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Elliptic Curves and the Weierstrass -function David Watson December 1, 2004 Abstract and Summary of the Presentation At this presentation I will define elliptic curves and examine how the Weierstrass -function illuminates the fundamental relationship between elliptic curves in 2 (the projective complex plane) and the field of elliptic functions in the complex plane. In addition, I will discuss the group structure on elliptic curves. 1. Introduction and Definition of Elliptic Curves 2. Lattices in and the Field of Elliptic Functions EL 3. The Weierstrass -function a. Definition b. Generator of EL c. Power series and differential equation d. Interpretation 4. The Weierstrass Cubic a. Definition; Weierstrass normal form b. The J-invariant for Lattices and Curves c. Interpretation 5. The Group Structure of Elliptic Curves a. Addition (geometric definition) b. Notes 6. Arithmetic on Elliptic Curves a. Mordell-Weil Theorem (statement only) b. Siegel Theorem (statement only) (*) indicates that the material may be presented, time permitting. Elliptic Curves and the Weierstrass -function David Watson December 1, 2004 1. Elliptic Curves An elliptic curve is the locus of points (X, Y) that satisfy an equation of the form f(X, Y) = Y2 - g(X) = 0 where g(X) K[X] is a cubic polynomial with coefficients in a field K (and char K ≠ 2), with distinct roots (perhaps in some extension of K). It is desirable to use projective coordinates, so the relation expressed by the above equation is homogenized: F(X0, X1, X2) = X0n f(X1/X0, X2/X0) = 0 where n is the degree of f(X, Y) (i.e., the maximum total degree of its monomials). For elliptic curves, the degree is 3. The solution set of the homogenized equation is the projective completion of f(X, Y) = 0. 2. Lattices in and the Field of Elliptic Functions EL A lattice L in the complex plane is the set of all integral linear combinations of two complex numbers 1 and 2, where 1 and 2 are linearly independent. That is, a lattice L is the set of all points l such that l = n1 + m2 for integers m, n. We can normalize this lattice by letting 2 / 1 in Im z > 0, so that we consider the lattice to be determined by 1 and τ. The fundamental parallelogram /L is defined as /L = {a1 + b2 0 a 1, 0 b 1} Because the fundamental parallelogram is topologically equivalent to a torus, it is sometimes referred to as such. For a given lattice L, a meromorphic function on is an elliptic function iff it is doubly periodic. (Recall that a meromorphic function is a quotient of analytic functions.) That is, if f is doubly periodic, then f(z + l) = f(z) for all l L. A doubly periodic function without any poles on the fundamental parallelogram is constant (by Liouville’s theorem). For a given lattice L, EL denotes the field of elliptic functions with respect to that lattice. EL is a subfield of M(), the field of meromorphic functions on , and is closed under differentiation. 3. The Weierstrass -function Remarkably, a given normalized lattice may be identified with a specific elliptic curve. In order to do this, it is necessary to define a special complex function, the Weierstrass -function. We define the Weierstrass function (z) with respect to a normalized lattice L in the complex plane as follows: 1 1 1 ( z ) 2 2, z lL,l 0 ( z l ) l 2 for l L. It can be shown that: 1. (z) converges uniformly and absolutely on compact subsets of /L; and 2. (z) EL, and its only poles are double poles on its lattice points. Pf: The convergence of (z) follows by comparison with the series 1/l3. Termwise differentiation shows that the derivative of (z) is clearly doubly periodic, and so '(z) EL. Then since '(z + i) - '(z) = 0, it follows that (z + i) - (z) = C. Then let z = -i/2 and note that (z) is even, so C = 0. Since (z) – (z – l)-2 is continuous at z = l, (z) is meromorphic with double poles only on l L. As (z) EL, this suffices to show the existence of elliptic functions for any lattice L. The Weierstrass function and its derivative generate EL in the following sense. Every f EL may be written as the sum of an even and an odd elliptic function. That is, f ( z ) f ( z ) f ( z ) f ( z ) f ( z) 2 2 If f is odd, f(z)'(z) is even, so it suffices to show that the field of even elliptic functions is generated by (z). If {ai} represent the distinct zeroes (poles) of f(z), and {mi} represent the order of those zeroes or poles (positive numbers represent the multiplicity of zeroes and negative numbers represent poles), then n f ( z ) C [( z ) (ai )] mi i 1 To see this, note that the quotient of f(z) by the product on the right hand side is elliptic without poles or zeroes on /L, and thus constant by Liouville. In order to derive some properties of the Weierstrass function, we consider the power series expansion of (z). Using standard techniques, one can show that: 1 ( z ) 2 (2n 1) s2 n 2 ( L) z 2 n , z n 1 where 1 sm ( L) , l 0 lm for l L. This expression may be re-written 1 ( z ) 3s4 z 2 5s6 z 4 ..., z2 so that 2 ( z ) 3 6s4 z 2 20s6 z 4 ... ' z It is then (relatively) easy to verify that (z) satisfies the differential equation (X')2 = 4X3 – g2X – g3, where g2 = 60s4 and g3 = 140s6. To see this, let Ψ(z) = '(z) – 4[(z)]3 + g2(z) + g3 and expand the power series expression for Ψ(z). Then the polar and constant terms cancel, so that Ψ(z) is elliptic without poles (thus constant by Liouville) but zero at the origin and so identically zero everywhere. If we let ei ( i / 2), for 1 and 2 on L, and 3 = 1 + 2, we see that, comparing zeroes and poles, [' ( z )] 2 4(( z ) e1 )(( z ) e2 )(( z ) e3 ) so that the cubic polynomial has three distinct roots and thus a nonzero discriminant. We can now define an analytic map from the fundamental parallelogram /L into 2, the complex projective plane. Define a map from /L into 2 so that z (1: (z) : '(z)) for z ≠ 0 and away from lattice points, z (1/'(z) : 1 : (z)/ '(z)) for z ≠ 0 and near lattice points, and 0 (1 : 0 : 0). Then the image of any nonzero point z of /L is a point on the elliptic curve defined by X 2 X 0 4 X 13 g 2 X 1 X 0 g 3 X 0 2 2 3 in 2. The map is one to one, because every x-value, except the roots of g(X) and , is taken twice on /L, and the y-values for a given z on /L are the (positive and negative) square roots of g((z)). If x is a root of g(x), there is only one z-value such that (z) = x and the corresponding y-coordinate is 0. This provides the desired identification of a normalized lattice L with a particular elliptic curve in 2. We will now establish the converse. 