Elliptic curve

A non-singular complete algebraic curve of genus 1. The theory of elliptic curves is the source of a large part of contemporary algebraic geometry. But historically the theory of elliptic curves arose as a part of analysis, as the theory of elliptic integrals and elliptic functions (cf. Elliptic integral; Elliptic function).

Examples. A non-singular plane projective cubic curve; the intersection of two non-singular quadrics in three-dimensional projective space; a two-sheeted covering of the projective line ramified at exactly four points; and also a one-dimensional Abelian variety are elliptic curves.

The geometry of an elliptic curve.

Let $X$ be an elliptic curve over an algebraically closed field $k$. Then $X$ is biregularly isomorphic to a plane cubic curve (see [1], [9], [13]). If ${\rm char k} \ne 2,3$, then in the projective plane ${\mathbb P}^2$ there is an affine coordinate system in which the equation of $X$ is in normal Weierstrass form: $$y^2=x^3+ax+b$$

The curve $ $ is non-singular if and

only if the polynomial $_$ does not have multiple zeros, that is, if the discriminant $_$. In $_$ the curve (1) has a unique point at infinity, which is denoted by $_$; $_$ is a point of inflection of (1), and the tangent at $_$ is the line at infinity. The $_$-invariant of an elliptic curve $_$, $$_$$

does not depend on the choice of the

coordinate system. Two elliptic curves have the same $_$-invariant if and only if they are biregularly isomorphic. For any $_$ there is an elliptic curve $_$ over $_$ with $_$.

The group structure on an elliptic curve.

Let $_$ be a fixed point on an elliptic curve $_$. The mapping $_$ assigning to a point $_$ the divisor $_$ on $_$ establishes a one-to-one correspondence between $_$ and the group $_$ of divisor classes of degree $_$ on $_$, that is, the Picard variety of $_$. This correspondence endows $_$ with the structure of an Abelian group that is compatible with the structure of an algebraic variety and that turns $_$ into a one-dimensional Abelian variety $_$; here $_$ is the trivial element of the group. This group structure has the following geometric description. Let $_$ be a smooth plane cubic curve. Then the sum of two points $_$ and $_$ is defined by the rule $_$, where $_$ is the third point of intersection of $_$ with the line passing through $_$ and $_$. In other words, the sum of three points on $_$ vanishes if and only if the points are collinear.

An elliptic curve as a one-dimensional Abelian variety.

Let $_$ denote the endomorphism of multiplication by $_$ in $_$. If $_$ is an elliptic curve with distinguished point $_$, then any rational mapping $_$ has the form $_$, where $_$ and $_$ is a homomorphism of Abelian varieties. Here $_$ is either a constant mapping at $_$ or is an isogeny, that is, there is a homomorphism of Abelian varieties $_$ such that $_$ and $_$ for some $_$ (see [1], [6]).

The automorphism group of an elliptic curve $_$ acts transitively on $_$, and its subgroup $_$ of automorphisms leaving $_$ fixed is non-trivial and finite. Suppose that $_$ is not $_$ or $_$. When $_$ is neither 0 nor $_$, then $_$ consists of the two elements $_$ and $_$. The order of $_$ is 4 when $_$ and 6 when $_$ (see [1], [6], [13]).

An important invariant of an elliptic curve is the endomorphism ring $_$ of the Abelian variety $_$. The mapping $_$ defines an imbedding of $_$ in $_$. If $_$, one says that $_$ is an elliptic curve with complex multiplication. The ring $_$ can be of one of the following types (see [1], [9], [13]): I) $_$; II) $_$, where $_$ is the ring of algebraic integers of an imaginary quadratic field $_$ and $_$; or III) $_$ is a non-commutative $_$-algebra of rank 4 without divisors of zero. In this case $_$ and $_$ is a maximal order in the quaternion algebra over $_$ ramified only at $_$ and $_$. Such elliptic curves exist for all $_$ and are called supersingular; elliptic curves in characteristic $_$ that are not supersingular are said to be ordinary.

The group $_$ of points of an elliptic curve $_$ with orders that divide $_$ has the following structure: $_$ when $_$. For $_$ and ordinary elliptic curves $_$, while for supersingular elliptic curves $_$. For a prime number $_$ the Tate module $_$ is isomorphic to $_$.

Elliptic curves over non-closed fields.

Let $_$ be an elliptic curve over an arbitrary field $_$. If the set of $_$-rational points $_$ of $_$ is not empty, then $_$ is biregularly isomorphic to a plane cubic curve (1) with $_$ ($_$). The point at infinity $_$ of (1) is defined over $_$. As above, one can introduce a group structure on (1), turning $_$ into a one-dimensional Abelian variety over $_$ and turning the set $_$ into an Abelian group with $_$ as trivial element. If $_$ is finitely generated over its prime subfield, then $_$ is a finitely-generated group (the Mordell–Weil theorem).

