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# Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections. 

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

## Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Alfred Menezes

Alfred Menezes is co-author of several books on cryptography, including the Handbook of Applied Cryptography, and is a professor of mathematics at the University of Waterloo in Canada.

## Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

## Algebraic curve

In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

## Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

## Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

## Andrew Wiles

Sir Andrew John Wiles (born 11 April 1953) is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory.

## Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

## Arithmetic dynamics

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory.

## Atkin–Lehner theory

In mathematics, Atkin–Lehner theory is part of the theory of modular forms, in which the concept of newform is defined in such a way that the theory of Hecke operators can be extended to higher level.

## Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.

## Barry Mazur

Barry Charles Mazur (born December 19, 1937) is an American mathematician and a Gerhard Gade University Professor at Harvard University.

## Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve.

## Carl Ludwig Siegel

Carl Ludwig Siegel (December 31, 1896 – April 4, 1981) was a German mathematician specialising in number theory and celestial mechanics.

## Carl Pomerance

Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American number theorist.

## Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

## Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Peterborough, New Hampshire, United States.

## Cohen structure theorem

In mathematics, the Cohen structure theorem, introduced by, describes the structure of complete Noetherian local rings.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Complex multiplication

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules).

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Complex projective plane

In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space.

## Conductor of an abelian variety

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is.

## Congruent number

In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Counting points on elliptic curves

An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve.

## Cryptography

Cryptography or cryptology (from κρυπτός|translit.

## Cubic plane curve

In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation.

## Cusp (singularity)

In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward.

## Degree of a field extension

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension.

## Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.

## Discrete logarithm

In the mathematics of the real numbers, the logarithm logb a is a number x such that, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logb a is an integer k such that.

## Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

## Doubling-oriented Doche–Icart–Kohel curve

In mathematics, the doubling-oriented Doche–Icart–Kohel curve is a form in which an elliptic curve can be written.

## Dual abelian variety

In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.

## Dual EC DRBG

Dual_EC_DRBG (Dual Elliptic Curve Deterministic Random Bit Generator) is an algorithm that was presented as a cryptographically secure pseudorandom number generator (CSPRNG) using methods in elliptic curve cryptography.

## EdDSA

In public-key cryptography, Edwards-curve Digital Signature Algorithm (EdDSA) is a digital signature scheme using a variant of Schnorr signature based on Twisted Edwards curves.

x^2+y^2.

## Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

## Elliptic algebra

In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2.

## Elliptic Curve Digital Signature Algorithm

In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic curve cryptography.

## Elliptic curve primality

In mathematics elliptic curve primality testing techniques are among the quickest and most widely used methods in primality proving.

## Elliptic function

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions.

## Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

## Elliptic surface

In 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 almost all fibers are smooth curves of genus 1.

## Elliptic-curve cryptography

Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.

## Elliptic-curve Diffie–Hellman

Elliptic-curve Diffie–Hellman (ECDH) is an anonymous key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Euclidean algorithm

. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.

## Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

## Factorization

In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

## Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Field extension

In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Finite morphism

In algebraic geometry, a morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi.

## Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.

## Finitely generated group

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

## Functional equation

In mathematics, a functional equation is any equation in which the unknown represents a function.

## Fundamental pair of periods

In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane.

## Galois module

In mathematics, a Galois module is a ''G''-module, with G being the Galois group of some extension of fields.

## Generalized Riemann hypothesis

The Riemann hypothesis is one of the most important conjectures in mathematics.

## Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.

## Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Gerhard Frey

Gerhard Frey (born 1944) is a German mathematician, known for his work in number theory.

## Glossary of arithmetic and diophantine geometry

This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.

## Hasse's theorem on elliptic curves

Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below.

## Hasse–Weil zeta function

In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function.

## Hesse configuration

In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by, is a configuration of 9 points and 12 lines with three points per line and four lines through each point.

## Hessian form of an elliptic curve

In geometry, the Hessian curve is a plane curve similar to folium of Descartes.

## Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.

## Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

## Inflection point

In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuously differentiable plane curve at which the curve crosses its tangent, that is, the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.

## Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

## Inventiones Mathematicae

Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.

## J-invariant

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

## J-line

In the study of the arithmetic of elliptic curves, the j-line over any ring R is the coarse moduli scheme attached to the moduli problem &Gamma;(1): with the j-invariant normalized a la Tate: j.

## Jacobian curve

In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one (Weierstrass equation).

## Jean-Pierre Serre

Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.

## John Tate

John Torrence Tate Jr. (born March 13, 1925) is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.

## Ken Ribet

Kenneth Alan "Ken" Ribet (born June 28, 1948) is an American mathematician, currently a professor of mathematics at the University of California, Berkeley.

## Lenstra elliptic-curve factorization

The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves.

## List of computer algebra systems

The following tables provide a comparison of computer algebra systems (CAS).

## Loïc Merel

Loïc Merel (born 13 August 1965) is a French mathematician.

## Local zeta-function

In number theory, the local zeta function Z(V,s) (sometimes called the congruent zeta function) is defined as where N_m is the number of points of V defined over the degree m extension \mathbf_ of \mathbf_q, and V is a non-singular n-dimensional projective algebraic variety over the field \mathbf_q with q elements.

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

## Louis J. Mordell

Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.

## Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

## Modular curve

In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).

## Modular form

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

## Modular lambda function

In mathematics, the elliptic modular lambda function &lambda;(&tau) is a highly symmetric holomorphic function on the complex upper half-plane.

## Montgomery curve

In mathematics the Montgomery curve is a form of elliptic curve, different from the usual Weierstrass form, introduced by Peter L. Montgomery in 1987.

## Mordell–Weil theorem

In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group.

## Nagell–Lutz theorem

In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.

## Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

## Noam Elkies

Noam David Elkies (born August 25, 1966) is an American mathematician and professor of mathematics at Harvard University.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

## Plane curve

In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane.

## Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

## Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

## Projective variety

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.

## Proof by infinite descent

In mathematics, a proof by infinite descent is a particular kind of proof by contradiction that relies on the least integer principle.

In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

## Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

## Rank of an abelian group

In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.

## Rational function

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.

## Rational mapping

In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties.

## Rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Richard Crandall

Richard E. Crandall (December 29, 1947 – December 20, 2012) was an American physicist and computer scientist who made contributions to computational number theory.

## Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.

## Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

## Riemann–Hurwitz formula

In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other.

## Ring of integers

In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.

## Sato–Tate conjecture

In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If Np denotes the number of points on Ep and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np.

## Schoof's algorithm

Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields.

## Scott Vanstone

Scott A. Vanstone was a mathematician and cryptographer in the University of Waterloo Faculty of Mathematics.

## Serge Lang

Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.

## Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter.

## Singular point of an algebraic variety

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined.

## Singularity (mathematics)

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

## Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Square-free integer

In mathematics, a square-free integer is an integer which is divisible by no perfect square other than 1.

## Square-free polynomial

In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, a unique factorization domain) that does not have as a factor any square of a non-unit factor.

## Stella octangula number

In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Supersingular isogeny key exchange

Supersingular isogeny Diffie–Hellman key exchange (SIDH) is a post-quantum cryptographic algorithm used to establish a secret key between two parties over an otherwise insecure communications channel.

## Symmetry

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.

## Torsion conjecture

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field.

## Torsion subgroup

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A).

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

## Tripling-oriented Doche–Icart–Kohel curve

The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve.

## Tunnell's theorem

In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.

## Twisted Edwards curve

In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008.

## Twisted Hessian curves

In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have strongly unified arithmetic.

## Twists of curves

In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny.

## Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.

## Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

## Upper half-plane

In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.

## Weierstrass's elliptic functions

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass.

## Weil conjectures

In mathematics, the Weil conjectures were some highly influential proposals by on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.

## References

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