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Mathematical concept
A modular elliptic curve is an elliptic curve E that admits a parametrization X0(N) → E by a modular curve. This is not the same as a modular curve that
Modular_elliptic_curve
Algebraic curve in mathematics
mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over
Elliptic_curve
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
Algebraic variety
complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending
Modular_curve
Approach to public-key cryptography
cryptosystems based on modular exponentiation in finite fields, such as the RSA cryptosystem and ElGamal cryptosystem. Elliptic curves are applicable for
Elliptic-curve_cryptography
1995 publication in mathematics
British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Unproved conjecture in mathematics
, as a consequence of the modularity theorem in 2001.[citation needed] Finding rational points on a general elliptic curve is a difficult problem. Finding
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Result concerning properties of Galois representations associated with modular forms
associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives
Ribet's_theorem
17th-century conjecture proved by Andrew Wiles in 1994
Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at
Fermat's_Last_Theorem
In mathematics, the conductor of an elliptic curve over the field of rational numbers (or more generally a local or global field) is an integral ideal
Conductor of an elliptic curve
Conductor_of_an_elliptic_curve
Analytic function on the upper half-plane with a certain behavior under the modular group
compactified modular curve. Another way to phrase the definition of modular functions is to use elliptic curves: every lattice Λ determines an elliptic curve C/Λ
Modular_form
Type of algebraic equation
the moduli problem, which are the points of the modular curve not corresponding to honest elliptic curves but degenerate cases, may be difficult to read
Modular_equation
Elliptic curve associated with a Fermat triple
In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve y 2 = x ( x − α ) ( x + β ) {\displaystyle y^{2}=x(x-\alpha )(x+\beta )}
Frey_curve
Mathematical concept
In arithmetic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 {\displaystyle p>0}
Supersingular_elliptic_curve
Algorithm for integer factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Algebraic invariant of topological spaces
mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. Historically
Elliptic_cohomology
semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. Suppose E is an elliptic curve
Semistable_abelian_variety
Orientation-preserving mapping class group of the torus
connection between the modular group and elliptic curves. Each point z {\displaystyle z} in the upper half-plane gives an elliptic curve, namely the quotient
Modular_group
American mathematician (born 1972)
Iwasawa–Greenberg main conjectures for a large class of modular forms. As a consequence, for a modular elliptic curve over the rational numbers, they prove that the
Christopher_Skinner
Algebraic stack in mathematics
In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M e l l {\displaystyle {\mathcal {M}}_{\mathrm
Moduli stack of elliptic curves
Moduli_stack_of_elliptic_curves
Plane algebraic curve
classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here
Classical_modular_curve
Prime number with a certain relationship to an elliptic curve
supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E {\displaystyle E} is defined
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
Modular function in mathematics
{\displaystyle j} -invariant was studied as a parameterization of elliptic curves over C {\displaystyle \mathbb {C} } , but it also has surprising connections
J-invariant
Theory of a class of elliptic curves
the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with
Complex_multiplication
Special point on a modular curve in mathematics
Shou-Wu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties (Zhang 2001, 2004, Yuan, Zhang & Zhang 2009)
Heegner_point
topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This
Topological_modular_forms
36 mathematical problems stated in 1955
algebraic geometry, number theory, and the connections between modular forms and elliptic curves. Taniyama's twelfth and thirteenth problems were the precursor
Taniyama's_problems
Mathematical operation on points on an elliptic curve
Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic
Elliptic curve point multiplication
Elliptic_curve_point_multiplication
Class of mathematical functions
with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice
Weierstrass_elliptic_function
Mathematical concept
mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that
Elliptic_surface
is the multiplication by n. See also: modular curve#Examples, moduli stack of elliptic curves. Siegel modular form Rigidity (mathematics) Local rigidity
Level structure (algebraic geometry)
Level_structure_(algebraic_geometry)
Special function defined by an integral
naming conventions. For expressing one argument: α, the modular angle k = sin α, the elliptic modulus or eccentricity m = k2 = sin2 α, the parameter Each
Elliptic_integral
Method of exchanging cryptographic keys
communications. Elliptic-curve Diffie–Hellman key exchange Supersingular isogeny key exchange Forward secrecy Diffie–Hellman problem Modular exponentiation
Diffie–Hellman_key_exchange
Class of periodic mathematical functions
properties of elliptic functions 30 years earlier but never published anything on the subject. Elliptic integral Elliptic curve Modular group Theta function
Elliptic_function
Specific class of fifteen prime numbers
theory) Supersingular elliptic curve Monstrous moonshine Monster group Ogg 1980. Silverman 2009, pp. V §4. Ogg, A. P. (1980). "Modular Functions". In Cooperstein
Supersingular prime (moonshine theory)
