Elliptic curve

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A catalog of elliptic curves. The region shown is x, y ∈ [−3,3].
(For (a, b) = (0, 0) the function is not smooth and therefore not an 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 K2, the Cartesian product of K with itself. If the field's characteristic is 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) for:

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 4a3 + 27b2 ≠ 0, that is, being square-free in x.) 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 § Elliptic curves over a general field 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 y2 = 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 in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane . Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with j ≤ 1, and any ellipse in described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the identity on each trajectory curve.[1][2]

Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.

Elliptic curves over the real numbers[edit]

Graphs of curves y2 = x3x and y2 = x3x + 1

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, and said to be in Weierstrass form, or Weierstrass normal form.

The definition of elliptic curve also requires that the curve is 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.)[3]

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[edit]

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 O at infinity (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. , as it is the identity element.

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 P + Q 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 P + O = P = O + P, making O the identity of the group. Next, if P and Q are opposites of each other, we define P + Q = O. Lastly, if P = Q we only have one point, thus we cannot 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.

ECClines.svg

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 (that is, 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). E(K) is 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, 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 groups can be described algebraically as well as geometrically. Given the curve y2 = x3 + 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 xPxQ (case 1). Let y = sx + d be the equation of the line that intersects P and Q, which has the following slope:

The line equation and the curve equation intersect at the points xP, xQ, and xR, so the equations have identical y values.

which is equivalent to

We know that this equation has its roots at exactly the same x values as

We equate the coefficient for x2 and solve for xR.

yR follows from the line equation

and is an element of K. This therefore defines R = (xR, yR) = −(P + Q).

If xP = xQ, then there are two options: if yP = −yQ (case 3), including the case where yP = yQ = 0 (case 4), 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 = (xR, yR) = −(P + P) = −2P = −2Q (case 2 using P as R). The slope is given by the tangent to the curve at (xP, yP).

Elliptic curves over the rational numbers[edit]

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.

Integral points[edit]

This section is concerned with points P = (x, y) of E such that x is an integer.

For example, the equation y2 = x3 + 17 has eight integral solutions with y > 0 :[4][5]

(x, y) = (−2, 3), (−1, 4), (2, 5), (4, 9), (8, 23), (43, 282), (52, 375), (5234, 378661).

As another example, Ljunggren's equation, a curve whose Weierstrass form is y2 = x3 − 2x, has only four solutions with y ≥ 0 :[6]

(x, y) = (0, 0), (−1, 1), (2, 2), (338, 6214).

The structure of rational points[edit]

Rational points can be constructed by the method of tangents and secants detailed above, starting with a finite number of rational points. More precisely[7] 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 the theorem[8] involves two parts. The first part shows that for any integer m > 1, the quotient group E(Q)/mE(Q) is finite (this is the weak Mordell–Weil theorem). Second, introducing a height function h on the rational points E(Q) defined by h(P0) = 0 and h(P) = log max(|p|, |q|) if P (unequal to the point at infinity P0) 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[9] and relies on the repeated application of Euclidean divisions on E: let PE(Q) be a rational point on the curve, writing P as the sum 2P1 + Q1 where Q1 is a fixed representant of P in E(Q)/2E(Q), the height of P1 is about 1/4 of the one of P (more generally, replacing 2 by any m > 1, and 1/4 by 1/m2). Redoing the same with P1, that is to say P1 = 2P2 + Q2, then P2 = 2P3 + Q3, 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.

The theorem however doesn't provide a method to determine any 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 the currently largest exactly-known rank is

y2 + xy + y = x3x2244537673336319601463803487168961769270757573821859853707x + 961710182053183034546222979258806817743270682028964434238957830989898438151121499931

It has rank 20, found by Noam Elkies and Zev Klagsbrun in 2020. Curves of rank higher than 20 have been known since 1994, with lower bounds on their ranks ranging from 21 to 28, but their exact ranks are not known and in particular it is not proven which of them have higher rank than the others or which is the true "current champion".[10]

As for the groups constituting the torsion subgroup of E(Q), the following is known:[11] 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/2NZ 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.[12]

The Birch and Swinnerton-Dyer conjecture[edit]

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 , reducing the coefficients modulo p defines an elliptic curve over the finite field Fp (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 Fp is, in some sense, a generating function assembling the information of the number of points of E with values in the finite field extensions Fpn of Fp. It is given by[13]

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[14] is defined to be the difference between the 'expected' number and the number of points on the elliptic curve over , viz.

or equivalently,

.

