Elliptic curve

In 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 a field "K" and describes points in "K"2, the Cartesian product of "K" with itself. If the field has characteristic different from 2 and 3 then the curve can be described as a plane algebraic curve which, after a linear change of variables, consists of solutions ("x","y") to:
for some coefficients "a" and "b" in "K". The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition formula_2.) It is always understood that the curve is really sitting in the projective plane, with the point "O" being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see below.)
An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and "O" serves as the identity element. 
If "y"2 = "P"("x"), where "P" is any polynomial of degree three in "x" with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If "P" has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity.
Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism.
Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization.
An elliptic curve is "not" an ellipse: see elliptic integral for the origin of the term. Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.
Elliptic curves over the real numbers.
Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry.
In this context, an elliptic curve is a plane curve defined by an equation of the form
after a linear change of variables ("a" and "b" are real numbers). This type of equation is called a Weierstrass equation.
The definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, or isolated points. Algebraically, this holds if and only if the discriminant
is not equal to zero. (Although the factor −16 is irrelevant to whether or not the curve is non-singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.)
The (real) graph of a non-singular curve has "two" components if its discriminant is positive, and "one" component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368.
The group law.
When working in the projective plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have an additional point at infinity, "O", at the homogeneous coordinates [0:1:0] which serves as the identity of the group.
Since the curve is symmetrical about the x-axis, given any point "P", we can take −"P" to be the point opposite it. We take −"O" to be just "O".
If "P" and "Q" are two points on the curve, then we can uniquely describe a third point, "P" + "Q", in the following way. First, draw the line that intersects "P" and "Q". This will generally intersect the cubic at a third point, "R". We then take to be −"R", the point opposite "R".
This definition for addition works except in a few special cases related to the point at infinity and intersection multiplicity. The first is when one of the points is "O". Here, we define , making "O" the identity of the group. Next, if "P" and "Q" are opposites of each other, we define . Lastly, if we only have one point, thus we can't define the line between them. In this case, we use the tangent line to the curve at this point as our line. In most cases, the tangent will intersect a second point "R" and we can take its opposite. However, if "P" happens to be an inflection point (a point where the concavity of the curve changes), we take "R" to be "P" itself and "P" + "P" is simply the point opposite itself.
For a cubic curve not in Weierstrass normal form, we can still define a group structure by designating one of its nine inflection points as the identity "O". In the projective plane, each line will intersect a cubic at three points when accounting for multiplicity. For a point "P", −"P" is defined as the unique third point on the line passing through "O" and "P". Then, for any "P" and "Q", "P" + "Q" is defined as −"R" where "R" is the unique third point on the line containing "P" and "Q".
Let "K" be a field over which the curve is defined (i.e., the coefficients of the defining equation or equations of the curve are in "K") and denote the curve by "E". Then the "K"-rational points of "E" are the points on "E" whose coordinates all lie in "K", including the point at infinity. The set of "K"-rational points is denoted by "E"("K"). It, too, forms a group, because properties of polynomial equations show that if "P" is in "E"("K"), then −"P" is also in "E"("K"), and if two of "P", "Q", and "R" are in "E"("K"), then so is the third. Additionally, if "K" is a subfield of "L", then "E"("K") is a subgroup of "E"("L").
The above group can be described algebraically as well as geometrically. Given the curve "y"2 = "x"3 + "ax" + "b" over the field "K" (whose characteristic we assume to be neither 2 nor 3), and points "P" = ("xP", "yP") and "Q" = ("xQ", "yQ") on the curve, assume first that "xP" ≠ "xQ" (first pane below). Let "y = sx + d" be the line that intersects "P" and "Q", which has the following slope:
Since "K" is a field, "s" is well-defined. The line equation and the curve equation have an identical "y" in the points "xP", "xQ", and "xR".
which is equivalent to formula_7. We know that this equation has its roots in exactly the same "x"-values as
We equate the coefficient for "x2" and solve for "xR". "yR" follows from the line equation. This defines "R" = ("xR", "yR") = −("P" + "Q") with
If "xP" = "xQ", then there are two options: if "yP" = −"yQ" (third and fourth panes below), including the case where "yP" = "yQ" = 0 (fourth pane), then the sum is defined as 0; thus, the inverse of each point on the curve is found by reflecting it across the "x"-axis. If "yP" = "yQ" ≠ 0, then "Q" = "P" and "R" = ("x""R", "y""R") = −("P" + "P") = −2"P" = −2"Q" (second pane below with "P" shown for "R") is given by
Elliptic curves over the complex numbers.
