Rational Periodic Sequences for the Lyness Recurrence
aa r X i v : . [ m a t h . D S ] A p r Rational Periodic Sequences for the Lyness Recurrence
Armengol Gasull (1)
V´ıctor Ma˜nosa (2) and Xavier Xarles (1) (1)
Dept. de Matem`atiques, Facultat de Ci`encies,Universitat Aut`onoma de Barcelona,08193 Bellaterra, Barcelona, [email protected], [email protected] (2)
Dept. de Matem`atica Aplicada III (MA3),Control, Dynamics and Applications Group (CoDALab)Universitat Polit`ecnica de Catalunya (UPC)Colom 1, 08222 Terrassa, [email protected]
Abstract
Consider the celebrated Lyness recurrence x n +2 = ( a + x n +1 ) /x n with a ∈ Q . Firstwe prove that there exist initial conditions and values of a for which it generates periodicsequences of rational numbers with prime periods 1 , , , , , , , ,
10 or 12 and thatthese are the only periods that rational sequences { x n } n can have. It is known thatif we restrict our attention to positive rational values of a and positive rational initialconditions the only possible periods are 1 , a, positive 9-period rational sequences occur. This last result is our main contribution andanswers an open question left in previous works of Bastien & Rogalski and Zeeman. Wealso prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves witha point of order n, n ≥ , including n infinity. This fact implies that the Lyness map isa universal normal form for most birrational maps on elliptic curves. Keywords:
Lyness difference equations, rational points over elliptic curves, periodic points,universal family of elliptic curves.
The dynamics of the Lyness recurrence x n +2 = a + x n +1 x n , (1)1pecially when a > , has focused the attention of many researchers in the last years and itis now completely understood in its main features after the independent research of Bastien& Rogalski [3] and Zeeman [21], and the later work of Beukers & Cushman [5]. See also[2, 11]. In particular all possible periods of the recurrences generated by (1) are known andfor any a / ∈ { , } infinitely many different prime periods appear.However there are still some open problems concerning the dynamics of rational points.With the computer experiments in mind, and following [3, 21], it is interesting to know theexistence of rational periodic sequences. The Lyness map F a ( x, y ) = ( y, ( a + y ) /x ) , (2)associated to (1), leaves invariant the elliptic curves a C a,h := { ( x + 1)( y + 1)( x + y + a ) − hxy = 0 } and the map action can be described in terms of the group law action of them. In con-sequence several tools for studying the rational periodic orbits on them are available. Inparticular, from Mazur’s Torsion Theorem (see for example [19]), we know that, under theabove hypotheses, the rational periodic points can only have (prime) periods 1 , , . . . , , . Our first result proves that almost all these periods appear for the Lyness recurrencefor suitable a ∈ Q + and rational initial conditions. Theorem 1.
For any n ∈ { , , , , , , , , , } there are a ∈ Q + ∪ { } and rationalinitial conditions x , x such that the sequence generated by (1) is n -periodic. Moreoverthese values of n are the only possible prime periods for rational initial conditions and a ∈ Q . Notice that the value n = 4 is the only one in Mazur’s list that is not in the list givenin the theorem. Following [21] it is possible to interpret that this period corresponds to thecase a = ∞ , see Remark 7 in Section 3.Concerning the rational periodic points in Q + × Q + for the Lyness map with a > , itis proved in [3] that it only can have periods 1 , . Taking a = n − n and x = x = n ∈ N we obtain trivially 1-periodic integer sequences.The existence of positive rational periodic points of period 5 is well known and simple:they only exist when a = 1 and in this case all rational initial conditions give rise to thembecause the recurrence (1) is globally 5-periodic. On the other hand, as far as we know,the case of period 9 has resisted all previous analysis. In particular, the Conjecture b a For some concrete values of h and a the curve is not elliptic. We study these values separately. b Conjecture 1 of Zeeman was about the monotonicity of certain rotation number function associated tothe invariant ovals of the Lyness map and was proved in [5]. c a for which the Lyness recurrences (1) havepositive rational periodic sequences of period 9 and, even more, that this happens forinfinitely many values of a, see more details in Theorem 2.It is known, see again [3, 21], that the periodic points of period 9 of F a are on the ellipticcurve a ( x + 1)( y + 1)( x + y + a ) − ( a − a − a + 1) xy = 0 , (3)and they have positive coordinates only when a > a ∗ ≃ . a ∗ is the biggestroot of a − a + 3 a + 1 = 0 , see also Subsection 2.1.By using MAGMA ([6]) d , and after several trials, we have found some positive rationalpoints on the above curve proving that the Zeeman’s Conjecture 2 has a negative answer.The simplest one that we have obtained is ( a ; x, y ) = (7; 3 / , / a = 7 and the initial condition x = 3 / x = 5 / , gives32 , , , , , , , , , , , . . . Other positive rational points that we have found are (cid:18)
11; 2982 , (cid:19) , (cid:18)
13; 158467661133 , (cid:19) , (cid:18)
19; 425969716150 , (cid:19) and many others with much bigger entries. Our main result proves that there are infinitelymany positive rational values of the parameter a giving rise to 9–periodic positive rationalorbits. Theorem 2.
