Elliptic Curves. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec.

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1 Elliptic Curves Akhil Mathew Department of Mathematics Drew University Math 155, Professor Alan Candiotti 10 Dec Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

2 What is an Elliptic Curve? An elliptic curve is the locus of solutions of an equation of the form y 2 = x 3 + Ax + B, where for nonsingularity 4A B 2 0. There is also a point at infinity (not shown). Figure: The elliptic curve y 2 = x 3 x Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

3 The Addition Law To add points M 1,M 2, draw the line D through them. Find the third intersection P of the line with the curve. Flip P over the x-axis to get M 3 = M 1 + M 2. Figure: The group law on an elliptic curve from [1] Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

4 Properties of the Group Law Theorem Addition is commutative and associative. In fact under the addition law, with the point at infinity as zero, an elliptic curve is an abelian group. Proof. Addition in an elliptic curve corresponds roughly to addition in the divisor class group (i.e. using algebraic geometry, cf. [4]). In the complex case, we will show another proof. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

5 Examples of the Addition Law Example The origin O is the point at infinity. Example If P is a point, then P is P flipped over the x-axis. P P Figure: An illustration of the preceding examples Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

6 Elliptic Curves over the Complex Numbers Let S 1 = R/Z be the unit circle. Theorem An elliptic curve E over the complex numbers is group-isomorphic to the torus S 1 S 1, cf. [3]. Figure: A torus from [2] Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

7 Lattices A lattice L C is a discrete free abelian subgroup of rank 2. Then C/L is a torus and a complex Riemann surface. Figure: A lattice One can construct a lattice L C such that E = C/L topologically, analytically, and group-theoretically. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

8 An Overview of the Proof Overview of Proof. The isomorphism is given by Weierstrass -functions: z ( (z;l), (z;l)). The Weierstrass -function is defined specifically as: (z;l) = 1 z 2 + ω L,ω 0 1 (z ω) 2 1 ω 2. functions are doubly periodic and meromorphic (i.e. elliptic). Hence they are defined as a map of C/L S 2 (S 2 being the Riemann sphere). The addition law for Weierstrass- functions is basically the theorem. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

9 Torsion Points over Elliptic Curves E is an elliptic curve over C, E = C/L = S 1 S 1 as groups. We want points P E of order m, i.e. such that mp = 0 and np 0 if 0 < n < m. The points in S 1 S 1 of order dividing m are of the form (z 1,z 2 ) for z 1,z 2 m-th roots of unity. Hence: Theorem There are m 2 points of order dividing m, and they form a group E[m] isomorphic to Z/mZ Z/mZ. In general, this is true over any algebraically closed field. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

10 Example: 2-Torsion Points If 2P = 0, then P = P, so P has y-coordinate zero or P is the point at infinity. This characterizes all 2-torsion points. There are three ways for y = 0 in the equation y 2 = P(x),P a cubic polynomial (Fundamental Theorem of Algebra), and we throw in the point at infinity to get: Theorem There are 4 points of order dividing 2. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

11 Points over Number Fields Let F be a number field, i.e. of finite degree over Q. We have an elliptic curve E : y 2 = x 3 + Ax + B, A,B F, and we want points (x,y) F 2. The set of such points forms a subgroup E(F) E. Theorem (Mordell-Weil) The group E(F) is finitely generated if F is a number field. Hence there exist x 1,...,x n E(F) which span E(F), i.e. generate the group. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

12 Rational Torsion Points Let E be an elliptic curve over Q, of the form y 2 = x 3 + Ax + B, A,B Z. By the Mordell-Weil theorem, the subgroup of rational torsion points on E is finite. Theorem (Nagell-Lutz; not too difficult) All rational torsion points (x,y) have integral coordinates. Also, if y 0, then y 2 4A B 2. In general, Theorem (Mazur; very difficult) The subgroup of rational torsion points has order 12. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

13 Integral Points Fix an elliptic curve: E : y 2 = x 3 + Ax + B, A,B Q. We want integral points (x,y) Z 2 E. Theorem (Siegel) Let E be an elliptic curve as above. Then E Z 2 is finite. Hence, if k Z, the number of integral solutions (x,y) to y 2 = x 3 + k, k 0 is finite. If k = 2, only the solutions (±5,3) exist, as shown by Euler. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

14 Extensions An algebraic number x Q al is called integral if x satisfies an equation x n + a 1 x n a n = 0, a j Z. Let K be a number field, and let E be an elliptic curve defined over K. Theorem (Siegel) If K is a number field, and B the set of integral elements, then E B 2 is finite. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

15 Roth s Theorem The proof of Siegel s theorem uses an approximation theorem of Roth: Theorem (Roth) If α is algebraic irrational and C,ɛ > 0, then there exist only finitely many (m,n) Z 2 with α m 1 n Cn 2+ɛ. There is an easier weaker result: Theorem (Liouville) Let α be irrational. Let d = deg α = [Q(α) : Q]. Then there exists η = η(α) > 0 such that for all m,n Z (say n > 0) α m η n n d. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

16 The Proof of Liouville s Theorem Proof. Let f be the irreducible polynomial of α over Q. If m n is close to α, we have roughly ( m ) ( f m ) f = f (α) m n n n α by the mean value theorem (f (α) 0). Hence n d m ( n α n d m ) f, n and the latter is a nonzero integer so 1. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

17 Why Diophantine Approximation is Important Consider the equation x 3 2y 3 = 1, x,y Z, or, with ρ a primitive cube root of unity, ( x y 3 )( x 2 y ρ 3 )( x 2 y ) ρ2 3 2 = 1 y 3. If y gets large, then one of the terms must get small, and the others are bounded below, so for some k,c > 0 x y ρk 3 2 c y 3, which proves that y cannot get arbitrarily large, and that the number of solutions to the initial equation is finite. This is the idea behind one proof of Siegel s theorem. Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

18 Conclusion Elliptic curves appear in diverse contexts: A commutative and associative addition law that makes an elliptic curve an abelian group Over C, isomorphic to tori Over number fields F, finite generation of F-rational points (Mordell-Weil) Over number fields, finite number of rational torsion points (Nagell-Lutz and Mazur) At most finitely many integral points (Siegel, using Diophantine approximation) Further extensions: Elliptic curves over finite fields: estimates on the number of points Elliptic curves over local (e.g. p-adic) fields Endomorphisms of elliptic curves and complex multiplication Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19

19 Sources Neal Koblitz. Introduction to Elliptic Curves and Modular Forms. Springer, Joseph Silverman. The Arithmetic of Elliptic Curves. Springer, Akhil Mathew (Department of Mathematics Drew UniversityElliptic MathCurves 155, Professor Alan Candiotti) 10 Dec / 19