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Lyness Cycles, Elliptic Curves, and Hikorski Triples

Lyness Cycles, Elliptic Curves, and Hikorski Triples. Jonny Griffiths, Maths Dept Paston Sixth Form College. Open University, June 2012. MSc by Research, UEA, 2009-12 (Two years part-time). Supervisors: Professor Tom Ward Professor Graham Everest Professor Shaun Stevens. Mathematics.

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Lyness Cycles, Elliptic Curves, and Hikorski Triples

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  1. Lyness Cycles, Elliptic Curves,and Hikorski Triples Jonny Griffiths, Maths Dept Paston Sixth Form College Open University, June 2012

  2. MSc by Research, UEA, 2009-12(Two years part-time) Supervisors: Professor Tom Ward Professor Graham Everest Professor Shaun Stevens

  3. Mathematics

  4. The Structure of this Talk1. Lyness cycles (periodic recurrence relations)2. An introduction to elliptic curves3. The link between Lyness cycles and elliptic curves4. Hikorskitriples5. Cross-ratio-type functions6. Conclusions

  5. Part 1: Lyness Cycles

  6. un+1 = un + un-1, u0 = 0, u1 = 1 The Fibonacci sequence z = x + y x, y, x + y, x + 2y, 2x + 3y, 3x + 5y, ... x y Can this be periodic? x = 2x + 3y, y = 3x + 5y x = 0, y = 0. Order-2, periodic for these starting values, (locally periodic).

  7. Can we have a recurrence relation that is periodic for (almost) all starting values? Globally periodic x1, x2,..., xn, f(x1,...,xn), f(x2,...,f(x1,...xn))..., x1, x2... xn+1, xn+2 ......... xm+1, xm+2 Order-n, period-m Globally periodic behaviour is very atypical of difference equations, and accordingly only a very restrictive class of functions f(x1, x2, ...) exhibit this behaviour. Mestel.

  8. Robert Cranston Lyness, 1909-1997

  9. Mathematical Gazette, 1942

  10. Globally periodic for x and y non-zero, order-2, period-5.

  11. Imagine you have series of numbers such that if you add 1 to any number, you get the product of its left and right neighbours. Then this series will repeat itself at every fifth step! The difference between a mathematician and a non-mathematician is not just being able to discover something like this, but to care about it and to be curious why it's true, what it means and what other things it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, and the Schrodinger equation of quantum mechanics. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful. Zagier

  12. Regular Lyness Cycles Order 1 Order 2 Order 3

  13. Order-1 regular Lyness cycles: what periods are possible?

  14. For what values of n does have solutions for a, b, c,d and k in Q? Answer: n = 1, 2, 3, 4, 6. Related question: what are the finite subgroups of GL2(Q)/N, Where N = { }?

  15. But that is it for rational coefficients...

  16. ..... so  is a root of K(x), AND a rational quadratic equation. Proof: Cull, Flahive, Robson • Since K(x) is irreducible, • (K) = 1 or 2 (where  is the totient function). • (1) = 1, (2) = 1, (3) = 2, (4) =2, (5) = 4,(6) =2, (n) > 2 for n > 6.

  17. Regular Lyness cycles order-2: what periods are possible? All coefficients rational. Possible periods: 2, 3, 4, 5, 6, 8, 12

  18. 2 3 4 5 6 Can add constants easily to these Pseudo-cycle

  19. Pseudo-cycle Pseudo-cycle

  20. But that is it for rational coefficients... Symmetric QRT maps (Quispel, Roberts and Thompson). Tsuda has given a theorem that restricts the periods for periodic symmetric QRT maps to 2, 3, 4, 5, 6.

  21. Note: x, y, ky - x,...can have any period if you are choosing k from R. k = 1/, period 5, k = √2, period 8, k = , period 10. x, y, |y| - x, ... is period 9.

  22. Part 2: An Introduction to Elliptic Curves

  23. ax + by + c = 0 Straight line ax2 + bxy + cy2 + dx + ey + f = 0 Conics Circle, ellipse, parabola, hyperbola, pair of straight lines

  24. ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0 Elliptic curves UNLESS The curve has singularities; a cusp or a loop or it factorises into straight lines... y2 = x4 + ax3 + bx2 + cx + d can be elliptic too...

  25. Any elliptic curve can be transformed into Weierstrass Normal Form Y2 = X3 + aX + b using a birational map; that is, you can get from the original curve to this normal form and back again using rational maps; The curves are said to be ISOMORPHIC.

  26. Y2 = X3 + aX + b a = -2.5 b = 1 a = 2.5 b = 1

  27. For example... Transforming to Normal Form

  28. This becomes...

  29. Where does a straight line cross our elliptic curve in normal form? We are solving simultaneously y = mx + c, y2 = x3 + ax + b which gives x3 - m2x2 + x(a - 2cm) + b - c2 = 0

  30. This is a cubic equation with at most three real roots. Note; if it has two real roots, it must have a third real root. So if we pick two points on the curve, the line joining them MUST cut the curve in a third point.

  31. P+Q+R=0 P+Q=-R

  32. We can form multiples of a point by taking the tangent at that point.

  33. Sometimes we find that kP = 0.In this case we say that P is a torsion point. y2=x3+1 6P=0 P is of order 6

  34. Amazing fact... The set of points on the curve together with this addition operation form a group. Closed – certainly. We want P and –P to be inverses. So P + -P = 0, and we define 0, the identity here, as the point at infinity.

  35. Associativity? Geogebrademonstration

  36. Notice also that if a, b are rational, then the set of rational points on the curve form a group. Closed – certainly. y = mx + c connects two rational points, so m and c must be rational. x3 - m2x2 + x(a - 2cm) + b - c2 = 0 If two roots are rational, the third must be. Inverses and identity as before

  37. Mordell Theorem (1922)Let E be an elliptic curve defined over Q. Then E(Q) is a finitely generated Abelian group.(Mordell-Weil Theorem [1928]generalises this.)

  38. Siegel’s Theorem (1929) If a, b and c are rational, (and if x3 + ax2 + bx + c = 0 has no repeated solutions), then there are finitely many integer points on y2 = x3 + ax2 + bx + c.

  39. Mordell’s Theorem implies that E(Q) is isomorphic to Etorsion(Q)  Zr The number r is called the RANK of the elliptic curve. How big can the rank be? Nobody knows.

  40. Largest rank so far found; 18 by Elkies (2006) y2 + xy = x3 − 2617596002705884096311701787701203903556438969515x + 5106938147613148648974217710037377208977 9103253890567848326775119094885041. Curves of rank at least 28 exist.

  41. Mazur’s Theorem (1977)The torsion subgroup of E(Q) is isomorphic to Z/nZfor some n in {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12}or to Z/2nZ  Z/2Zfor some n in {1, 2, 3, 4}.

  42. Part 3: The link between Lyness cycles and elliptic curves

  43. x = 3, k = -14 Other roots are -7, -2, -1/3.

  44. x = 3, k = -19/3 Other roots are -7, -2, -1/3.

  45. An elliptic curve!

  46. Note: simplifies to exactly the same set of curves. Additional note: This curve has 5-torsion

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