Supersingular_prime_(moonshine_theory)
Symmetric holomorphic function
branch points of a ramified double cover of the projective line by the elliptic curve C / ⟨ 1 , τ ⟩ {\displaystyle \mathbb {C} /\langle 1,\tau \rangle }
Modular_lambda_function
Algorithm in the theory of elliptic curves
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over Q {\displaystyle \mathbb {Q} } , or more
Tate's_algorithm
Iwasawa–Greenberg main conjectures for a large class of modular forms. As a consequence, for a modular elliptic curve over the rational numbers, they prove that the
Eric_Urban
Geometric space
genus g = 1 {\displaystyle g=1} curves having a marked point (elliptic curve groups) is the (classical) modular curve. For g > 1 {\displaystyle g>1}
Moduli_of_algebraic_curves
British-American mathematician (born 1962)
deprecated parameter |citeseerx= (help) Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem". Annals of Mathematics. 141 (3): 443–551
Richard Taylor (mathematician)
Richard_Taylor_(mathematician)
British mathematician who proved Fermat's Last Theorem
there in 1974, he worked on unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur's generalizations of Iwasawa
Andrew_Wiles
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
382–406. ISBN 978-3-642-08142-2. "Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)". The L-functions and modular forms database. The function F
Lemniscate_constant
Mathematical concept
introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and
Euler_system
Major type of automorphic form in mathematics
Siegel modular forms are a type of automorphic form that generalize conventional elliptic modular forms, which are closely related to elliptic curves. The
Siegel_modular_form
Mathematical structure
In mathematics, an Igusa curve is (roughly) a coarse moduli space of elliptic curves in characteristic p with a level p Igusa structure, where an Igusa
Igusa_variety
An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve. There have been several approaches to do
Counting points on elliptic curves
Counting_points_on_elliptic_curves
Branch of algebraic geometry
Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms. This connection would ultimately lead to
Arithmetic_geometry
some sense. Katz's definition of a p-adic modular form is similar, except that E is now an elliptic curve over some algebra R (with p nilpotent) over
P-adic_modular_form
Mathematical functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied
Lemniscate_elliptic_functions
ISBN 978-0-8218-4476-2, MR 2498060 Cremona, J.E. (1997), Algorithms for modular elliptic curves (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-59820-6
Modular_symbol
Over the open subscheme where q is invertible, the Tate curve is an elliptic curve. The Tate curve can also be defined for q as an element of a complete
Tate_curve
Mathematical function associated to algebraic varieties
function. For elliptic curves over the rational numbers, the Hasse–Weil conjecture follows from the modularity theorem: each elliptic curve E over Q {\displaystyle
Hasse–Weil_zeta_function
Mathematical concept
In the study of the arithmetic of elliptic curves, the j-line over a ring R is the coarse moduli scheme attached to the moduli problem sending a ring
J-line
Mathematical equation
{\displaystyle g_{2}} and g 3 {\displaystyle g_{3}} the modular invariants of the elliptic curve in Weierstrass form: y 2 = 4 x 3 − g 2 x − g 3 . {\displaystyle
Picard–Fuchs_equation
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
of the main conjectures for a large class of modular forms. As a consequence, for a modular elliptic curve over the rational numbers, they prove that the
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Group in arithmetic geometry
this for some elliptic curves of rank at most 1 with complex multiplication. Victor A. Kolyvagin extended this to modular elliptic curves over the rationals
Tate–Shafarevich_group
Algebraic curve
= 2 (a conic) and genus 1 only for n = 3 (an elliptic curve). The Jacobian variety of the Fermat curve has been studied in depth. It is isogenous to
Fermat_curve
Bicuspid curve Cassini oval Cubic curve Elliptic curve Watt's curve Butterfly curve (algebraic) Elkies trinomial curves Hyperelliptic curve Klein quartic
Gallery_of_curves
Conjecture in number theory
between the torsion conjecture for elliptic curves over the rationals and the theory of classical modular curves. In the early 1970s, the work of Gérard
Torsion_conjecture
ISBN 978-0-387-97329-6. Silverman, J.; Tate, J. (1992). Rational Points on Elliptic Curves. New York, New York: Springer-Verlag. p. 110. ISBN 978-0-387-97825-3
List of publications in mathematics
List_of_publications_in_mathematics
Prime number of the form that allows fast modular reduction
primes as moduli for elliptic curve cryptography: curve p-192 uses modulus 2 192 − 2 64 − 1 {\displaystyle 2^{192}-2^{64}-1} curve p-224 uses modulus 2
Solinas_prime
Problem of inverting exponentiation in groups
Digital Signature Algorithm) and cyclic subgroups of elliptic curves over finite fields (see Elliptic curve cryptography). While there is no publicly known
Discrete_logarithm
Topics referred to by the same term
characteristic The elliptic modular function, giving the modular invariant of an elliptic curve. This disambiguation page lists mathematics articles associated
Modular_invariant
Mathematical function
{\displaystyle m} , or as the elliptic modulus k {\displaystyle k} , where k 2 = m {\displaystyle k^{2}=m} , or in terms of the modular angle α {\displaystyle
Jacobi_elliptic_functions
Hypotrochoid Lissajous curve Poinsot's spirals Rational normal curve Rose curve Bicuspid curve Cassinoide Cubic curve Elliptic curve Watt's curve Bolza surface
List_of_curves
Mathematical conjecture about elliptic curves
conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost
Sato–Tate_conjecture
American mathematician
