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Multiplicative Group Law on the Folium of Descartes

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Multiplicative Group Law on the Folium of Descartes Multiplicative group law on the folium of Descartes Stelut¸a Pricopie and Constantin Udri¸ste Abstract. The folium of Descartes is still studied and understood today. Not only did it provide for the proof of some properties connected to Fermat’s Last Theorem, or as Hessian curve associated to an elliptic curve, but it also has a very interesting property over it: a multiplicative group law. While for the elliptic curves, the addition operation is compatible with their geometry (additive group), for the folium of Descartes, the multiplication operation is compatible with its geometry (multiplicative group). The original results in this paper include: points at infinity described by the Legendre symbol, a group law on folium of Descartes, the fundamental isomorphism and adequate algorithms, algorithms for algebraic computa- tion. M.S.C. 2010: 11G20, 51E15. Key words: group of points on Descartes folium, algebraic groups, Legendre symbol, algebraic computation. 1 Introduction An elliptic curve is in fact an abelian variety G that is, it has an addition defined algebraically, with respect to which it is a group G (necessarily commutative) and 0 serves as the identity element. Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles (assisted by Richard Taylor), of Fermat’s Last Theorem. They also find applications in cryptography and integer factorization (see, [1]-[5]). Our aim is to show that the folium of Descartes also has similar properties and perhaps more. Section 2 refers to the folium of Descartes and its rational parametriza- tion. Section 3 underlines to connection between the points at infinity for this curve and the Legendre symbol. Section 4 gives the geometrical meaning of the multi- plicative group law on folium of Descartes, excepting the critical point. The law was proposed by the second author and to our knowledge is the first time it appears in Balkan∗ Journal of Geometry and Its Applications, Vol. 18, No. 1, 2013, pp. 54-70. c Balkan Society of Geometers, Geometry Balkan Press 2013. Multiplicative group law 55 the mathematical literature. Section 5 spotlights an isomorphism between the group on the folium of Descartes and a multiplicative group of integers congruence classes. Section 6 formulates to conclusions and open problems. 2 Folium of Descartes The folium of Descartes is an algebraic curve defined by the equation x3 + y3 − 3axy = 0. Descartes was first to discuss the folium in 1638. He discovered it in an attempt to challenge Fermat’s extremum-finding techniques. Descartes challenged Fermat to find the tangent straightline at arbitrary points and Fermat solved the problem easily, something Descartes was unable to do. In a way, the folium played a role in the early days of the development of calculus. If we write y = tx we get x3 + t3x3 − 3atx2 = 0. Solving this for x and y in terms of t (t3 6= −1) yields the parametric equations 3at 3at2 x(t) = , y(t) = . 1 + t3 1 + t3 The folium is symmetrical about the straightline y = x, forms a loop in the first quadrant with a double point at the origin and has an asymptote given by x+y+a = 0. The foregoing parametrization of folium is not defined at t = −1. The left wing is formed when t runs from −1 to 0, the loop as t runs from 0 to +∞ , and the right wing as t runs from −∞ to −1. It makes sense now why this curve is called ”folium”- from the latin word folium which means ”leaf”. It’s worth mentioning that, although Descartes found the correct shape of the curve in the positive quadrant, he believed that this leaf shape was repeated in each quadrant like the four petals of a flower. 3 Points at infinity According to Eves [1], infinity has been introduced into geometry by Kepler, but it was Desargues, one of the founders of projective geometry, who actually used this idea. Addition of the points and the straightline at infinity metamorphoses the Euclidean plane into the projective plane. Projective geometry allows us to make sense of the fact that parallel straightlines meet at infinity, and therefore to interpret the points at infinity. Even if in projective geometry no two straightlines are parallel, the essential fea- tures of points and straightlines are inherited from Euclidean geometry: any two distinct points determine a unique straightline and any two distinct straightlines de- termine a unique point. Let K be a field. The two-dimensional affine plane over K is denoted 2 AK = {(x, y) ∈ K × K}. 