There are infinitely many values a ∈ Q + for which there exist initial conditions x ( a ) , x ( a ) ∈ Q + such that the sequence given by the Lyness recurrence (1) is –periodic.Furthermore, the closure of these values of a contains the real interval [ a , + ∞ ) , where a ≃ . is the biggest root of a − a − a − . Notice that there is a small gap between the values a ∗ and a where we do not know ifthere are or not rational values of a for which the Lyness recurrence has positive rationalperiodic orbits of period 9. As we will see in the proof of the theorem the gap is provokedby our approach and it seems to us that it is not intrinsic to the problem, see the commentsin Section 4, after the proof of the theorem.From the classical results of Mordell, see for example [18, Ch.VIII], it is well known thatthe set of rational points on an elliptic curve E over Q , together with the point at infinity, c Problem 1 of [3] is about rational 5-periodic points and it is recalled and solved in Section 5. d It is also possible to use SAGE ([16]). E ( Q ) and E ( Q ) ∼ = Z r × Φ , where r ∈ N is called the rank of E ( Q ) and Φ is the torsion of the group. Notice that r is a measure of the amount of rational points that the curve contains. The torsion partΦ is well understood from the results of Mazur already quoted, and it contains at most16 points. For the elliptic curve (3) it is easy to check that Φ = Z / Z . Fixed any Φ , among all the allowed possibilities, it is not known if the rank of the elliptic curves having E ( Q ) ∼ = Z r × Φ , for some r ∈ N , has an upper bound, but it is believed that it has not (seethe related Conjecture VIII.10.1 in [18]). As far as we know, nowadays when Φ = { } thehighest known rank is greater or equal than 28 while when Φ = Z / Z the highest knownrank is 4, see [9]. We have found values of a for which the algebraic curve (3) has ranks0 , , , a = 6 and the value 4 happens for a = 408 / . Akey point to obtain these results is the following theorem, which extends the results of [4],given for the cases of torsion points with order 5 or 6, and proves the universality of thelevel curves invariant for the Lyness map.
Theorem 3. (Lyness normal form) The family of elliptic curves C a,h , ( x + 1)( y + 1)( x + y + a ) − hxy = 0 , together with the points O = [1 : − and Q = [1 : 0 : 0] , is the universal family ofelliptic curves with a point of order n, n ≥ (including n = ∞ ). This means that for anyelliptic curve E over any field K (not of characteristic 2 or 3) with a point R ∈ E ( K ) oforder n , there exist unique values a ( E,R ) , h ( E,R ) ∈ K and a unique isomorphism between E and C a ( E,R ) ,h ( E,R ) sending the zero point of E to O and the point R to Q . Moreover, givena finite n , the relation between a and h can be easily obtained from the computations madein Subsection 2.1. From a dynamical viewpoint the above result implies that the Lyness Map is an affinemodel for most birrational maps on elliptic curves. This result is similar to the one describedby Jogia, Roberts and Vivaldy in [13, Theorem 3], where they use the Weierstrass normalform. Moreover, notice that in our result the sum operation takes the extremely easy form˜ F ([ x : y : z ]) = [ xy : az + yz : xz ] (see Section 2.1).Once for some value a ∈ Q + a periodic orbit in Q + × Q + of period 9 for F a is obtained,it is not difficult to obtain infinitely many different 9-periodic orbits by using the group lawon the curve. For instance, for a given a, we know that if a point P = ( x, y ) is on (3) then(2 k + 1) P, k ∈ Z is also on it. In particular for a = 7 if we write 3 P = ( z, w ) we get z = 2(6727 x + 913 y + 913)(90272 x − y − x + 858451819 xy − y − x − y − w can be obtained for instance by plugging x = z in (3) and choosing a suitablesolution y. Taking P = (3 / , /
7) we find3 P = (cid:18) , (cid:19) , which also gives rise to a 9-periodic orbit in Q + × Q + . Taking now 3 P as starting point forthe procedure we obtain a new initial condition for a 9-periodic orbit:9 P = (cid:18) , (cid:19) , and so on. In general we have the following result, see Subsection 2.1. Proposition 4. If a ∈ Q + is a value for which there exist an initial condition ( x , x ) ∈ Q + × Q + such that the sequence (1) is –periodic, then there exist infinite many differentrational initial conditions giving rise to –periodic sequences. Moreover the points corre-sponding to these initial conditions fill densely the elliptic curve (3). Observe that such a point ( x , x ) ∈ Q + × Q + always gives rise to a point on the ellipticcurve given by (3) which is not a torsion point, because all the 9-torsion points are not in Q + × Q + , see again Subsection 2.1.The case of 5-periodic points is studied in Section 5, proving that for a = 1 there aresome elliptic curves C ,h , h ∈ Q , with rank 0 and, as a consequence, ovals without pointswith rational coordinates.Finally, last section is devoted to give rational values of a for which the Lyness map F a has as many periods as possible, of course with rational initial conditions.The structure of this paper is the following. In Subsection 2.1 we recall some knownresults which describe the action of the Lyness map in terms of a linear translation overelliptic curves. Subsection 2.2 is devoted to prove Theorem 3 about the Lyness normalform. In Section 3 we prove Theorem 1, while the proof our main result, Theorem 2, andall our outcomes on rational 9-periodic points of (2) are presented in Section 4. Section 5studies the number of rational points on the elliptic curves invariant for F and the lastsection deals with the problem of finding a concrete F a with as many periods as possible. The results of this subsection are well known, we refer the reader to references [3, 13, 21]for instance to get more details. 5s mentioned above, the phase space of the discrete dynamical system defined by themap (2) is foliated by the family curves C a,h := { ( x + 1)( y + 1)( x + y + a ) − hxy = 0 } . This family is formed by elliptic curves except for a few values of h . When h = 0 it is aproduct of straight lines; when h = a − a = 1, it is the formed by a straight line and anhyperbola; and when h = h ± c := 2 a + 10 a − ± (4 a + 1) √ a + 12 a , (4)with h ± c = a − ,