The formulation of elliptic curves as the embedding of a torus in the complex projective plane follows naturally from a curious property of Weierstrass's elliptic functions. These functions and their first derivative are related by the formula
Here, "g"2 and "g"3 are constants; formula_12 is the Weierstrass elliptic function and formula_13 its derivative. It should be clear that this relation is in the form of an elliptic curve (over the complex numbers). The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus "T" = C/Λ. This torus may be embedded in the complex projective plane by means of the map
This map is a group isomorphism of the torus (considered with its natural group structure) with the chord-and-tangent group law on the cubic curve which is the image of this map. It is also an isomorphism of Riemann surfaces from the torus to the cubic curve, so topologically, an elliptic curve is a torus. If the lattice Λ is related by multiplication by a non-zero complex number "c" to a lattice "c"Λ, then the corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the j-invariant.
The isomorphism classes can be understood in a simpler way as well. The constants "g"2 and "g"3, called the modular invariants, are uniquely determined by the lattice, that is, by the structure of the torus. However, all real polynomials factorize completely into linear factors over the complex numbers, since the field of complex numbers is the algebraic closure of the reals. So, the elliptic curve may be written as
One finds that
and
so that the modular discriminant is
Here, λ is sometimes called the modular lambda function.
Note that the uniformization theorem implies that every compact Riemann surface of genus one can be represented as a torus.
This also allows an easy understanding of the torsion points on an elliptic curve: if the lattice Λ is spanned by the fundamental periods ω1 and ω2, then the "n"-torsion points are the (equivalence classes of) points of the form
for "a" and "b" integers in the range from 0 to "n"−1.
If formula_20 is an elliptic curve over the complex numbers and formula_21, formula_22 and formula_23, then a pair of fundamental periods of formula_24 can be calculated very rapidly by formula_25 and formula_26 where formula_27 is the arithmetic–geometric mean of formula_28 and formula_29. At each step of the arithmetic–geometric mean iteration, the signs of formula_30 arising from the ambiguity of geometric mean iterations are chosen such that formula_31 where formula_32 and formula_30 denote the individual arithmetic mean and geometric mean iterations of formula_28 and formula_29, respectively. When formula_36, there is an additional condition that formula_37. 
Over the complex numbers, every elliptic curve has nine inflection points. Every line through two of these points also passes through a third inflection point; the nine points and 12 lines formed in this way form a realization of the Hesse configuration.
Elliptic curves over the rational numbers.
A curve "E" defined over the field of rational numbers is also defined over the field of real numbers. Therefore, the law of addition (of points with real coordinates) by the tangent and secant method can be applied to "E". The explicit formulae show that the sum of two points "P" and "Q" with rational coordinates has again rational coordinates, since the line joining "P" and "Q" has rational coefficients. This way, one shows that the set of rational points of "E" forms a subgroup of the group of real points of "E". As this group, it is an abelian group, that is, "P" + "Q" = "Q" + "P".
The structure of rational points.
The most important result is that all points can be constructed by the method of tangents and secants starting with a "finite" number of points. More precisely the Mordell–Weil theorem states that the group "E"(Q) is a finitely generated (abelian) group. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups.
The proof of that theorem rests on two ingredients: first, one shows that for any integer "m" > 1, the quotient group "E"(Q)/"mE"(Q) is finite (weak Mordell–Weil theorem). Second, introducing a height function "h" on the rational points "E"(Q) defined by "h"("P"0) = 0 and if "P" (unequal to the point at infinity "P"0) has as abscissa the rational number "x" = "p"/"q" (with coprime "p" and "q"). This height function "h" has the property that "h"("mP") grows roughly like the square of "m". Moreover, only finitely many rational points with height smaller than any constant exist on "E".