4. The Weierstrass Cubic An equation of the form Y2 = g(X), where g(x) is a general cubic, may be put into the reduced form Y2 = 4X3 - pX – q by an appropriate translation of the X-coordinate. We will make the (naïve) association of p and q with g2 and g3. We will show that it is sensible to identify g2 = 60s4 and g3 = 140s6, as discussed above. We then describe the cubic as being rewritten in Weierstrass normal form: Y2 = 4X3 - g2X – g3, where g2 = 60s4 and g3 = 140s6 as before. We assume also that the elliptic curve is smooth, and therefore has a nonzero discriminant. The discriminant for a curve in Weierstrass normal form is: g 2 27 g 3 . 3 2 Note that if we are able to establish the identification of the coefficients in the general equation with the values of g2 and g3 determined by a particular lattice, we will be able to identify elliptic curves with normalized lattices. Now, it is possible to define a map from a normalized lattice L defined by (1,τ) into , as follows: 3 g2 J ( ) 3 g 2 27 g32 where g2 = 60s4 and g3 = 140s6, and s4 and s6 are defined with respect to the lattice, as above. It can be shown that this function J(τ) is bijective from the set of normalized lattices to . Now, we can then look for an invariant property of a particular elliptic curve. The J-invariant of a smooth Weierstrass cubic is 3 3 g2 g2 J(g2, g3): 3 g 2 27 g 32 . It can be shown that two smooth Weierstrass cubic curves are equivalent under maps of one curve into the other that preserve (0 : 1 : 0), if only if their J-invariants are equal. Each such map has the form X c2X, Y c3Y, where c . We can associate the set of elliptic curves with the same J-invariant to a particular normalized lattice, completing the correspondence. 5. The Group Structure of Elliptic Curves Bezout’s theorem specifies that the number of intersections in 2 of two distinct irreducible curves of degrees m and n, respectively, is the product mn. Accordingly, an elliptic curve (being of degree 3) intersects a line (being of degree 1) in three points (counting multiplicities) in the complex projective plane. This leads to a natural way of defining a “group operation” on the elliptic curve, since any two points of intersection between the curve and the line determine a third. We can define an “additive law” on an elliptic curve E as follows: Define O as (1:0:0). Let P, Q E, and let L be the line connecting P and Q. (L is a tangent to E at P if P = Q.) Let R be the third point of intersection of L with E. Let L' be the line connecting R and O. Then P + Q is the point such that L' intersects E at R, O and P + Q. These points on an elliptic curve form an abelian group, for: 1) If L intersects E at the (not necessarily distinct) points P, Q, and R, then (P + Q) + R = 0 2) P + 0 = P, for all P in E. 3) P + Q = Q + P, for all P, Q in E. 4) For P in E, there is a (-P) such that P + (-P) = 0. 5) (P + Q) + R = P + (Q + R) The verification of these properties is not difficult, although verification of the associativity property requires laborious manipulations of explicit expressions of the additive law. Accordingly, we shall omit the verifications. Note that, with this group structure, an elliptic curve is both a compact Riemann surface and a complex Lie group. It is also true that any smooth curve with a group structure is an elliptic curve. 6. Arithmetic on Elliptic Curves There are many interesting results on the arithmetic on elliptic curves, two of the most important of which are the Mordell-Weil Theorem and the Siegel Theorem. (Perhaps this should not be too surprising, as we might expect nice relations among the points on the lattice!) A rational point on an elliptic curve E is a point with coordinates in . The set of all such points forms a group, typically denoted E(). In 1921, Louis Mordell proved that the group of all rational points on an elliptic curve (over ) is finitely generated. This is a special case of what is known as the Mordell-Weil Theorem, which states that for any number field K and an elliptic curve E, the Mordell-Weil group E(K) is finitely generated. E(K) may be expressed E ( K ) Etors ( K ) r While the torsion subgroup can be computed, the rank r is difficult to find in general. It is still an open question as to whether there exist elliptic curves of arbitrarily large rank. Another natural question is that of determining whether a finite number of rational points have integral coordinates. This was answered in the affirmative by Siegel in the 20s, using techniques of Diophantine approximation. Serge Lang has conjectured a relation between the number of integral points and the rank of the Mordell-Weil group. For further references, see J. Silverman, The Arithmetic of Elliptic Curves. Elliptic curves have various other applications, including cryptography, algebraic coding and superstring theory.

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