0, it is a rational cubic, having an isolated real singularity. In fact the values h ± c correspond to the level sets containing the fixed points of F a , ((1 ± √ a + 1) / , (1 ±√ a + 1) / Q = { ( x, y ) , x > , y > } is invariant under the action of F a and itis foliated by ovals with energy h > h + c . In summary, on most energy levels, F a is a birational map on an elliptic curve andtherefore it can be expressed as a linear action in terms of the group law of the curve[13, Theorem 3]. Indeed, taking homogeneous coordinates on the projective plane P R thecurves C a,h have the form˜ C a,h := { ( x + z )( y + z )( x + y + az ) − hxyz = 0 } , and F can be seen as the map ˜ F ([ x : y : z ]) = [ xy : az + yz : xz ]. Taking the point O = [1 : − C a,h is an abelian group withrespect the sum defined by the usual secant–tangent chord process (i.e. if a line intersectsthe curve in three points P , Q , and R then P + Q + R = O ). According to these operationthe Lyness map can be seen as the linear action˜ F : P −→ P + Q, (5)where Q = [1 : 0 : 0]. When a ( a − = 0 it is not difficult to get that 2 Q = [ − , Q = [0 : − a : 1] , Q = (cid:20) − a : ah − a + 1 a − (cid:21) , Q = (cid:20) ah − a + 1 a − − a − ah + 2 a − a ( a −
1) : 1 (cid:21) , Q = (cid:20) − a − ah + 2 a − a ( a −
1) : a − a − ah + 2 a − a ( ah − a + 1) : 1 (cid:21) , Q = (cid:20) − a − ah + 2 a − a ( a −
1) : − a h + a h + a + a h − a − ah + 3 a − a h − a + a h − a h + 3 a + 2 ah − a + 1 : 1 (cid:21) , Q = [0 : 1 : 0] , − Q = [0 : − , − Q = [ − a : 0 : 1] , − Q = (cid:20) ah − a + 1 a − − a : 1 (cid:21) , − Q = (cid:20) − a − ah + 2 a − a ( a −
1) : ah − a + 1 a − (cid:21) , see also [3].In terms of the recurrence, the linear action (5) can be seen as follows: taking the initialconditions P := [ x : x : 1] then [ x n +1 : x n +2 : 1] = P + ( n + 1) Q , where + is the groupoperation. Notice also that, from this point of view, the condition of existence of rationalperiodic orbits is equivalent to the condition that Q is in the torsion of the group given bythe rational points of ˜ C a,h , which, as we have already commented, is described by Mazur’sTheorem.Hence, from the above expressions of kQ we can obtain the values of h correspondingto a given period. For instance, for period 9 we impose that 4 Q = − Q , or equivalently9 Q = O , which gives − a = ( − a − ah + 2 a − / ( a ( a − h = ( a − a − a + 1) a , which corresponds to the elliptic curve (3). For these values of h the points correspondingto the torsion subgroup are: Q =[1 : 0 : 0] , [ − , [0 : − a : 1] , [ − a : a ( a −
1) : 1] , [ a ( a −
1) : − a : 1] , [ − a : 0 : 1] , [0 : − , [0 : 1 : 0] , O = [1 : − . It is also important from a dynamical point of view the following well know property ofthe secant–tangent chord process defined on any real non-singular elliptic curve E . Let P be a point of E such that kP, for k ∈ Z , is never the neutral element O . Recall that whenthe elliptic curve is defined over Q there are at most sixteen points P with rational entriesnot satisfying this property due the Mazur classification of the torsion subgroup of E ( Q ).Then the adherence of the set { kP } k ∈ Z is: • either all the curve E , when P belongs to the connected component of E which doesnot contains the neutral element O ; or • the connected component containing O when P belongs to it.This result is due to the fact that there is a continuous isomorphism between E ∪ O withthis operation and the group { e it : t ∈ [0 , π ) } × { , − } , with the operation ( u, v ) · ( z, w ) =( uz, vw ) , see for instance Corollary 2.3.1 of [17, Ch. V.2]. Clearly, by using this construction,Proposition 4 follows. 7otice that when an elliptic curve is given by C a,h the bounded component never con-tains O . Proof of Theorem 3.
It is known that any elliptic curve having a point R that is not a 2 ora 3 torsion point can be written in the so called Tate normal form Y + (1 − c ) XY − bY = X − bX , where R is sent to (0 , C a,h can be transformed into the ones of the Tate normal form. In projectivecoordinates these curves write as Y Z + (1 − c ) XY Z − bY Z = X − bX Z and the curves C a,h as ( x + z )( y + z )( x + y + az ) − hxyz = 0 . With the change of variables X = − bc + 1 z, Y = − bcc + 1 ( y + z ) , Z = − cc + 1 ( x + y ) − z and the relations h = − bc , a = c + c − bc , both families of curves are equivalent and the theorem follows. Observe that the case c = 0corresponds to a curve with a 4–torsion point. As we will see in the proof of Theorem 2one advantage of this normal form is that it is symmetric with respect to x and y. From Theorem 3 we have that all the known results on elliptic curves with a point oforder greater than 4 can be applied to the corresponding Lyness curves. In particular wefind inside C a,h , the curves with high rank and prescribed torsion given in [9] or we can usethe list of “Elliptic Curve Data” for curves in Cremona form ([8]), taking advantage of theMAGMA or SAGE softwares that allow to identify a given elliptic curve in it. As was explained in Subsection 2.1 the curves C a,h are elliptic for all values of a and h except for h ∈ H := { , a − , h ± c } , with h ± c given in (4). On the curves corresponding to8hese values there could be, for the rational periodic orbits, some periods that are not inthe list given by Mazur’s theorem. In this section we prove that no new period appears. Lemma 5.