For any elliptic curve $_$ there is defined the Jacobi variety $_$, which is a one-dimensional Abelian variety over $_$, and $_$ is a principal homogeneous space over $_$. If $_$ is not empty, then the choice of $_$ specifies an isomorphism $_$ under which $_$ becomes the trivial element of $_$. In general, $_$ and $_$ are isomorphic over a finite extension of $_$ (see [1], [4], [13]).

Elliptic curves over the field of complex numbers.

An elliptic curve over $_$ is a compact Riemann surface of genus 1, and vice versa. The group structure turns $_$ into a complex Lie group, which is a one-dimensional complex torus $_$, where $_$ is a lattice in the complex plane $_$. Conversely, any one-dimensional complex torus is an elliptic curve (see [3]). From the topological point of view, an elliptic curve is a two-dimensional torus.

The theory of elliptic curves over $_$ is in essence equivalent to the theory of elliptic functions. An identification of a torus $_$ with an elliptic curve can be effected as follows. The elliptic functions with a given period lattice $_$ form a field generated by the Weierstrass $_$-function (see Weierstrass elliptic functions) and its derivative $_$, which are connected by the relation $$_$$

The

mapping $_$ ($_$) induces an isomorphism between the torus $_$ and the elliptic curve $_$ with equation $_$. The identification of $_$ given by (1) with the torus $_$ is effected by curvilinear integrals of the holomorphic form $_$ and gives an isomorphism $_$.

The description of the set of all elliptic curves as tori $_$ leads to the modular function $_$. Two lattices $_$ and $_$ determine isomorphic tori if and only if they are similar, that is, if one is obtained from the other by multiplication by a complex number. Therefore it may be assumed that $_$ is generated by the numbers 1 and $_$ in $_$. Two lattices with bases $_$ and $_$ are similar if and only if $_$ for an element $_$ of the modular group $_$. The modular function $$_$$

is

also called the absolute invariant; $_$ if and only if $_$ for some $_$, and the function $_$ produces a one-to-one correspondence between the classes of isomorphic elliptic curves over $_$ and the complex numbers. If $_$, then $_$.

An elliptic curve $_$ has complex multiplication if and only if $_$ is an imaginary quadratic irrationality. In this case $_$ is a subring of finite index in the ring of algebraic integers of the imaginary quadratic field $_$. Elliptic curves with complex multiplication are closely connected with the class field theory for imaginary quadratic fields (see [4], [8]).

Arithmetic of elliptic curves.

Let $_$ be an elliptic curve over the finite field $_$ with $_$ elements. The set $_$ is always non-empty and finite. Hence $_$ is endowed with the structure of a one-dimensional Abelian variety over $_$, and $_$ with that of a finite Abelian group. The order $_$ of $_$ satisfies $_$. The characteristic polynomial of the Frobenius endomorphism acting on the Tate module $_$, $_$, is $_$. Its roots $_$ and $_$ are complex-conjugate algebraic integers of modulus $_$. For any finite extension $_$ of $_$ of degree $_$, the order of $_$ is $_$. The zeta-function of $_$ is $$_$$

For any algebraic

integer $_$ of modulus $_$ in some imaginary quadratic field (or in $_$) one can find an elliptic curve $_$ over $_$ such that the order of $_$ is $_$.

Let $_$ be the field $_$ of $_$-adic numbers or a finite algebraic extension of it, let $_$ be the ring of integers of $_$, let $_$ be an elliptic curve over $_$, and suppose that $_$ is non-empty. The group structure turns $_$ into a commutative compact one-dimensional $_$-adic Lie group (cf. Lie group, $_$-adic). The group $_$ is Pontryagin-dual to the Weil–Châtelet group $_$. If $_$, then $_$ is a Tate curve (see [1], [5]) and there exists a canonical uniformization of $_$ analogous to the case of $_$.

Let $_$ be an elliptic curve over $_$ for which $_$ is not empty. Then $_$ is biregularly isomorphic to the curve (1) with $_$. Of all curves of the form (1) that are isomorphic to $_$ with integers $_$ and $_$, one chooses the one for which the absolute value of the discriminant $_$ is minimal. The conductor $_$ and the $_$-function $_$ of $_$ are defined as formal products of local factors: $$_$$

over all prime

numbers $_$ (see [1], [5], [13]). Here $_$ is some power of $_$, and $_$ is a meromorphic function of the complex variable $_$ that has neither a zero nor a pole at $_$. To determine the local factors one considers the reduction of $_$ modulo $_$ ($_$), which is a plane projective curve $_$ over the residue class field $_$ and is given in an affine coordinate system by the equation $$_$$

Let $_$ be the number of

$_$-points on $_$. If $_$ does not divide $_$, then $_$ is an elliptic curve over $_$, and one puts $$_$$

If $_$ divides $_$, then the

polynomial $_$ has a multiple root, and one puts $$_$$

(depending on

whether it is a triple or a double root). The product (2) converges in the right half-plane $_$. It has been conjectured that $_$ has a meromorphic extension to the whole complex plane and that the function

$$_$$

(where $_$ is the

gamma-function) satisfies the functional equation $_$ with $_$ (see [5], [3]). This conjecture has been proved for elliptic curves with complex multiplication.

