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 be an elliptic curve over an algebraically closed field . Then is biregularly isomorphic to a plane cubic curve (see [1], [9], [13]). If , then in the projective plane there is an affine coordinate system in which the equation of is in normal Weierstrass form:

(1)

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:

(2)

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)