The periods of the rational periodic orbits of F a lying on the curves C a,h for h ∈ H are , , , , and 12.Proof. It is well known that the case a = 0 is globally 6-periodic. So from now on weconsider that a = 0 . We start the study of the rational cubic curves C a,h ± c , where h ± c aregiven in (4) and moreover h ± c
6∈ { , a − } . By setting b = ±√ a + 1, we have( b + 3) b + 1) = h + c when b ≥ , b = 1 ,h − c when b ≤ , b / ∈ {− , − , − } . Note that a = 0 implies that b = ± h ± c = a − = 0 implies that b = −
2; and h ± c = 0implies that b = −
3. Observe also that since a and h ± c are in Q then b ∈ Q . By using again a computer algebra software we obtain the following joint parametrizationof both curves C a,h ± c : t → ( x ( t ) , y ( t )) = (cid:18) (3 t + tb −
2) (2 tb + 4 t − b − b + 1) ( t − , − (3 t + tb − b −
1) (2 tb + 4 t − − b )2 t ( b + 1) (cid:19) . Moreover t = x ( t ) − ( b + 1) / x ( t ) + y ( t ) − ( b + 1) . (6)On each curve C a,h ± c the Lyness map F a can be seen as F a | C a,h ± c : (cid:0) x ( t ) , y ( t ) (cid:1) −→ (cid:18) y ( t ) , ( b − / y ( t ) x ( t ) (cid:19) = (cid:0) x ( f ( t )) , y ( f ( t )) (cid:1) , where f ( t ) has to be determined. By using (6) we obtain that f is the linear fractionaltransformation f ( t ) = t − ( b + 1) / (2 b + 4) t , where recall that b ∈ Q . Therefore the dynamics of F a on each of the curves C a,h ± c iscompletely determined by the dynamics of the maps f ( t ).It is a well-known fact that if a linear fractional map, g ( t ) = ( At + B ) / ( t + D ), hasa periodic orbit of prime period p , p > , then it is is globally p -periodic. Moreover thishappens if and only if either: • ∆ := ( D − A ) + 4 B > A = − D and in this case g is 2-periodic; or • ∆ < ξ := ( A + D − p | ∆ | i ) / ( A + D + p | ∆ | i ) is a primitive p -root of the unityand in this case g is p -periodic. 9ence, apart of the fixed points, the maps f ( t ) can have p -periodic solutions, p >
1, ifand only if∆ = − bb + 2 < ξ = 1 b + 1 − b + 2 b + 1 r bb + 2 i is a primitive p -root of the unity.The condition that ξ is a primitive p -root of the unity, implies thatcos (cid:18) π qp (cid:19) = 1 b + 1 ∈ Q , for some q ∈ Z . It is also a well-known fact that the only rational values of cos( x ), where x is a rationalmultiple of π, are 0, ± / ± b ∈{− , − , , } . The only allowed valued is b = 0, which implies that ∆ = 0 , and so thecorresponding map f is not periodic. Hence, only the fixed points of F a appear on thisfamily of curves.Concerning the case h = 0, observe that C a, = { ( x + 1)( y + 1)( a + x + y ) = 0 } . When a = 1 there are no periodic orbits on this level set. When a = 1 , the three straight linesforming this set are mapped one into the other in cyclical order by F a , and so they areinvariant under F a . The cyclical order determined by F a is: { x + 1 = 0 } → { a + x + y = 0 } → { y + 1 = 0 } → { x + 1 = 0 } . The restriction of F a on { x + 1 = 0 } is given by F a ( − , y ) = (cid:18) − , − ay + a (cid:19) . Hence the dynamics of F a is determined by the dynamics of the linear fractional map f ( y ) = 1 − ay + a . One of its fixed points corresponds to the continua of three periodic points of F a given inTable 1. By using again the characterization of the periodicity of the linear fractional mapswe obtain that f is periodic only when a = 0 . It remains to study the case h = a − , a = 1. In this situation C a,a − = { ( x + y + 1)( a + x + y + xy ) = 0 } and F a sends the straight line to the hyperbola and vice versa. So bothlevel sets are invariant under F a . The restriction of F a on { x + y + 1 = 0 } is given by F a ( x, − − x ) = (cid:18) − x + a − x , − ax (cid:19) . Hence the dynamics of F a is determined by the linear fractional map f ( x ) = − x + a − x , a ≥ / . They give rise to 2-periodic points of F a when a > / a = 3 / . Arguing as in the case h = h ± c , the map f is p -periodic, p ≥ , only when∆ = 4 a − < ξ = 1 − a − i √ − a a − p − root of the unity.This happens if and only if a < / − a ) / (2 a − ∈ { , ± / , ± } , or equivalentlywhen a ∈ { / , / } (recall that the case a = 0 is already considered). The case a =1 / f and a = 2 / F a on C a,a − . Proof of Theorem 1.