There are two points to note about this quantity. First, these are not to be confused with the in the definition of the curve 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 , with replacing everywhere.

The L-function of E over Q is then defined by collecting this information together, for all primes p. It is defined by

where N is the conductor of E, i.e. the product of primes with bad reduction, 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.

At s=1 (the conductor product can be discarded as it is finite), the L-function becomes

The Birch and Swinnerton-Dyer conjecture relates the arithmetic of the curve to the behaviour of this 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:

  • A congruent number is defined as an odd square-free integer n which is the area of a right triangle with rational side lengths. It is known that n is a congruent number if and only if the elliptic curve has a rational point of infinite order; assuming BSD, this is equivalent to its L-function having a zero at s = 1. Tunnell has shown a related result: assuming BSD, n is a congruent number if and only if the number of triplets of integers (x, y, z) satisfying is twice the number of triples satisfying . The interest in this statement is that the condition is easy to check.[15]
  • In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the critical strip for certain L-functions. Admitting BSD, these estimations correspond to information about the rank of families of the corresponding elliptic curves. For example: assuming the generalized Riemann hypothesis and BSD, the average rank of curves given by is smaller than 2.[16]

Elliptic curves over finite fields[edit]

Set of affine points of elliptic curve y2 = x3x over finite field F61.

Let K = Fq be the finite field with q elements and E an elliptic curve defined over K. While the precise number of rational points of an elliptic curve E over K is in general difficult to compute, Hasse's theorem on elliptic curves gives the following inequality:

(which includes the point at infinity).

In other words, the number of points on the curve grows proportionally to the number of elements in the field. This fact can be understood and proven with the help of some general theory; see local zeta function and étale cohomology for example.

Set of affine points of elliptic curve y2 = x3x over finite field F89.

The set of points E(Fq) is a finite abelian group. It is always cyclic or the product of two cyclic groups, depending whether q is even or odd. For example,[17] the curve defined by

over F71 has 72 points (71 affine points including (0,0) and one point at infinity) over this field, whose group structure is given by Z/2Z × Z/36Z. The number of points on a specific curve can be computed with Schoof's algorithm.

Studying the curve over the field extensions of Fq is facilitated by the introduction of the local zeta function of E over Fq, defined by a generating series (also see above)

where the field Kn is the (unique up to isomorphism) extension of K = Fq of degree n (that is, Fqn).

The zeta function is a rational function in T. To see this, there is an integer such that

with complex number and its complex conjugate. We can choose so that its absolute value is , that is , and that . (This proves Hasse's Theorem as minimizes ).

The local zeta function then becomes

Then , so finally

The functional equation is

For example,[18] the zeta function of E : y2 + y = x3 over the field F2 is given by

which follows from:

Set of affine points of elliptic curve y2 = x3x over finite field F71.

The Sato–Tate conjecture is a statement about how the error term 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,[19] 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 Fq). Also, the group structure of elliptic curves is generally more complicated.

Elliptic curves over a general field[edit]

Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K.

If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form

after a linear change of variables. Here p and q are elements of K such that the right hand side polynomial x3pxq does not have any double roots. If the characteristic is 2 or 3, then more terms need to be kept: in characteristic 3, the most general equation is of the form

for arbitrary constants b2, b4, b6 such that the polynomial on the right-hand side has distinct roots (the notation is chosen for historical reasons). In characteristic 2, even this much is not possible, and the most general equation is

provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable linear change of variables.

One typically takes the curve to be the set of all points (x,y) which satisfy the above equation and such that both x and y are elements of the algebraic closure of K. Points of the curve whose coordinates both belong to K are called K-rational points.

Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to Loïc Merel shows that for a given integer d, there are (up to isomorphism) only finitely many groups that can occur as the torsion groups of E(K) for an elliptic curve defined over a number field K of degree d. More precisely,[20] there is a number B(d) such that for any elliptic curve E defined over a number field K of degree d, any torsion point of E(K) is of order less than B(d). The theorem is effective: for d > 1, if a torsion point is of order p, with p prime, then

As for the integral points, Siegel's theorem generalizes to the following: Let E be an elliptic curve defined over a number field K, x and y the Weierstrass coordinates. Then there are only finitely many points of E(K) whose x-coordinate is in the ring of integers OK.

The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation.

Elliptic curves over the complex numbers[edit]

An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ω1 and ω2. The four-torsion is also shown, corresponding to the lattice 1/4Λ containing Λ.

