Torsion of elliptic curves over number fields ( 수체 위에서 타원곡선의 위수구조 )

1. Torsion of elliptic curves over number fields (수체 위에서 타원곡선의 위수구조) 발표: 김창헌(한양대학교) 전대열 (공주대학교), Andreas Schweizer (KAIST) 박사와의 공동연구임 김창헌

2. The main object of arithmetic geometry: finding all the solutions of Diophantine equations Examples: Find all rational numbers X and Y such that Diophantine equation 김창헌

3. Pythagorean Theorem Pythagoras lived approx 569-475 B.C. 김창헌

4. Pythagorean Triples Triples of whole numbers a, b, c such that 김창헌

5. Enumerating Pythagorean Triples Line of Slope t Circle of radius 1 김창헌

6. Enumerating Pythagorean Triples If then is a Pythagorean triple. 김창헌

7. Quadratic equations with rational coefficients • Why does the secant method works? • We have a solution • Any straight line cuts the circle in 0,1 or 2 points • Fact: If we have a quadratic equation with rational coefficients and we know one solution, then there are infinite number of solutions and they can be parametrized in terms of one parameter. 김창헌

8. what happens with the cubic equations? • Claude Gasper Bachet de Méziriac (1581-1638) : • Let c be a rational number. Suppose that (x,y) is a rational solution of Y2 = X3+c. Then is also a rational solution. Bachet 김창헌

9. Cubic Equations & Elliptic Curves A great bookon elliptic curves by Joe Silverman Cubic algebraic equations in two unknowns xand y. 김창헌

10. The Secant Process 김창헌

11. The Tangent Process 김창헌

12. Elliptic curves • Consider a non-singular elliptic curve Y2 = X3+aX2+bX+c • Suppose we know a rational solution (x,y). • Compute the tangent line of the curve at this point. • Compute the intersection with the curve. • The point you obtain is also a rational solution. 김창헌

13. Rational points on elliptic curves • Formula: If (x,y) is a rational solution, then (x,y) is another rational solution, where • x = • it seems that we have found a procedure to compute infinitely many solutions if we know one. But this is not true! x4-2bx2-8cx+b2-4ac 4y2 김창헌

14. Torsion points x4-2bx2-8cx+b2-4ac 4y2 • x = • Problem: • If y = 0, x is not defined (or better, it is equal to infinite). • If x = x, and y = y, we get no new point. • What else could happen? 김창헌

15. Torsion points • Beppo Levi (1875-1961) conjectured in 1908 that there is only a finite number of possibilities, and gave the exact list. Beppo Levi 김창헌

16. Torsion points • B. Mazur proved this conjecture in 1977 in a cellebrated paper. • Theorem (Mazur) Let (x,y) be a rational point in an elliptic curve. Compute x, x, x and x. If you can do it, and all of them are different, then the formula before gives you infinitely many different points. Barry Mazur 김창헌

17. (x,y) = (1,0) x= x, y= -y 김창헌

18. Torsion points • In modern language Mazur’s Theorem says: If (x,y) is a rational torsion point of order N in an elliptic curve over Q, then N <= 12 and N is not equal to 11. 김창헌

19. Mordell’s Theorem The rational solutions of a cubic equation are all obtainable from a finite number of solutions, using a combination of the secant and tangent processes. 1888-1972 김창헌

20. Mordell-Weilgroup (Mordell-Weil group)

21. Mordell-Weil Theorem K: number field, The Mordell-Weil group E(K) is finitely generated. Mordell(1888-1972) E(K)tors: torsion subgroup of EoverK. Weil(1906-1998)

22. Mazur’s Theorem There are 15 group structures of Etors(Q) of elliptic curves y2 = x3 + ax + b for any two rational a and b.

23. Mazur’s Theorem The curve X1(N) is of genus 0 iff N = 1–10,12.

24. Modular curves • The curve X1(N) is a parametrization of the elliptic curves with a torsion point of order N.

25. Modular curves • Tate normal form • E(b,c) satisfies the following: • - P = (0,0): K-rational point, • - ord(P) ≠ 2,3.

26. Modular curves • Modular curveX1(N) FN(b,c) = 0: the formula arising from the conditionNP = 0. (b,c) satisfiesFN(b,c) = 0if and only if E(b,c) is an elliptic curve with a torsion point P = (0,0) of order N. X1(N): FN(b,c) = 0.

27. Modular curves

28. Modular curves

29. : the equation of a projective line, i.e., X1(5) is of genus 0. • Modular curveX1(5) Modular curves

30. Modular curves • Modular curveX1(11) X1(11) : y2 + y = x3 –x2 is an elliptic curve, i.e., X1(11) is of genus 1.

31. Genus table of modular curves

32. Mazur’s Theorem The curve X1(N) is of genus 0 iff N = 1-10, 12.

33. Infinitely many rational points • X1(N) contains infinitely many rational points if N = 1–10, 12. • There exist infinitely manyelliptic curvesdefined over Q with rational torsion points of order N for N = 1–10, 12.

34. ⇒ E(b,b) is an elliptic curvedefined over Q with a rational torsion point of order 5. Infinitely many rational points • When does a modular curve has infinitely many K-rational points with a number fieldK?

35. Infinitely many rational points • (Mordell-Faltings) Any smooth projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points. • When does a modular curve has infinitely many K-rational points with number fieldsK of a fixed order?

36. K : quadratic number fields Kamienny, Mazur • X1(N): of genus 0(rational) iff N = 1–10, 12. • X1(N): of genus 1(elliptic) iff N = 11, 14, 15. • X1(N): hyperelliptic iff N = 13, 16, 18.

37. Each of these groups occurs infinitely often as . Kamienny, Mazur There exist infinitely many K-rational points of X1(N) defined over quadratic number fields K for N=1-16,18.

38. Infinitely many rational points • If there exist a map f : X → P1 of degree d, then X is called d-gonal. • If X is 2-gonal and g(X) > 1, then X is called hyperelliptic.

39. Infinitely many rational points • (Mestre) X1(N) is hyperelliptic for N =13, 16, 18.

40. (Jeon-Kim-Schweizer) X1(N) is 3-gonal iff • N = 1–16, 18, 20 iff • is infinite. Infinitely many rational points

41. Jeon, Kim, Schweizer K : cubic number fields The group structure that occurs infinitely often as :

42. (Jeon-Kim-Park) X1(N) is 4-gonal iff • N = 1–18, 20, 21, 22, 24 iff • is infinite. Infinitely many rational points

43. Jeon, Kim, Park K : quartic number fields The group structure that occurs infinitely often as :

44. Further Studies • Theorem (1996, L. Merel) For any integer d >=1, there is a constant Bd such that for any field K of degree d over Q and any elliptic curve over K with a torsion point of order N, one has that N <= Bd . 김창헌

45. Torsion subgroups 김창헌

46. Torsion subgroups 김창헌

47. Jeon, Kim, Schweizer K : cubic number fields The group structure that occurs infinitely often as :

48. Jeon, Kim, Park K : quartic number fields The group structure that occurs infinitely often as :

49. Further Studies • If d=1, then Bd=12. • If d=2, then Bd=18. • If d=3, then Bd=20? • If d=4, then Bd=24? 김창헌

50. 감사합니다.