From Lemma 5 we know that when h ∈ H the possible periods on C a,h are in the list given in the statement. For those points on the elliptic curves C a,h for allvalues of a and h / ∈ H we can apply Mazur’s theorem and we obtain that the only possibleperiods are the ones of the statement together with the period 4. The points of (prime)period 4 can be discarded by observing that in Subsection 2.1 we prove that 4 Q = O . Itis also possible to perform a direct study with resultants of the system F a ( x, y ) = ( x, y ).From this study we get that that its only solutions are the ones corresponding to fix or2-periodic points.These results together with the ones presented on Table 1 prove the theorem.Period a x x Comments1 u − u u u u ∈ Q \ { } u + u + 1 u − u − u ∈ Q \ {− , } a − − a ∈ Q \ { } x y Almost for all x , y in Q x y Almost for all x , y in Q u − u − u − u − u +1 − x Almost for all u in Q u − u +2 u − u − u +1 − x Almost for all u in Q emark 6. (i) The continua of sequences of periods 1 and 2 are not the most general ones,we have chosen simple one-parameter families.(ii) It is already known that there exist one-parameter families of elliptic curves havingpoints of 7 (or 8) torsion and rank 1. By using Theorem 3 we know that they should appearalso in the Lyness normal form. Nevertheless, we have got them by a direct study, onlysearching points satisfying x + y = 0 . (iii) The values corresponding to periods 10 and 12 are obtained with MAGMA. By usingsimilar arguments to the ones used to study the case of period 9 we can prove the existenceof rational periodic points, with these periods, for infinitely many rational values of a. It isnot known the existence of a continua of them.
Remark 7.
By introducing y n := x n / √ a, n ∈ N , the Lyness recurrence (1) writes as y n +2 = (1 + y n +1 / √ a ) /y n . When a tends to infinity we obtain the recurrence y n +2 = 1 /y n ,which is globally 4-periodic. In the proof of Theorem 2 we will use the following Lemma, see for example [1].
Lemma 8.
The curve K = A + w A + w A + w is isomorphic to the elliptic curve Y = X − (cid:18) w
48 + w (cid:19) X + w
64 + w − w w , where the change of variables is given by X = 12 (cid:16) A + K + w (cid:17) , Y = A (cid:16) A + K + w (cid:17) + w . Proof of Theorem 2.
As we have already explained in the introduction, following [3, 21]we already know that all the real positive initial conditions that give rise to 9-periodicrecurrences (1) correspond to the points of the positive oval of the elliptic curves (3), S a := { ( a ; x, y ) : a ( x + 1)( y + 1)( x + y + a ) − ( a − a − a + 1) xy = 0 , x > , y > , a > a ∗ } . We want to find infinitely many points ( x ( a ) , y ( a ) , a ) ∈ ( Q + × Q + × Q + ) ∩ S a . As wewill see bellow, we will find first one point ( a ; x, y ), satisfying x + y = 23 / , and from it wewill construct infinitely many via a multiplication process.We start by applying the transformation S = x + y , P = xy to each surface S a . Weobtain, a (1 + S + P ) ( a + S ) − ( a − (cid:0) a − a + 1 (cid:1) P = 0 ,
12r equivalently, P = a (1 + S ) ( a + S ) a − a + (2 − S ) a − . (7)It is easy to see that if ( S, P ) ∈ Q + × Q + and ∆ := S − P is a perfect square thenthe corresponding ( x, y ) ∈ Q + × Q + . So, we want to find a value of S such that ∆ has asuitable expression that facilitates to find values of a for which ∆ is a perfect square.By using (7), ∆ = S − P writes as∆ = S a − (cid:0) a − a − a − (cid:1) S + (cid:0) a + 4 a (cid:1) S + 4 a a − a + (2 − S ) a − . We will fix a value of S such that the value of the discriminant with respect a of thedenominator of the last expression vanishes. The discriminant is (4 S − S + 1) , so wefix S = 23 /
4, obtaining∆ := ∆ | S =23 / = 116 2116 a − a − a − a −
4) (1 + 2 a ) . In order to avoid the terms which are perfect squares and to have an algebraic expression,we consider∆ := (cid:18) a )( a − (cid:19) ∆ = ( a − (cid:18) a − a − a − (cid:19) , and we try to find rational values of a on the quartic k = ∆ , that is k = ( a − (cid:18) a − a − a − (cid:19) . In order to apply Lemma 8 to the above equation we perform the following translation A = a − / , obtaining the new elliptic quartic K = A − A − A + 100962485825724920047612231936 , (8)which, by Lemma 8, is isomorphic to the elliptic cubic Y = X − X + 8775405707427303177500672 . (9)Consider the additive group of rational points on the elliptic curve (9): E ( Q ) = { ( X, Y ) ∈ Q × Q satisfying (9) } ∪ O where O is the usual point at infinity acting as a neutral element, by using MAGMA wehave found the following rational point R = ( , ), which lies on the compact oval of(9), see Figure 1. 