The group $_$ is isomorphic to $_$, where $_$ is a finite Abelian group and $_$ is a free Abelian group of a certain finite rank $_$. $_$ is isomorphic to one of the following 15 groups (see [11]): $_$, $_$ or $_$, and $_$, $_$. The number $_$ is called the rank of the elliptic curve over $_$, or its $_$-rank. Examples are known of elliptic curves over $_$ of rank $_$. There is a conjecture (see [1], [13]) that over $_$ there exist elliptic curves of arbitrary large rank.

In the study of $_$ one uses the Tate height $_$, which is a non-negative definite quadratic form on $_$ (see [1], [3], [8], and also Height, in Diophantine geometry). For any $_$ the set $_$ is finite. In particular, $_$ vanishes precisely on the torsion subgroup of $_$.

An important invariant of an elliptic curve is its Tate–Shafarevich group $_$ (see Weil–Châtelet group). The non-trivial elements of $_$, an elliptic curve without $_$-points, provide examples of elliptic curves for which the Hasse principle fails to hold. The group $_$ is periodic and for every $_$ the subgroup of its elements of order dividing $_$ is finite. For a large number of elliptic curves it has been verified that the 2- and $_$-components of $_$ are finite (see [1], [4], [5]). There is a conjecture that $_$ is finite.

A conjecture of Birch and Swinnerton-Dyer asserts (see [5], [13]) that the order of the zero of the $_$-function $_$ at $_$ is equal to the $_$-rank of $_$. In particular, $_$ has a zero at $_$ if and only if $_$ is infinite. So far (1984) the conjecture has not been proved for a single elliptic curve, but for elliptic curves with complex multiplication (and $_$) it has been established that when $_$ is infinite, then the $_$-function has a zero at $_$ (see [14]). The conjecture of Birch and Swinnerton-Dyer gives the principal term of the asymptotic expansion of the $_$-function as $_$; in it there occur the orders of the groups $_$ and $_$ and the determinant of the Tate height [1]. It can be restated in terms of the Tamagawa numbers (cf. Tamagawa number, see [7]).

There is a conjecture of Weil that an elliptic curve $_$ has a uniformization by modular functions relative to the congruence subgroup $_$ of the modular group $_$ (see [5] and also Zeta-function in algebraic geometry). This conjecture has been proved for elliptic functions with complex multiplication. It is known (see [15]) that every algebraic curve over $_$ can be uniformized (cf. Uniformization) by modular functions relative to some subgroup of $_$ of finite index.

References

[1]	J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291
[2]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968)
[3]	D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)
[4]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[5]	Yu.I. Manin, "Cyclotomic fields and modular curves" Russian Math. Surveys , 26 : 6 (1971) pp. 6–78 Uspekhi Mat. Nauk , 26 : 6 (1971) pp. 7–71
[6]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91
[7]	S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer conjecture" Invent. Math. , 58 (1980) pp. 65–76
[8]	S. Lang, "Elliptic curves; Diophantine analysis" , Springer (1978)
[9]	S. Lang, "Elliptic functions" , Addison-Wesley (1973)
[10]	B. Mazur, "Rational isogenies of prime degree" Invent. Math. , 44 (1978) pp. 129–162
[11]	J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , Modular functions of one variable. 4 , Lect. notes in math. , 476 , Springer (1975)
[12]	J.F. Mestre, "Construction d'une courbe elliptique de rang $_$" C.R. Acad. Sci. Paris Sér. 1 , 295 (1982) pp. 643–644
[13]	J. Tate, "The arithmetic of elliptic curves" Invent. Math. , 23 (1974) pp. 197–206
[14]	J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" Invent. Math. , 39 (1977) pp. 223–251
[15]	G.V. Belyi, "On Galois extensions of a maximal cyclotomic field" Math. USSR Izv. , 14 : 2 (1980) pp. 247–256 Izv. Akad. Nauk SSSR Ser. Mat. , 43 (1979) pp. 267–276

References

[a1]	B. Mazur, "Modular curves and the Eisenstein ideal" Publ. Math. IHES , 47 (1978) pp. 33–186
[a2]	J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986)