Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence x n+2 = (a + x n+1)/x nwith a ε Q. First we prove that there exist initial conditions and values of a for which it generates periodic sequences of rational numbers with prime periods 1, 2, 3, 5, 6, 7, 8, 9, 10 or 12 and that these are the only periods that rational sequences {x n} n can have. It is known that if we restrict our attention to positive rational values of a and positive rational initial conditions the only possible periods are 1, 5 and 9. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of a, positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastion & Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lynoss map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order n, n ≥ 5, including n infinity. This fact implies that the Lyness map is a universal normal form for most birational maps on elliptic curves.