13igure 1: The elliptic curve Y = X − X + .Figure 2: The elliptic quartic k = ∆ = ( a − (cid:0) a − a − a − (cid:1) . It is easy to check that R is not in the torsion of E ( Q ) . In fact, E ( Q ) has trivial torsion.Then, as has been explained in Section 2.1, we know that the set of rational points { kR } k ∈ Z ,obtained by the secant-tangent chord process, fill densely the full elliptic curve (9). So, theirimages, trough the transformations given in our proof form a dense set of rational pointson the quartic (8). The corresponding projections give a set of rational values of a in( −∞ , a ] ∪ [ a , a ] ∪ [ a , + ∞ ) such that the corresponding map F a has 9-periodic rationalpoints e . Here a < a < a = 4 < a are the four roots of ∆ , see Figure 2. Of course, e Notice that not all the rational points of on (9) are good seeds for obtaining the density of values of a in [ a , + ∞ ) by the secant-tangent chord process. For instance, although the rational point ( X, Y ) =(23947 / , / a ; x ( a ) , y ( a )) = (cid:18) , (cid:19) , a belongs to the first twointervals. For instance one of two points corresponding to R is( a ; x ( a ) , y ( a )) = (cid:18) , − (cid:19) . Finally we prove that the rational points found on the quartic (8) corresponding tovalues of a ∈ [ a , + ∞ ) give rise to positive rational 9-periodic orbits. Observe that for S = 23 /
4, the value of P given by equation (7) is P = 274 a (4 a + 23)( a − a ) . Hence for all values of a ≥ a ∗ , the value P is positive and therefore the corresponding valuesof x and y are also positive. Hence the theorem follows.Notice that our proof of Theorem 2 searches positive rational points ( a ; x, y ) in S a , suchthat x + y = 23 /
4. Hence a first necessary condition for their existence is that both curvesintersect in the first quadrant of the ( x, y )-plane. It is easy to prove that when a ≥ a ∗ thisonly happens when a ≥ a > a ∗ . Some values of a in ( a ∗ , a ) with not large numerators and denominators f are:142432632 , , , , , , , , , , . Notice that a − a ∗ ≃ . × − . For the above values we have not been able to findrational points on the corresponding elliptic curve (3). Nevertheless it is not difficult, byusing again MAGMA or SAGE, to compute the root numbers associated to these ellipticcurves, obtaining − Remark 9.
The elliptic curve given by the equation (9) has minimal model Y + XY + Y = X − X − X + 376423337 . which is a counterexample to Zeeman’s Conjecture 2, it is not useful for our purposes because it lies on theunbounded connected component of (9). f We have obtained them by computing several convergents of the expansion in continuous fractions ofsame points in the interval. computation with either SAGE or MAGMA reveals that this curve has rank . As faras we know, this is the highest known rank for the base elliptic curve of a family of ellipticcurves with a torsion point parameterized by such curve, see [9] or [15, App. B.5]. Next, we show the non-existence of rational 9-periodic points for a some F a , a > a ∗ . Proposition 10. (i) For a ∈ { , } there are no initial conditions in Q × Q such that thesequence (1) has period 9.(ii) For a = 9 there are no initial conditions in Q + × Q + such that the sequence (1) hasperiod 9, but there are infinitely many in Q × Q .Proof. Consider the elliptic curves (3) for a = 6 , a = 9 we havethat the rank of the elliptic curve is 1, so there are infinitely many rational points on it.But a generator of the free part is ( − / , − / a = 6 of the above proposition can also be proved in different ways.It is possible to use the 2-descent method, or, even better, since our curves have 9-torsionpoints and, hence, 3-torsion points, one can instead use the descent via 3-isogeny method. Itis also possible to use an indirect method: showing that the value of its L -function at s = 1is not zero. Then, by the known true cases of the Birch and Swinnerton-Dyer Conjecture(see [7], Theorem 8.1.8), we can deduce its rank is 0. By using MAGMA or SAGE, oneshows that L ( E, ≃ . E has rank 0. The same argument applies for a = 8 . By using Theorem 3, together with the known example of an elliptic curve of rank 4with a torsion point of order 9, see [9], we get an example of rank 4 for the curves C a . Thiscurve gives also a counterexample of Zeeman’s Conjecture 2 with many (positive) rationalperiodic orbits. Corollary 11.
For a = 408 / , the curve C a has rank over Q , that is C a ( Q ) ∼ = Z × Z / Z . Moreover the non-torsion points are generated by (cid:18) , (cid:19) , (cid:18) , (cid:19) , (cid:18) − , (cid:19) , (cid:18) , (cid:19) . Period 5 points
When a = 1 , the map F ( x, y ) = ( y, (1 + y ) /x ) is the celebrated, globally 5-periodic, Lynessmap. We define E h := C ,h as the family of elliptic curves E h := { ( x + 1)( y + 1)( x + y + 1) − h xy = 0 } . It is clear that all them are filled of 5-periodic orbits. Following [3] we know the torsionof the group of rational points of each of the curves, E h ( Q ) is either Z / Z or Z / Z . Inthat paper it is proved that (1 , ∈ E or that (1 , ∈ E / and that they are not in thecorresponding torsion subgroups. Hence the corresponding elliptic curves have infinitelymany positive rational points and moreover the ranks of both groups are greater or equalthan 1. In [3, Prob. 1] the authors ask the following question: Is the rank of every E h ( Q )for h ∈ Q , h > h + c always positive? Here h + c = (11 + 5 √ / ≃ .
09 is the value where E h starts to have points in the first quadrant. Next result proves that the answer to the abovequestion is negative. Proposition 12.