56 Stelut¸a Pricopie and Constantin Udri¸ste 2 The two-dimensional projective plane PK over K is given by equivalence classes of triples (x, y, z) with x, y, z ∈ K and at least one of x, y, z nonzero. Two triples (x1, y1, z1) and (x2, y2, z2) are said to be equivalent if there exists a nonzero element λ ∈ K∗ such that (x1, y1, z1) = (λx2, λy2, λz2). We denote by (x : y : z) the equivalence class whose representative is the triple (x, y, z). 2 2 The points in PK with z 6= 0 are the ”finite” points in PK (because they can be identified with points in the affine plane). If z = 0 then, considering that dividing by 2 zero gives ∞, we call the points (x : y : 0) ”points at infinity” in PK . Now let’s return to our cubic given by x3 + y3 − 3axy = 0. Its homogeneous form is x3 + y3 − 3axyz = 0. The points (x, y) on the affine space correspond to the points (x : y : 1) in the projective space. To see what points lie at infinity, we set z = 0 and obtain x3+y3 = 0, which is equivalent with (x + y)(x2 − xy + y2) = 0. We now have two possibilities: 1. x + y = 0. In this case we find the point at infinity ∞1 = (−1 : 1 : 0). 2. x2 − xy + y2 = 0. We consider this as an equation in the unknown x, with the parameter y, and we compute the discriminant ∆ = −3y2. If there is an element α ∈ K such that α2 = −3, then, rescaling by y, we have the following two points at infinity ! ! 1 + α 1 − α ∞ = : 1 : 0 , ∞ = : 1 : 0 . 2 2 3 2 Remark If we have 3 points at infinity, the sum of their x coordinates is zero. For example, when K = R, since (−1 : 1 : 0) = (1 : −1 : 0), we can imagine that the point at infinity ∞1 lies on the “bottom” and on the “top” of every straightline that has slope −1. 3.1 Legendre symbol Of course, we want to know when we have just one point at infinity and when we have 3 points at infinity. To decide we need the Legendre symbol which helps us decide whether an integer a is a perfect square modulo a prime number p. ! a For an odd prime p and an integer a such that p 6 |a, the Legendre symbol p is defined by ! a 1, if a is a quadratic residue of p = p −1, otherwise. The Legendre symbol has the following properties: 1. ( Euler’s Criterion) Let p be an odd prime and a any integer with p 6 |a. Then ! a p−1 ≡ a 2 mod (p). p Multiplicative group law 57 2. Let p be an odd prime, and a and b be any integers with p 6 |ab. Then ! ! a b (a) If a ≡ b mod (p), then ≡ . p p ! ! ! a b ab (b) = . p p p ! a2 (c) = 1. p 3. If p is an odd prime, then ! −1 1, if a ≡ 1 mod (4) = p −1, if a ≡ −1 mod (4). 4. If p is an odd prime, then ! 2 1, if a ≡ ±1 mod (8) = p −1, if a ≡ ±3 mod (8). 5. (Law of Quadratic Reciprocity) Let p and q be distinct odd primes. Then ! ! p q p−1 q−1 = (−1) 2 2 . q p Now we can give the following result. Proposition 3.1. Let K be a finite field of prime cardinal p 6= 2, 3 and let α2 = −3 (so α might lie in an extension of K). 1. If p = 3r + 1, then α ∈ K. 2. If p = 3r + 2 , then α∈ / K. Proof. First let us make the following two remarks: when p = 3r + 1, the element r cannot be an odd integer and when p = 3r + 2, the element r cannot be an even integer. −3 Now, the proof actually consists in computing the Legendre symbol ( p ), in the sense that if this Legendre symbol is 1, then α ∈ K and if it equals −1, then α∈ / K. 1. Consider p = 3r + 1, where the element r is an even integer. We first compute the Legendre symbol ! ! ! ! −3 −1 3 p−1 p p−1 · 3−1 3r = · = (−1) 2 · · (−1) 2 2 = (−1) = 1. p p p 3 Consequently, there is an element α ∈ K such that α2 = −3 mod (p). 58 Stelut¸a Pricopie and Constantin Udri¸ste 2. Consider p = 3r + 2, where r is an odd integer. We have ! ! ! ! −3 −1 3 p−1 p p−1 · 3−1 3r+2 = · = (−1) 2 · · (−1) 2 2 = (−1) = −1. p p p 3 So in this case α∈ / K. 4 The group law on folium of Descartes We shall define and study a multiplicative group on the non-singular points of folium of Descartes in an analogous way as the additive group on an elliptic curve. The group law is called “multiplication” and it is denoted by the symbol ? (we underline that the operation over elliptic curves, motivated by geometry, is represented by the addition sign +).
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