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, g2 and g3 are constants; ℘(z) is the Weierstrass elliptic function and (z) 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. 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 cΛ, 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 g2 and g3, 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

with j-invariant j(τ) and λ(τ) is sometimes called the modular lambda function. For example, let τ = 2i, then λ(2i) = (−1 + 2)4 which implies g2, g3, and therefore g23
− 27g32
of the formula above are all algebraic numbers if τ involves an imaginary quadratic field. In fact, it yields the integer j(2i) = 663 = 287496.

In contrast, the modular discriminant

is generally a transcendental number. In particular, the value of the Dedekind eta function η(2i) is

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 integers a and b in the range 0 ≤ (a, b) < n.

If

is an elliptic curve over the complex numbers and

then a pair of fundamental periods of E can be calculated very rapidly by

M(w, z) is the arithmetic–geometric mean of w and z. At each step of the arithmetic–geometric mean iteration, the signs of zn arising from the ambiguity of geometric mean iterations are chosen such that |wnzn| ≤ |wn + zn| where wn and zn denote the individual arithmetic mean and geometric mean iterations of w and z, respectively. When |wnzn| = |wn + zn|, there is an additional condition that Im(zn/wn) > 0.[21]

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.

Algorithms that use elliptic curves[edit]

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:

Alternative representations of elliptic curves[edit]

See also[edit]

Notes[edit]

  1. ^ Sarli, J. (2012). "Conics in the hyperbolic plane intrinsic to the collineation group". J. Geom. 103: 131–148. doi:10.1007/s00022-012-0115-5. S2CID 119588289.
  2. ^ Sarli, John (2021-10-22). "The Elliptic Curve Decomposition of Central Conics in the Real Hyperbolic Plane". doi:10.21203/rs.3.rs-936116/v1. {{cite journal}}: Cite journal requires |journal= (help)
  3. ^ Silverman 1986, III.1 Weierstrass Equations (p.45)
  4. ^ T. Nagell, L'analyse indéterminée de degré supérieur, Mémorial des sciences mathématiques 39, Paris, Gauthier-Villars, 1929, pp. 56–59.
  5. ^ OEIS: https://oeis.org/A029728
  6. ^ Siksek, Samir (1995), Descents on Curves of Genus 1 (Ph.D. thesis), University of Exeter, pp. 16–17, hdl:10871/8323.
  7. ^ Silverman 1986, Theorem 4.1
  8. ^ Silverman 1986, pp. 199–205
  9. ^ See also J. W. S. Cassels, Mordell's Finite Basis Theorem Revisited, Mathematical Proceedings of the Cambridge Philosophical Society 100, 3–41 and the comment of A. Weil on the genesis of his work: A. Weil, Collected Papers, vol. 1, 520–521.
  10. ^ Dujella, Andrej. "History of elliptic curves rank records". University of Zagreb.
  11. ^ Silverman 1986, Theorem 7.5
  12. ^ Silverman 1986, Remark 7.8 in Ch. VIII
  13. ^ The definition is formal, the exponential of this power series without constant term denotes the usual development.
  14. ^ see for example Silverman, Joseph H. (2006). "An Introduction to the Theory of Elliptic Curves" (PDF). Summer School on Computational Number Theory and Applications to Cryptography. University of Wyoming.
  15. ^ Koblitz 1993
  16. ^ Heath-Brown, D. R. (2004). "The Average Analytic Rank of Elliptic Curves". Duke Mathematical Journal. 122 (3): 591–623. arXiv:math/0305114. doi:10.1215/S0012-7094-04-12235-3. S2CID 15216987.
  17. ^ See Koblitz 1994, p. 158
  18. ^ Koblitz 1994, p. 160
  19. ^ Harris, M.; Shepherd-Barron, N.; Taylor, R. (2010). "A family of Calabi–Yau varieties and potential automorphy". Annals of Mathematics. 171 (2): 779–813. doi:10.4007/annals.2010.171.779.
  20. ^ Merel, L. (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres". Inventiones Mathematicae (in French). 124 (1–3): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. S2CID 3590991. Zbl 0936.11037.
  21. ^ Wing Tat Chow, Rudolf (2018). "The Arithmetic-Geometric Mean and Periods of Curves of Genus 1 and 2" (PDF). White Rose eTheses Online. p. 12.

References[edit]

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.

External links[edit]

This article incorporates material from Isogeny on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.