There exist rational values of h, h > h + c , such that E h ( Q ) has rank .Proof. We use the same method that in the proof of Proposition 10. Consider E h with h = 13 . Its corresponding number in Cremona’s list ([8]) is 325e1, obtaining rank 0 andtorsion Z / Z . Hence the result follows. Remark 13. (i) Arguing as in the previous proof, for the values h = 12 , and weobtain ranks 0,0 and 1, and torsions Z / Z , Z / Z and Z / Z , respectively. In fact, in the first integers values of h > , there are values with rank 0 (and values with rank1, and values of rank 2).(ii) For the case h = 12 , one can also use the existence of a point with 2-torsion to givean easy proof that the rank of E ( Q ) is 0, by using the descent via 2-isogeny method.(iii) Recall that the Lyness map can be written as P → P + Q , where Q = [1 : 0 : 0] andthis point has 5-torsion. Thus when the initial point P has 2-torsion we obtain that P + (2 m + 1) Q = (2 m + 1)( P + Q ) , m ≥ and so the odd multiples of P + Q by the secant–tangent chord process coincide with thepoints generated by the Lyness map.(iv) The case h = 12 is also interesting because, although the rang of E is zero, thereis a unique rational (integer) orbit corresponding to half of the torsion points. This orbit isgiven by the initial conditions x = 1 , x = 1 and is , , , , , , , . . . s it is explained in item (iii), this situation happens because the point (1 , has 2-torsion.(iv) By using once more the universality of the level sets of the Lyness curves C a,h andthe results given in [9], where four examples of elliptic curves with torsion Z / Z and rank8 appear, we have obtained the following values of h : , , , (and also the ones corresponding to − /h ) for which the rank of E h is . Since all thesevalues are in (0 , none of them satisfies h > h + c . (v) By using the same method than in the previous item we know that E ˜ h and E − / ˜ h for ˜ h = 12519024 / ≃ . > h + c have rank 7. Looking at Theorem 1 it is natural to wonder if there exists some rational fixed value of a forwhich there are rational initial conditions such that the corresponding sequences generatedby (1) has all the allowed periods { , , , , , , , , , } given the theorem. It is easyto see that the answer is no, because period 5 (resp. 6) only happens when a = 1 (resp. a = 0). Moreover we will prove in next proposition that rational sequences with primeperiods 1 and 12 or 2 and 12 can not coexist for a given value of a . On the other hand, asit is shown in Table 2, for each a = 1, the point ( − , −
1) always gives rise to a 3-periodicorbit for F a .Hence the interesting problem is to find values of a ∈ Q such that F a has one of thefollowing sets of prime periods realized for rational periodic points: P = { , , , , , , } or P = { , , , , , } . Table 2 shows that for a = 21 /
37 the set of periods of F a is P . The search of rationalpoints with periods 9 and 10 is quite involved. We give some details of how we have obtainedthem in the end of this section.This table also shows that for the integer value a = 20 the set of achieved periods is P \ { } and it is easy to see that period 2 does not appear. To find explicit rational values a , candidates to have as set of periods with rational entries the full set P , is not verydifficult. First we give a rational parametrization of the values of a for which rational fixedand period 2-points appear. After, among these values, we search for “small” values forwhich the root numbers of the elliptic curves corresponding to the points of period 7 , , −
1. We have obtained the list8840118496 , , , , , , . a , if one believes once more in the trueness of the Birch andSwinnerton-Dyer Conjecture or of the Parity Conjecture, the set of periods of the cor-responding F a should be P . Unfortunately for none of these values of a we have been ableto find explicitly periodic points of all the periods.Period ( x , x ) for a = 20 ( x , x ) for a = 21 /
371 (5 ,
5) -2 - -3 ( − , −
1) ( − , − − , − ) ( , − )8 ( − , − ) ( , − )9 ( , − ) ( − , )10 ( − , − ) ( , − )12 - ( − , − )Table 2. Values of a and rational initial conditions for the recurrences (1)with periodic sequences of several periods. Proposition 14.
Given a ∈ Q , the Lyness map F a has not simultaneously rational periodicpoints with prime periods 1 and 12, or 2 and 12.Proof. From the results of Subsection 3.1 we know that these couple of periods do notcoexist when the corresponding curves C a,h are not elliptic curves. So, from now one wecan assume that the 12 periodic points lie on an elliptic curve C a,h . First of all, given a ∈ Q ,it is easy to obtain that there is a rational periodic point with period 1 if and only if 4 a + 1is a square in Q and that there is one with period 2 if and only if 4 a − Q .By using the expressions of kQ given in Section 2.1 (or by using the known results for theTate curve [1]), we get that the explicit parameterization of the values of a and h for which Q can have order 12 is a = 2 t (1 + t )3 t + 1 and, h = − ( t − ( t + 1) t (1 + t )(3 t + 1) , for some t = 0 , ±
