1. 2015-10-12 김창헌 1 Torsion of elliptic curves over number fields ( 수체 위에서 타원곡선의 위수 구조 ) 김창헌 발표 : 김창헌 ( 한양대학교 ) 전대열 ( 공주대학교 ), Andreas Schweizer (KAIST) 박사와의 공동연구임

2. 2015-10-12 김창헌 2 Diophantine equation The main object of arithmetic geometry: finding all the solutions of Diophantine equations Examples: Find all rational numbers X and Y such that

3. 2015-10-12 김창헌 3 Pythagorean Theorem Pythagoras lived approx 569-475 B.C.

4. 2015-10-12 김창헌 4 Pythagorean Triples Triples of whole numbers a, b, c such that

5. 2015-10-12 김창헌 5 Enumerating Pythagorean Triples Circle of radius 1 Line of Slope t

6. 2015-10-12 김창헌 6 If then is a Pythagorean triple. Enumerating Pythagorean Triples

7. 2015-10-12 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.

8. 2015-10-12 김창헌 8 what happens with the cubic equations? Claude Gasper Bachet de M é ziriac (1581-1638) : Let c be a rational number. Suppose that (x,y) is a rational solution of Y 2 = X 3 +c. Then is also a rational solution. Bachet

9. 2015-10-12 김창헌 9 Cubic Equations & Elliptic Curves Cubic algebraic equations in two unknowns x and y. A great book on elliptic curves by Joe Silverman

10. 2015-10-12 김창헌 10 The Secant Process

11. 2015-10-12 김창헌 11 The Tangent Process

12. 2015-10-12 김창헌 12 Elliptic curves Consider a non-singular elliptic curve Y 2 = X 3 +aX 2 +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.

13. 2015-10-12 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! x 4  2bx 2  8cx+b 2  4ac 4y 2

14. 2015-10-12 김창헌 14 Torsion points 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? x 4  2bx 2  8cx+b 2  4ac 4y 2

15. 2015-10-12 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

16. 2015-10-12 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

17. 2015-10-12 김창헌 17 (x,y) = (1,0) x  x  y  y

18. 2015-10-12 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.

19. 2015-10-12 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

20. Mordell-Weil group (Mordell-Weil group)

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

22. Mazur’s Theorem There are 15 group structures of E tors ( Q ) of elliptic curves y 2 = x 3 + ax + b for any two rational a and b.

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

24. Modular curves The curve X 1 (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. (b,c) satisfies F N (b,c) = 0 if and only if E(b,c) is an elliptic curve with a torsion point P = (0,0) of order N. Modular curves Modular curve X 1 (N) F N (b,c) = 0: the formula arising from the condition NP = 0. X 1 (N): F N (b,c) = 0.

27. Modular curves

29. Modular curve X 1 (5) : the equation of a projective line, i.e., X 1 (5) is of genus 0.

30. X 1 (11) : y 2 + y = x 3 – x 2 is an elliptic curve, i.e., X 1 (11) is of genus 1. Modular curves Modular curve X 1 (11)

31. Genus table of modular curves Ng1(N)g1(N)Ng1(N)g1(N) 10111 20120 30132 40141 50151 60162 70175 80182 90197 100203

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

33. Infinitely many rational points X 1 (N) contains infinitely many rational points if N = 1–10, 12. There exist infinitely many elliptic curves defined over Q with rational torsion points of order N for N = 1–10, 12.

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

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 fields K of a fixed order?

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

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

38. Infinitely many rational points If there exist a map f : X → P 1 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) X 1 (N) is hyperelliptic for N = 13, 16, 18.

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

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

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

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

44. 2015-10-12 김창헌 44 Further Studies Theorem (1996, L. Merel) For any integer d  1, there is a constant B d 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 <= B d.

45. 2015-10-12 김창헌 45 Torsion subgroups

46. 2015-10-12 김창헌 46 Torsion subgroups

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. 2015-10-12 김창헌 49 Further Studies If d=1, then B d =12. If d=2, then B d =18. If d=3, then B d =20? If d=4, then B d =24?

50. 감사합니다.