In particular
Euler's_criterion
Identifies Fourier coefficients of some modular forms with the value of an L-series
mathematics, Waldspurger's theorem, introduced by Jean-Loup Waldspurger (1981), is a result that identifies Fourier coefficients of modular forms of half-integral
Waldspurger's_theorem
Monster and modular connection
Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized
Monstrous_moonshine
Mathematical function
construct a basis for a vector space of modular forms and cusp forms. A useful theorem for limiting the number of modular eta quotients to consider states that
Dedekind_eta_function
Transformations induced by a mathematical group
known as the orbit–stabilizer theorem. If G is finite then the orbit–stabilizer theorem, together with Lagrange's theorem, gives | G ⋅ x | = [ G : G x
Group_action
Type of group in group theory
stronger results (Ratner's theorems) were later obtained by Marina Ratner. In another direction the classical topic of modular forms has blossomed into
Arithmetic_group
Symmetric holomorphic function
the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard
Modular_lambda_function
Conditions under which the congruence x^3 equals p (mod q) is solvable
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q)
Cubic_reciprocity
Existence of group elements of prime order
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number
Cauchy's theorem (group theory)
Cauchy's_theorem_(group_theory)
MODULARITY THEOREM
MODULARITY THEOREM
Girl/Female
African, American, Arabic, Danish, Gujarati, Hindu, Indian, Kannada, Latin, Sanskrit, Tamil
Blessing of God; Origin; Popularity; Variants
Girl/Female
Tamil
Hashmitha | ஹஷà¯à®®à¯€à®¤à®¾
Popularity
Hashmitha | ஹஷà¯à®®à¯€à®¤à®¾
Female
English
 English adopted use of German Avis ("refuge in war"). But its popularity in the Middle Ages was due to its association with the Latin noun avis, AVIS means "bird."Â
Boy/Male
Arabic, British, English, French, German, Hindu, Indian, Muslim, Sindhi
Long-living; Builder; Popularity; Constructor; Religious Person; Long of Age; The Maker
Girl/Female
Hindu, Indian
Fame; Reputation; Popularity; Famous
Girl/Female
Indian
Popularity
Boy/Male
Arabic, Muslim
Popularity
Boy/Male
Bengali, Celebrity, Gujarati, Hindu, Indian, Kannada, Sanskrit, Tamil, Telugu, Traditional
Anything; God of the Universe; Son of Emperor Ashok; Popularity; Numerology; The Person who can See the Beauty; Lotus
Girl/Female
Indian
Popularity
Girl/Female
Tamil
Hasmitha | ஹஸà¯à®®à¯€à®¤à®¾Â
Popularity
Hasmitha | ஹஸà¯à®®à¯€à®¤à®¾Â
Girl/Female
Indian, Latin
Popularity; Golden
Girl/Female
Indian
Popularity
Boy/Male
Muslim
Popularity
Girl/Female
Indian, Marathi, Telugu
Goddess Parvati; Popularity
Boy/Male
Hindu, Indian
Popularity
Girl/Female
Tamil
Hasmita | ஹஸà¯à®®à¯€à®¤à®¾Â
Popularity
Hasmita | ஹஸà¯à®®à¯€à®¤à®¾Â
Girl/Female
Indian
Popularity
Girl/Female
Hindu, Indian
Fame; Reputation; Popularity; Famous
Girl/Female
Indian
Popularity
Girl/Female
Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Oriya, Sindhi, Telugu
Fame; Reputation; Popularity; Famous
MODULARITY THEOREM
MODULARITY THEOREM
Boy/Male
Hindu, Indian, Tamil, Traditional
Aim; Peaceful; Satisfaction
Male
English
 English surname transferred to forename use, derived from the city name Chester, from an Old English form of Latin castra, CHESTER means "legionary camp."Â
Boy/Male
Tamil
A yadava
Boy/Male
American, English
Respelling of Zachary
Boy/Male
Tamil
Wealth
Boy/Male
Hindu
Joy of God, Son of God
Girl/Female
Tamil
Friend, See also ervin
Female
Babylonian
, a goddess of abundance; consort to Nin-girsu.
Girl/Female
Indian, Punjabi, Sikh
Eternal Infinite; Without Beginning or End
Boy/Male
British, English, Swedish
Form of Reginald; Counsel Power; Advice; Decision Ruler
MODULARITY THEOREM
MODULARITY THEOREM
MODULARITY THEOREM
MODULARITY THEOREM
MODULARITY THEOREM
n.
Public sentiment; general passion.
n.
The quality or state of being adapted or pleasing to common, poor, or vulgar people; hence, cheapness; inferiority; vulgarity.
n.
The quality or state of being popular; popularity.
v. t.
To formulate into a theorem.
n.
One who constructs theorems.
a.
Advancing or increasing amid noisy excitement; as, booming prices; booming popularity.
n.
Jesting; merriment.
pl.
of Popularity
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
Facetiousness; jocularity.
n.
State of being current; currency; popularity.
n.
Something which obtains, or is intended to obtain, the favor of the vulgar; claptrap.
n.
The act of courting the favor of the people.
a.
Theorematic.
n.
The manner or action of a wag; mischievous merriment; sportive trick or gayety; good-humored sarcasm; pleasantry; jocularity; as, the waggery of a schoolboy.
n.
A numerical coefficient in any particular case of the binomial theorem.
n.
The quality or state of being popular; especially, the state of being esteemed by, or of being in favor with, the people at large; good will or favor proceeding from the people; as, the popularity of a law, statesman, or a book.
a.
Alt. of Theorematical
adv.
In a popular manner; so as to be generally favored or accepted by the people; commonly; currently; as, the story was popularity reported.
n.
The quality or state of being scurrile or scurrilous; mean, vile, or obscene jocularity.