1. Hence, to have a rational periodic points with periods 1 and 12, we needrational numbers t such that t (1+ t )3 t +1 +1 is a square in Q . We will show that this only happensfor the values t = 0 and 1, which do not give prime period 12. Multiplying by (3 t + 1) , weneed to search for rational solutions of the equation z = (3 t + 1)(11 t + 8 t + 1). A standardchange of variables, similar to the one used used in Section 4, shows that this genus onecurve is isomorphic to the elliptic curve with corresponding number in the Cremona’s list19qual to 15a8, which has only four rational points. These points correspond to the pointswith t = 0 and t = − t such that t (1+ t )3 t +1 − Q . Arguing as in the previous case we arriveto the equation z = (3 t + 1)( − t + 8 t −
3) which number in Cremona’s list is 39a4 andhas only the two rational points corresponding to t = 1. So the result follows. F / In order to find 9 or 10 rational periodic points for F / , one cannot just naively searchfor points, since their coordinates are too big. So we use the following strategy. First, withthe same formulas that in the proof of Theorem 3, we transform the equations C / ,h , for their corresponding h = − and h = − , to a Weierstrass equation. We need tofind non-torsion points on these elliptic curves. Since the Weierstrass equations have toobig coefficients, we apply 2-descent procedure with MAGMA in order to get an equivalentquartic equation with smaller coefficients for the corresponding elliptic curves. We get thatthey are equivalent respectively to y = − x + 66683940 x + 800552377 x − x + 209901456and y = − x + 116312038 x + 178646017 x − x − . These are still not sufficient simple to be able to find (non-torsion) rational points, so wedo another transformation. In the first case, we apply 4-descent in order to obtain anotherform, this case as intersection of two quadrics (in the projective space). We get the equations15 X + 104 XY − Y + 12 XZ − Y Z + 18 Z + 62 XT − Y T − ZT + 22 T = 0 , X − XY − Y − XZ − Y Z − Z + 122 XT − Y T + 688 ZT + 321 T = 0 . Finally, an easy search finds the point given in projective coordinates by [ − : : : 1].Using the transformation rules given by MAGMA one gets the corresponding point in C / ,h , shown in Table 2. For the second equation, corresponding to the 10 rationalperiods, we directly transform the quartic equation to an intersection of two quadrics, andthen we apply an algorism due to Elkies ([10]) to search for rational points in this type ofcurves (as implemented in MAGMA). 20 cknowledgements The authors are partially supported by MCYT through grants MTM2008-03437 (first au-thor), DPI2008-06699-C02-02 (second author) and MTM2009-10359 (third author). Theauthors are also supported by the Government of Catalonia through the SGR program.
References [1] A. O. L. Atkin, F. Morain.
Finding suitable curves for the elliptic curve method of fac-torization , Math. Comp. 60 (1993), no. 201, 399–405.[2] E. Barbeau, B. Gelbord and S. Tanny.
Periodicities of solutions of the generalized Lynessrecursion , J. Difference Equations Appl. 1 (1995), 291–306.[3] G. Bastien, M. Rogalski.
Global behavior of the solutions of Lyness’ difference equation u n +2 u n = u n +1 + a , J. Difference Equations Appl. 10 (2004), 977–1003.[4] A. Beauville. Les familles stables de courbes elliptiques sur P1 admettant quatre fibressinguli`eres , C. R. Acad. Sci. Paris, S´erie I 294 (1982), 657–660.[5] F. Beukers, R. Cushman.
Zeeman’s monotonicity conjecture , J. Differential Equations143 (1998), 191–200.[6] W. Bosma, J. Cannon and C. Playoust.
The Magma algebra system. I. The user lan-guage,
J. Symbolic Comput. 24 (1997), 235–265.[7] H. Cohen. “Number Theory” Volume I: Tools and Diophantine Equations. Springer,GTM 239, 2007.[8] Web page “Elliptic Curve Data” by J. E. Cremona, University of Warwick, [9] Web page sith “Elliptic curve tables”, mantianed by A. Dujella, University of Zagreb, http://web.math.hr/ ∼ duje/ [10] N. Elkies, Rational Points Near Curves and Small Nonzero —x3 - y2— via LatticeReduction , pp. pages 33–63. In ANTS IV, volume 1838 of LNCS. Springer-Verlag, 2000.[11] J. Esch and T. D. Rogers.
The screensaver map: dynamics on elliptic curves arisingfrom polygonal folding , Discrete Comput. Geom. 25 (2001), 477–502.[12] D. Husemoller. “Elliptic curves”. With an appendix by Ruth Lawrence. Graduate Textsin Mathematics, 111. Springer-Verlag, New York, 1987.2113] D. Jogia, J. A. G. Roberts, F. Vivaldi,
An algebraic geometric approach to integrablemaps of the plane , J. Phys. A 39, 1133–1149 (2006).[14] I. Niven, H.S. Zukerman, H.L. Montgomery. “An introduction to the theory of num-bers” (Fifth ed). John Wiley, New York, 1991.[15] F. P. Rabarison. “Torsion et rang des courbes elliptiques d´efinies sur les corps de nom-bres alg´ebriques”, Th`ese de doctorat, Universit´e de Caen, 2008.[16] W. Stein et al., Sage: Open Source Mathematical Software (Version 4.0), TheSage Group, 2009, [17] J. Silverman. “Advanced topics in the arithmetic of elliptic curves”. Graduate Textsin Mathematics, 151. Springer-Verlag, New York, 1994.[18] J. Silverman. “The Arithmetic of Elliptic Curves” (2nd Ed.). Graduate Texts in Math-ematics, 106. Springer, Dordrecht, 2009.[19] J. Silverman, J. Tate. “Rational Points on Elliptic Curves”. Undergraduate Texts inMathematics. Springer, New York, 1992.[20] J. M. H. Olmsted.
Rational values of trigonometric functions , Amer. Math. Monthly 52(1945), 507–508.[21] E. C. Zeeman.
Geometric unfolding of a difference equation , Hertford Col-lege, Oxford (1996). Unpublished paper. A video of the distinguished lec-ture, with the same title, at PIMS on March 21, 2000, is aviable at: . The slides can be obtainedat: http://zakuski.utsa.edu/ ∼ gokhman/ecz/gu.htmlgokhman/ecz/gu.html