this curve has a complicated form, it is natural and conceptually significant in the number theory of elliptic curves. The equation describes a modular curve
Jennifer_Balakrishnan
German mathematician (1912–1992)
method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later
Martin_Eichler
modular group on the upper half-plane. Consequently, the period domain is the Riemann sphere. This is the usual parameterization of an elliptic curve
Period_mapping
Computation modulo a fixed integer
modular arithmetic directly underpins public key systems such as RSA and Diffie–Hellman, and provides finite fields which underlie elliptic curves, and
Modular_arithmetic
Special mathematical function
description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents
Nome_(mathematics)
American mathematician (born 1943)
November 22, 2025. Leroy P. Steele Prize 2023 Wiles, Andrew (1995). "Modular Elliptic Curves and Fermat's Last Theorem". The Annals of Mathematics. 141 (3):
Nick_Katz
Modular unit in mathematics
mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions
Elliptic_unit
Rational right triangles cannot have square area
ISBN 978-0-8218-0863-4 Koblitz, Neal (1993), Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, vol. 97 (2nd ed.), Springer-Verlag
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
Algebraic variety that is a moduli space for principally polarized abelian varieties
Siegel modular varieties are the most basic examples of Shimura varieties. Siegel modular varieties generalize moduli spaces of elliptic curves to higher
Siegel_modular_variety
Mathematical concept
number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by
Shimura_variety
Problem about mathematical number fields
corresponding elliptic curve. One interpretation of Hilbert's twelfth problem asks to provide a suitable analogue of exponential, elliptic, or modular functions
Hilbert's_twelfth_problem
American mathematician (1947–2020)
Montgomery ladder, which is used to protect against side-channel attacks in elliptic curve cryptography. Montgomery began his undergraduate career at the University
Peter Montgomery (mathematician)
Peter_Montgomery_(mathematician)
Weierstrass's elliptic functions Elliptic integral Complex multiplication Weil pairing Hyperelliptic curve Klein quartic Modular curve Modular equation Modular function
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Generalization of Fermat's Last Theorem and of Catalan's conjecture,
Théorie des Nombres de Bordeaux. 18: 315–321. Andrew Wiles (1995). "Modular Elliptic Curves and Fermat's Last Theorem". Annals of Mathematics. 142: 443–551
Fermat–Catalan_conjecture
}}_{E,3}} is modular. This residual modularity was one of the starting inputs in Wiles's proof of the modularity of semistable elliptic curves, and hence
Langlands–Tunnell_theorem
American-Turkish mathematician (born 1992)
graduating summa cum laude. His undergraduate thesis, The average elliptic curve has few integral points, was supervised by Jacob Tsimerman. After earning
Levent_Alpöge
American mathematician and cryptographer
Waterloo. He is the creator of hyperelliptic curve cryptography and the independent co-creator of elliptic curve cryptography. Koblitz received his B.A. in
Neal_Koblitz
American mathematician (born 1947)
Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4 Goldfeld, Dorian Modular elliptic curves and Diophantine problems. Number theory (Banff, AB, 1988), 157–175
Dorian_M._Goldfeld
American mathematician
algebraic geometry, including several foundational results on elliptic curves and modular curves. His 1975 observation connecting supersingular primes to the
Andrew_Ogg
Class of integer sequences in mathematics
nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by
Elliptic divisibility sequence
Elliptic_divisibility_sequence
varieties are a natural generalization of elliptic curves to higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved stack playing
Moduli_of_abelian_varieties
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map
Dixon_elliptic_functions
Projective variety that is also an algebraic group
of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for g > 1 {\displaystyle g>1} it has been
Abelian_variety
Connects non-singular algebraic curves with compact Riemann surfaces
modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular
Belyi's_theorem
Concept in mathematics
mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing
Drinfeld_module
Topics referred to by the same term
Discriminant of an elliptic curve Discriminant of a quadratic form Discriminant of a real-valued function Fundamental discriminant Modular discriminant Modified
Discriminant_(disambiguation)
calculating the number of points on an elliptic curve over a finite field. Its primary application is in elliptic curve cryptography. The algorithm is an extension
Schoof–Elkies–Atkin_algorithm
Polynomial equation whose integer solutions are sought
Springer, p. 117, ISBN 9781846280443. Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3):
Diophantine_equation
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
Topics referred to by the same term
mathematics Modular lattice a kind of partially ordered set Modularity theorem (formerly Taniyama–Shimura conjecture), a connection between elliptic curves and
Module
Conjecture in number theory
546. ISSN 1246-7405. Anni, Samuele; Siksek, Samir (2016-08-30). "Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ
Beal_conjecture
Monster and modular connection
moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular the j function. The initial numerical observation
Monstrous_moonshine
Mathematical function
3. ISBN 3-540-97127-0. Koblitz, Neal (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd ed.). Springer-Verlag
Dedekind_eta_function
Conjectures connecting number theory and geometry
main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized
Langlands_program
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