The proof of the theorem is thus a variant of the method of infinite descent and relies on the repeated application of Euclidean divisions on "E": let "P" ∈ "E"(Q) be a rational point on the curve, writing "P" as the sum 2"P"1 + "Q"1 where "Q"1 is a fixed representant of "P" in "E"(Q)/2"E"(Q), the height of "P"1 is about of the one of "P" (more generally, replacing 2 by any "m" > 1, and by ). Redoing the same with "P"1, that is to say "P"1 = 2"P"2 + "Q"2, then "P"2 = 2"P"3 + "Q"3, etc. finally expresses "P" as an integral linear combination of points "Qi" and of points whose height is bounded by a fixed constant chosen in advance: by the weak Mordell–Weil theorem and the second property of the height function "P" is thus expressed as an integral linear combination of a finite number of fixed points.
So far, the theorem is not effective since there is no known general procedure for determining representatives of "E"(Q)/"mE"(Q).
The rank of "E"(Q), that is the number of copies of Z in "E"(Q) or, equivalently, the number of independent points of infinite order, is called the "rank" of "E". The Birch and Swinnerton-Dyer conjecture is concerned with determining the rank. One conjectures that it can be arbitrarily large, even if only examples with relatively small rank are known. The elliptic curve with biggest exactly known rank is
It has rank 20, found by Noam Elkies and Zev Klagsbrun in 2020. Curves of rank higher than 20 were known since 1994, with lower bounds on their ranks ranging from at-least-21 to at-least-28, but their exact ranks are not currently known and in particular it is not proven which of them have higher rank than the others or which is the true "current champion".
As for the groups constituting the torsion subgroup of "E"(Q), the following is known: the torsion subgroup of "E"(Q) is one of the 15 following groups (a theorem due to Barry Mazur): Z/"NZ for "N" = 1, 2, ..., 10, or 12, or Z/2Z × Z/2"NZ with "N" = 1, 2, 3, 4. Examples for every case are known. Moreover, elliptic curves whose Mordell–Weil groups over Q have the same torsion groups belong to a parametrized family.
The Birch and Swinnerton-Dyer conjecture.
The "Birch and Swinnerton-Dyer conjecture" (BSD) is one of the Millennium problems of the Clay Mathematics Institute. The conjecture relies on analytic and arithmetic objects defined by the elliptic curve in question.
At the analytic side, an important ingredient is a function of a complex variable, "L", the Hasse–Weil zeta function of "E" over Q. This function is a variant of the Riemann zeta function and Dirichlet L-functions. It is defined as an Euler product, with one factor for every prime number "p".
For a curve "E" over Q given by a minimal equation
with integral coefficients formula_39, reducing the coefficients modulo "p" defines an elliptic curve over the finite field F"p" (except for a finite number of primes "p", where the reduced curve has a singularity and thus fails to be elliptic, in which case "E" is said to be of bad reduction at "p").
The zeta function of an elliptic curve over a finite field F"p" is, in some sense, a generating function assembling the information of the number of points of "E" with values in the finite field extensions F"pn" of F"p". It is given by
The interior sum of the exponential resembles the development of the logarithm and, in fact, the so-defined zeta function is a rational function:
where the 'trace of Frobenius' term formula_42 is defined to be the (negative of) the difference between the number of points on the elliptic curve formula_24 over formula_44 and the 'expected' number formula_45, viz.:
There are two points to note about this quantity. First, these formula_42 are not to be confused with the formula_39 in the definition of the curve formula_24 above: this is just an unfortunate clash of notation. Second, we may define the same quantities and functions over an arbitrary finite field of characteristic formula_50, with formula_51 replacing formula_50 everywhere.
The Hasse–Weil zeta function of "E" over Q is then defined by collecting this information together, for all primes "p". It is defined by
where "ε"("p") = 1 if "E" has good reduction at "p" and 0 otherwise (in which case "ap" is defined differently from the method above: see Silverman (1986) below).
This product converges for Re("s") > 3/2 only. Hasse's conjecture affirms that the "L"-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any "s", "L"("E", "s") to "L"("E", 2 − "s"). In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture, which asserts that every elliptic curve over "Q" is a modular curve, which implies that its "L"-function is the "L"-function of a modular form whose analytic continuation is known. One can therefore speak about the values of "L"("E", "s") at any complex number "s". 
The "Birch–Swinnerton-Dyer conjecture" relates the arithmetic of the curve to the behavior of its "L"-function at "s" = 1. It affirms that the vanishing order of the "L"-function at "s" = 1 equals the rank of "E" and predicts the leading term of the Laurent series of "L"("E", "s") at that point in terms of several quantities attached to the elliptic curve.
Much like the Riemann hypothesis, the truth of the BSD conjecture would have multiple consequences, including the following two:
The modularity theorem and its application to Fermat's Last Theorem.
The modularity theorem, once known as the Taniyama–Shimura–Weil conjecture, states that every elliptic curve "E" over Q is a modular curve, that is to say, its Hasse–Weil zeta function is the "L"-function of a modular form of weight 2 and level "N", where "N" is the conductor of "E" (an integer divisible by the same prime numbers as the discriminant of "E", Δ("E")). In other words, if one writes the "L"-function for Re("s") > 3/2 in the form
then the expression
defines a parabolic modular newform of weight 2 and level "N". For prime numbers "ℓ" not dividing "N", the coefficient "a"("ℓ") is equal to "ℓ" minus the number of solutions of the minimal equation of the curve modulo "ℓ".
For example, the elliptic curve formula_60, with discriminant (and conductor) 37, is associated to the form
For prime numbers ℓ not equal to 37, one can verify the property about the coefficients. Thus, for "ℓ" = 3, there are 6 solutions of the equation modulo 3: , , , , , ; thus .
The conjecture, going back to the 1950s, was completely proven by 1999 using ideas of Andrew Wiles, who proved it in 1994 for a large family of elliptic curves.
There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve "E" of conductor "N" can be expressed also by saying that there is a non-constant rational map defined over Q, from the modular curve "X"0("N") to "E". In particular, the points of "E" can be parametrized by modular functions.
For example, a modular parametrization of the curve formula_62 is given by
where, as above, "q" = exp(2π"iz"). The functions "x"("z") and "y"("z") are modular of weight 0 and level 37; in other words they are meromorphic, defined on the upper half-plane Im("z") > 0 and satisfy
and likewise for "y"("z"), for all integers "a, b, c, d" with "ad" − "bc" = 1 and 37|"c".
Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.
The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime "p" ≥ 5, the Fermat equation
has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice, the elliptic curve
of discriminant
cannot be modular. Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987.
Integral points.
This section is concerned with points "P" = ("x", "y") of "E" such that "x" is an integer. The following theorem is due to C. L. Siegel: the set of points "P" = ("x", "y") of "E"(Q) such that "x" is an integer is finite. This theorem can be generalized to points whose "x" coordinate has a denominator divisible only by a fixed finite set of prime numbers.
The theorem can be formulated effectively. For example, if the Weierstrass equation of "E" has integer coefficients bounded by a constant "H", the coordinates ("x", "y") of a point of "E" with both "x" and "y" integer satisfy:
The Sato–Tate conjecture is a statement about how the error term formula_69 in Hasse's theorem varies with the different primes "q", if an elliptic curve E over Q is reduced modulo q. It was proven (for almost all such curves) in 2006 due to the results of Taylor, Harris and Shepherd-Barron, and says that the error terms are equidistributed.
Elliptic curves over finite fields are notably applied in cryptography and for the factorization of large integers. These algorithms often make use of the group structure on the points of "E". Algorithms that are applicable to general groups, for example the group of invertible elements in finite fields, F*"q", can thus be applied to the group of points on an elliptic curve. For example, the discrete logarithm is such an algorithm. The interest in this is that choosing an elliptic curve allows for more flexibility than choosing "q" (and thus the group of units in F"q"). Also, the group structure of elliptic curves is generally more complicated.
Algorithms that use elliptic curves.
Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. For more see also:
References.
Serge Lang, in the introduction to the book cited below, stated that "It is possible to write endlessly on elliptic curves. (This is not a threat.)" The following short list is thus at best a guide to the vast expository literature available on the theoretical, algorithmic, and cryptographic aspects of elliptic curves.